Engineering Mechanics Solutions Manual by nareshk304

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									 Solutions Manual
        For

Engineering
Mechanics

Manoj Kumar Harbola
      IIT Kanpur




          1
                                              Chapter 1


1.1   Rotational speed of the earth is very small (about 7  10 5 radians per second). Its
      effect on particle motion over small distances is therefore negligible. This will not be
      true for intercontinental missiles.
1.2   The net force on the (belt+person) system is zero. This can be seen as follows. To
      pull the rope up, the person also pushes the ground and therefore the belt on which he
      is standing. This gives zero net force on the belt. For the person, the ground pushes
      him up on the feet but the belt pulls him down when he pulls it, giving a zero net
      force on him.
1.3   A vector between coordinates (x1, y1, z1) and (x2,y2,z2) is given by
                  ˆ                                ˆ
      ( x 2  x1 )i  ( y 2  y1 ) ˆ  ( z 2  z1 )k . Thus (i), (ii) and (iv) are equal.
                                   j

1.4                        ˆ     j    ˆ       ˆ     j ˆ                     ˆ
      The vectors are (i) 2i  3 ˆ  5k (ii) 4i  3 ˆ  k (iii) 2i  9 ˆ  5k (iv)
                                                                 ˆ     j
         ˆ          ˆ
       3i  3 ˆ  2k
               j

1.5                                  ˆ    ˆ ˆ               ˆ       ˆ    ˆ
      (i) The resultant vectors are 6i  6k , 4i  6 ˆ  10 k and  i  3k
                                                     j

                                      ˆ          ˆ ˆ             ˆ      ˆ    ˆ
      (ii) The resultant vectors are 2i  6 ˆ  4k , 2i  6 ˆ  4k and 7i  3k
                                            j               j
1.6
                                          
                                          A

                                                            
                        B                                   A B
                                                                            
                                       B A                                   B
                                                            
                                                            A



1.7   On each reflection, the sign of the vector component perpendicular to the reflecting
      mirror changes.




                                                  2
1.8
                     z




                                       y               
                                                       v




                                                                                 x
                         O


       The fly is flying along the vector from (2.5, 2, 0) to (5, 4, 4). This vector is
                                                                 ˆ          ˆ    ˆ          ˆ
                                                              2.5i  2 ˆ  4k 2.5i  2 ˆ  4k
                                                                       j               j
          ˆ          ˆ
       2.5i  2 ˆ  4k . The unit vector in this direction is
                j                                                                            .
                                                               6.25  4  16       26.25
                                                              ˆ          ˆ
                                                           2.5i  2 ˆ  4k
                                                                    j
       The velocity of the fly is therefore 0.5                                   ˆ               ˆ
                                                                              0.25i  .20 ˆ  0.39k .
                                                                                           j
                                                                26.25

                                                
1.9    After time t, the position vectors rA and rB of particles A and B, respectively, are
                                                 
                          ˆ
          rA  l sin t i  l cos t ˆ
                                      j                         ˆ
                                                  rB  l sin t i  l cos t ˆ
                                                                             j
                                
       Their velocities v A and v B are given by differentiating these vectors with respect to
       time to get
                                                
                            ˆ
           v A  l cos t i  l sin t ˆ
                                          j                      ˆ
                                                 v B  l cos t i  l sin t ˆ
                                                                               j
                                                                               
       Velocity v AB of A with respect to B is obtained by subtracting v B from v A
                                                                  ˆ
                                      v AB  v A  v B  2l cos t i

1.10   For rotation about the z-axis by an angle 
       v x '  v x cos  v y sin              v y '   v x sin   v y cos        v z'  v z

       It is given that   30 . Therefore

       v x' 
                1
                2
                 vx 3  vy                    v y' 
                                                         1
                                                         2
                                                            
                                                            vx  vy 3               v z'  v z




                                                   3
                             
1.11   Component of a vector A along an axis is given by its projection on that axis. This is
       obtained by taking the dot product of the vector with the unit vector along that axis.
       Thus
                                                                              
                ˆ
       Ax  A  i  A cos1       Ay  A  ˆ  A cos 2
                                           j                                   ˆ
                                                                      Az  A  k  A cos 3

       Also
                                               2
                                               A  Ax  Ay  Az2
                                                    2    2



       Substituting the expression for Ax, Ay and Az completes the proof.
                                              
1.12   (i) Dot product of two vectors A and B is
                                  
                                 A  B  Ax Bx  Ay B y  Az Bz

       This gives the dot product of the first vector of problem 1.4 each of the other vectors
       to be 4, 2 and 25.
                                                   
       (ii) Cross product between two vectors A and B is
                                        ˆ                                       ˆ
                A  B  (Ay Bz  Az By )i  (Az Bx  Ax Bz )ˆ  (Ax By  Ay Bx )k
                                                            j
                                                
       Taking A to be the fourth vector and B to be the first, second and the third vector
       gives the cross products to be
                                            ˆ           ˆ
                                          9i  11 ˆ  3k
                                                   j

                                            ˆ           ˆ
                                          9i  5 ˆ  21k
                                                  j
                                            ˆ            ˆ
                                         33i  11 ˆ  24k
                                                   j
1.13   If the angle between two vectors is , the cosine of this angle is given by
                
               A B
       cos    . Thus
               AB

                                         4
       Between (i) and (ii) cos                 0.127    82 .7 
                                        38 26
                                          2
       Between (i) and (iii) cos                    0.037    88 .2 
                                        38 110
                                         25
       Between (i) and (iv) cos                    0.865    149 .8 
                                        38 22




                                                4
                                             40
          Between (ii) and (iii) cos                         0.748    41 .6 
                                           26 110
                                            5
          Between (ii) and (iv) cos                      0.209    102 .1
                                           26 22
                                             11
          Between (iii) and (iv) cos                     0.380    67 .6 
                                            38 22


   1.14
                                     
                                                ˆ 
          Vector A can be written as A  A cos1i  cos 2 ˆ  cos 3 k
                                                           j          ˆ               
                     
                               ˆ           j           ˆ
          Similarly B  B cos 1i  cos  2 ˆ  cos  3 k        
                                                                                       
                                                                                      A B
          Taking the dot product between the two vectors and using the formula cos    ,
                                                                                      AB

          where  is the angle between the two vectors, we get the answer.


                                2 2      
   1.15   Magnitude A  B        A  B  2 A  B cos

                              2 2      
          Similarly A  B      A  B  2 A  B cos
                                  
          Equating the two gives A  B  0 which implies that the two vectors are perpendicular
          to each other.

            
1.16      B  C  By C z  Bz C y i  Bz C x  Bx C z  ˆ  Bx C y  By C x k
                                    ˆ                      j                     ˆ

                   
             
          A  B  C  Ax By C z  Bz C y   Ay Bz C x  Bx C z   Az Bx C y  By C x 

                                             
                                               
                                                          
          This comes out to be equal to C  A  B and B  C  A
                                                                   
                                                                            
1.17                                   
                                         
                                              
          From the expression for A  B  C it is clear that it is equal to the determinant


                                            Ax        Ay        Az
                                            Bx        By        Bz
                                            Cx        Cy        Cz




                                                      5
       Interchange of two rows in a determinant changes the sign of the determinant. This
       implies


                                Bx     By       Bz       Bx       By   Bz    Ax       Ay   Az
                                Cx    Cy     C z   Ax           Ay   Az  B x       By   Bz
                                Ax     Ay       Az       Cx       Cy   Cz    Cx       Cy   Cz


       thereby proving the equalities in problem 1.16.

        
1.18   B  C  By C z  Bz C y i  Bz C x  Bx C z  ˆ  Bx C y  By C x k
                                 ˆ                      j                     ˆ

       Therefore


                        
         
       A  B  C  Ay B x C y  B y C x   Az B z C x  B x C z i
                                                                      ˆ

                              Az B y C z  B z C y   Ax B x C y  B y C x ˆ
                                                                                  j
                              Ax B z C x  B x C z   Ay B y C z  B z C y k
                                                                                  ˆ


                            
       The x-component of A  B  C                 

       A B C
         y       x   y    B y C x   Az B z C x  B x C z   B x ( Ay C y  Az C z )  C x ( Ay B y  Az BZ )


       On the right hand side above, add and subtract Ax B x C x to get


       A B C
         y       x   y    B y C x   Az B z C x  B x C z   B x ( Ax C x Ay C y  Az C z )  C x ( Ax B x Ay B y  Az BZ )
                                                                                        
                                                                  B x ( A  C )  C x ( A  B)


       Do the same manipulation for the other components to get

                                              
                                                         
                                                           
                                            A B C  AC B  A B C            


                                                              6
1.19 ( i)
                                    
                                    ˆ j      
                                              ˆ    ˆ       
                                                         j ˆ
                              AB  ai  aˆ  ai  ak  a ˆ  k
                                    
                              AC  aˆ  ak  ai  ak  a ˆ  i
                                    j     ˆ  ˆ    ˆ       
                                                          j ˆ
                                        ˆ   ˆ j         
                                                         ˆ ˆ
                              BC  aˆ  ak  ai  aˆ  a k  i
                                    j


       (ii)                                                             ˆ j ˆ    
              Body diagonal from the origin to the opposite corner is a i  ˆ  k .
              This vector is perpendicular to the vectors AB, AC and BC in the plane, because
              its dot product with each one of them vanishes. This shows that the diagonal is
              perpendicular to the plane.


              Another way to see this is to find a vector perpendicular to the plane by taking
              cross product of any of the two vectors from AB, AC or BC, and show that it is
              parallel to the body diagonal. For example taking the cross product of AB and AC
              gives
                                                             
                                                          ˆ ˆ j
                                            AB  AC  a 2 k  i  ˆ       
              which is parallel to the body diagonal.
       (iii) If the angle between OA and AB is  then

                                cos 
                                                  ˆ   ˆ    
                                                            ˆ j
                                         OA  BA ai  ak  ak  aˆ 1
                                                                 
                                         OA BA             
                                                    a 2 a 2         2

              This gives   60 . Note that we have taken dot product with the vector BA rather
              than AB because we wish to keep  less than 90 . In the same manner we get the
              angle between OA and AC also to be 60 .


1.20     The problem is to be solved exactly in tha same manner as done in example 1.3 by
         replacing the position vector by the velocity vector and the velocity vector by the
         acceleration.


                                                        
                          
1.21                                        ˆ
         Position vector R (t )  R cos t i  sin t ˆ
                                                      j
                                   
                         
         Velocity vector v (t ) 
                                  dR(t )
                                   dt
                                             
                                          R  sin t i  cos t ˆ
                                                        ˆ         j   

                                                 7
                              
                    
       Acceleration a (t ) 
                             dv (t )
                              dt
                                                  
                                       2 R cost i  sin t ˆ
                                                    ˆ          j     

                                                                                
1.22   (i) In reference frame 2, the components V x 2 ,V y 2 , V z 2 of vector V at time t are given in

       terms of its components V x1 , V y1 , V z1 in frame 1 by formula (1.10) as

       V x 2 (t )  V x1 cost  V y1 sin t
       V y 2 (t )  V x1 sin t  V y1 cost
       V z 2  V z1
                                                                    
       Here V x1 , V y1 , V z1 are time-independent because vector V is constant in frame 1.

       Differentiating V x 2 ,V y 2 , V z 2 with respect to time, we get

       dVx 2 (t )
                    V x1 sin t  V y1 cos t  V y 2
         dt
       dV y 2 (t )
                    V x1 cos t  V y1 sin t  V x 2
         dt
       dVz 2
               0
        dt
       (ii) From (i) it is clear that
                                                 
                                                dV         
                                                       ˆ
                                                    k  V
                                                dt


1.23   Done in later chapters
                                      
1.24   Magnitude of a vector quantity A(t ) is fixed. This means
                                              
                                      A(t )  A(t )  constant
       Differentiating both sides with respect to time we get
                                                
                                               A(t )
                                        A(t )        0
                                                 dt
                                                                                              
                                                                                           dA(t )
       Since     A(t ) is not zero, the equation above implies that              A(t ) and          are
                                                                                              dt
       perpendicular to each other.




                                                      8
       An everyday example is a particle moving in a circle. The magnitude of its position
       vector is a constant and therefore its velocity, which is the time-derivative of its
       position vector, is perpendicular to the position vector.


                
       OA  R1 sin t i  cost ˆ
                      ˆ         j 
                                     
1.25
                         ˆ
       OB  R2  sin t i  cost ˆ
                                  j

       The area of triangle OAB is given as
                                                          RR
                       OA  OB  R1 R2 2 sin t cost   1 2 sin 2t
                     1          1
                     2          2                           2
                                                  
       This is maximum at 2t            or t 
                                      2            4
1.26   Let the angle between the z axis and the vector be . Then the component OB is

       OAsin  .                                ˆ
                    Thus the magnitude of OB is k  OA .           However, its direction is

       perpendicular to the plane containing the z axis and the vector OA. To get the proper
                                                ˆ           ˆ
       direction we again take cross product of k  OA with k . The component in the z
                             ˆ
       direction is given as k  OA . Thus the vector OA is        kˆ  OAkˆ  kˆ  OA kˆ .   In
               ˆ                    ˆ
       general k can be replaced by n .




                                              9
                                            Chapter 2


2.1   (i)



                                                  y
                                                              T(y)


                   L
                                                    y



                                                                         M
                                                              T(y+y)+     yg
                                                                         L



      (ii) Since the element of length y is in equilibrium, we have
                                                      M
                           T ( y)  T ( y  y)        yg
                                                      L
            Using Taylor series expansion for T(y+y), which gives
                                   dT      1 d 2T
            T ( y  y)  T ( y)     y       2
                                                  (y) 2  
                                   dy      2 dy
            And taking limit y  0 leads to the differential equation for T(y). The equation
      is


                                                  dT   M
                                                      g
                                                  dy   l
      Solution of this equation is
                                                  M
                                     T ( y)       gy  C
                                                  L
      where C is the integration constant. This is determined by using the fact that at the
      lose end (y=L) of the rope, the tension is zero. This gives




                                              10
                                                       M ( L  y)
                                C  Mg and T ( y)                g
                                                           L


                    
2.2      Torque   r  F
                              ˆ j
         It is given that r  2i  ˆ and the force has magnitude 50N and acts in the direction
                    ˆ
         of vector 3i  2 ˆ . Thus the force is 50 times the unit vector in the direction of the
                          j
         given vector. This gives
                                                      ˆ
                                                     3i  2 ˆ 
                                                             j
                                                     13 
                                              F  50          
                                                              

                                  
                                  ˆ j
         With this the torque is 2i  ˆ 50 ˆ
                                                  
                                              3i  2 ˆ 
                                                     j   
                                                         50 ˆ
                                                             k
                                           13             13
2.3      (a)




                                                                               100N




                100N



         (b) The centre of the rod is at (3, 2)
                                                ˆ
               The right end is at position r  8i  2 ˆ and the force at this end is
                                                       j

                       3         
              F  100      ˆ 1 ˆ
                        2 i  2 j
                                  
                                       ˆ
      The left end is at position r  2i  2 ˆ and the force at this end is
                                              j




                                                  11
                                         3          
                                 F  100   ˆ 1
                                             i      ˆ
                                                     j
                                          2    2
                                                     
    Torque with respect to the origin =

8iˆ  2 ˆj 100
                 
                 
                   3ˆ 1
                    i
                             
                               
                           ˆ    2i  2 ˆ  100 3 i  1 ˆ   10i  100 3 i  1 ˆ 
                           j       ˆ     j      
                                                   2
                                                      ˆ     j
                                                              
                                                                     ˆ
                                                                           
                                                                            2
                                                                               ˆ     j
                                                                                       
                 2    2                                2                      2 
                                                             500k  ˆ


    Torque with respect to the centre of the rod =
          3                      3                    3      
 8i  100
  ˆ
          2
             ˆ 1j
                2 
                          ˆ 
             i  ˆ    2i  100
                                   2
                                      ˆ 1j
                                      i  ˆ   10i  100
                                         2 
                                                  ˆ
                                                          2
                                                             ˆ 1j
                                                             i  ˆ
                                                                2 
                                                             
                                             500k ˆ

    Torque with respect to the left end of the rod =
                                                                   3        
                                                       10 i  100 
                                                          ˆ           ˆ 1 ˆ
                                                                   2 i  2 j
                                                                            
                                                        500 k ˆ

    Torque with respect to the right end of the rod =
                                                                      3          
                                                        10 i  100 
                                                            ˆ            ˆ 1     ˆ
                                                                      2 i2     j
                                                                                 
                                                        500 k ˆ

(c) Torques about all the points are equal because the net force on the rod is zero.




                                      12
2.4   (a)

                           TA                                          TB



                                                    3m
                       A                                                    B

                                 1m                                0.5m
                                                     30N

                                        70N                     120N



      (b)   F   y    0 gives

                                                TA  TB  220                          (i)
            Net torque about A is zero, which gives
                                 3TB  1  70  1.5  30  2.5  120
                                                                                       (ii)
                                        415

            Equation (ii) gives TB  138 N
            This substituted in equation (i) gives TA  82 N


2.5   Component of force in the plane perpendicular to the axis is F cos at a distance of
      R from the axis.
      Therefore the torque about the axis is RF cos


2.6   Free-body diagram of the block




                                               13
                       N1

                                                         N2




                                      W




N1, N2 and W are three forces in a plane. Thus they must pass through one common
point for equilibrium. So the equilibrium conditions are only the force conditions.

F  horizontal    0 gives

                                   N1 sin   N 2 cos

F  vertical    0 gives

                                 N1 cos  N1 sin   W
   Solution of these two equations is
                             N1  W cos    and N 2  W sin 




                                           14
2.7    Free-body diagram of the plank



                                         N                      2m

                  Ry
                                1m                                                      100N

                                              0.2m

                                     Rx


       Free-body diagram of the block


                                     N



                                                                             F


                                              Nground



If the rod makes angle  with the horizontal then
                                 sin   0.2 and cos  0.98
(a) To get the horizontal force F, we first calculate the normal force N on the rod. To do so,
we calculate the total torque about the hinged end of the plank  1  N  3  100 cos  and
equate it to zero. This gives
                                       N  3  100 cos
                                              3  98
                                              294N

Now we balance the horizontal forces    F     horizontal    0 on the block to get

                                          F  294 sin 
                                               0.2  294
                                               59 N



                                               15
(b) Force balance on the plank

                           F    horizontal    0  Rx  N sin   59N

                           F      vertical    0  Ry  N cos  100
This gives
                                     R y  100  288  188 N

Minus sign in front implies that the direction is opposite to that shown in the free-body
diagram above.


2.8    Free-body diagram of the rod



                                                  N2



                                                        N1


                                                             F
                                 W


Balancing the vertical forces gives N1 = W = 50N
Balancing the horizontal forces gives N2 = F
Balancing the torque about the centre of gravity gives
                                              F  8  50  0.5
leading to
                                                   25
                                              F         8.8 N
                                                    8
2.9    Free body diagram of the painting




                                                   16
                     T



                                                         N1           N2

             N1+N2


                                     Fx

                             W                                  W


Force balance equations give
                                    Fx  T and N1  N 2  W

N1 and N2 are equal because the component of torque perpendicular to the wall must vanish.
This gives
                                           N1=N2=25N
Balancing the component of torque parallel to the wall taken about the lower end of the
painting gives
                                        20 3  T  10  50
giving
                                               25
                                          T         14 .4 N
                                                3


2.10     We first calculate the forces at the ends of the rod. These forces are applied by the
         supports. After finding the forces on the rod, we then calculate the forces and the
         torques applied by the wall on the supports.


         Free body diagram of the rod




                                               17
                    N1                                                            N2
                                                  140cm

                                 60cm

                                                  35N



       Free body diagrams of the left and the right supports




                     F1                                          F2

           1                                           2
                          5cm                                          5cm
                                   N1                                        N2




       The forces on the rod satisfy   Fy    0 which gives

                                             N1  N 2  35

       Taking torque about the left end and using         0 gives
                                140  N 2  60  35  N 2  15 N
       This gives
                                               N1  20 N
       Now balancing vertical forces and the torque on the supports gives
       For the left support F1=20N and  1  0.05  20  1Nm
       For the right support F2=15N and  2  0.05  15  0.75 Nm


2.11




                                             18
                                Ry                  60cm


                         Rx

                                                                 40cm



                           N

                                                40N

To find the force applied by the plastic block, we balance torque about the upper left corner.
This leads to
40  N  30  40  N  30N
Balancing the vertical forces gives Ry = 40N
Balancing the horizontal forces gives Rx = 30N
Negative sign means that the direction of Rx is opposite to that assumed in the free-body diagram
above.
Free-body diagram of the pole




                                               19
                                                         30N

                           40cm
                                            40N



                     30N




                                            N




                                                


Balancing the vertical forces on the pole gives N = 40N
There is no net horizontal force and the two horizontal forces give a couple = 300.4 = 12Nm
Balancing the torques on the pole about the ground gives  = 300.4 = 12Nm


2.12       Free-body diagram of the table


                                                    90cm

           ˆ
           j
                                     Rx                                        Nx


                                                                     Ny
                               Ry
                iˆ                                  20N


          To find Ny, we balance the torque on the table about its left hand edge to get
                               90  Ny  45  20  Ny  10N


                                                    20
By balancing the vertical forces, we get Ry  10N . The negative sign tells us that the force is
direction opposite to that shown above.
Free-body diagram of one of the rods



                                    Ry
                                    2

                                          30
                 Rx
                 2


                                                                                 Sx




                                                                Sy


Free body diagram of the entire system

                                                 90cm

                                          30                                  Nx


                                                                      Ny


                                                20N
                                                             2Sy              2Sx




To get Nx, we balance the component of the torque coming out of the paper on the entire system
about the lower hinge. This gives
                           30 3  Nx  45  20        Nx  10 3N




                                                21
The negative sign again tells us that the direction of the force is opposite to that shown.
Balancing the horizontal component of the force on the table then gives Rx  10 3N
                                                               Rx ˆ Ry ˆ
Note: The net force on each rod on its upper end is              i          ˆ
                                                                       j  5 3i  5 ˆ which is along
                                                                                    j
                                                               2     2
the rod as it must be for the equilibrium of a rod held at its ends.
Balancing the horizontal and vertical components of forces on each rod gives
                                        Rx
                                 Sx        5 3N and Sy  5N
                                        2
Thus the net force on each rod is 10N compressive.


2.13 Free body diagrams of the two side portions and the portion AC over the pulley:


                                             N
                     TA                                                  TC




                                     TA
                                                             TC
                                                 RM
                                                       g
                                                  L


                                                                                   F
                                                                        L2 M
                                                                             g
                                                                         L
                        L1 M
                             g
                         L



Tension TA and TC at both ends of the portion over the pulley is the same because the torque
                                                           L1
about the centre must vanish. This gives TA  TC             Mg
                                                           L
Free body diagrams of the portion AB and BC




                                                  22
                            Neffy                                        Neffy


          Neffx                                                                       Neffx
                                           TB                    TB


                            RM                                         RM
                                   g                                          g
                            2L                                           2L



             TA                                                                     TC



Notice that Torque of the normal reaction about the centre of the cylinder vanishes because for
each small portion of the rope over the cylinder, the normal reaction is radial. Thus T A (or TC)
and TB cannot be equal because they together provide a torque to balance the torque due to the
weight of the rope. Balancing the torque about the centre on AB gives
                               R         RM                               R  M
                    R  TA                   g  R  TB     TB   L1       g
                                 2       2L                               2 2 L
Thus if the net force by the cylinder on the rope is Neff at an angle  from the horizontal then by
force balance

                                        R  M                                 R  M
                    N eff sin    L1      g             N eff cos   L1       g
                                         2  L                                2 2 L
Note that Neff acts at a point different from the centre of BC because on different infinitesimal
portions it is different.


2.14 The support does not apply any torque about the x-axis. All other components and torques
      are balanced by the support.


2.15 When forces are applied at two points of the rod, force balance demands that the force be
      equal and opposite. However two such forces acting at two different points will give rise to
      a couple moment. The couple moment is zero only if the forces point along the rod (see
      figure below)


                                                       23
             Couple moment non-zero                            Couple moment zero



2.16 Let cables OA and OC make angle 1 and OB and OD angle 2 with the vertical. Then
      balancing the vertical forces gives
                                      2T (sin 1  sin  2 )  45000
The sine of the angles is easily calculated to be
                                           1                  1            2
                                sin 1         sin  2               
                                            2               11 4          5
This gives T=14050N

                                                                         
2.17 The torque direction is given by the direction of cross product n  F , which is
                                                                     ˆ
                       ˆ                                                                        ˆ
      perpendicular to n . This implies there is no component of the torque in the direction of n .


                                      ˆ             ˆ       ˆ
2.18 The net force on the plate is 50 i  50 ˆ  70 i  120 i  50 ˆ
                                             j                     j
Therefore the force that must be applied to the plate to keep it in equilibrium is

         ˆ
F  120 i  50 ˆ . Since there are only three forces acting on the body, they must all pass
                j
through the same point so that their net torque is zero. This is shown in figure below.




                                                    24
               A                                 B




                              O




              D                                   C



                                                      50 
                   ˆ
The force F  120 i  50 ˆ is at an angle   tan 1 
                          j                                   22 .6 from the line DC. Thus it does
                                                       120 
not pass through O and intersects with side AD and diagonal BD of the square. Therefore:
   (i)       It is not possible to keep the square in equilibrium by applying the third force at O.
   (ii)      It is possible to keep the square in equilibrium by applying the third force at a point
             on BD.
          Equation of BD with O as origin is y  x

                                                                       a 5   3a 
          Equation of line along which the third force acts is y       x 
                                                                       2 12   2 
                                                      3
          Solving the two equations gives y  x        a
                                                     34
          This gives the distance of point O = 0.125a
                               1 3 
          And from B =     2    a  0.58 a
                                2 34 
   (iii)     It is clear that for equilibrium, the force can be applied only on AD and BC.




                                                  25
                                                      Chapter 3


      3.1 For the three trusses shown, m = 21, j = 12. Thus they all satisfy 2j-3 = m. Thus they are
          all simple trusses.
      3.2 Showing that pin E is in equilibrium
          There are five forces acting on E of which two (FE and ED) are horizontal, two (CE and
          the external load) are vertical and one (BE) is at an angle. We wish to check if the
          horizontal and vertical forces add up to zero. It is solved in 3.1 that

                                              N Tensile , FBE            N Tensile 
                                        5000                      5000 2
                                FFE 
                                          3                           3
                                                N Tensile , FDE          N Tensile 
                                        10 ,000                     10 ,000
                                FCE   
                                            3                           3
          Since all the forces are tensile, they all pull the pin. In addition there is the external load
          of 5000N vertically down.
          The net horizontal force is

                                      F   x      FFE  FBE cos 45  FDE
                                                      5000 1 5000 2 10000
                                                                
                                                       3    2   3     3
                                                 0
          Similarly the net vertical force is

                                       F   y    5000  FBE sin 45  FCE
                                                             1 5000 2 10000
                                                  5000            
                                                              2   3     3
                                                 0
3.3       (i) The truss has 4 members and 4 joints. Number of force balance equations therefore is
          8 (2number of joints).         On the other hand, number of forces available is only 7
          (3+number of members), which implies that the truss will not be stable and will collapse.
          In terms of stability condition 2 j  3  m which implies that the truss will collapse.
         (ii) If we add one more member to the truss, i.e make m = 5, then 2 j  3  m is satisfied
         and the truss becomes stable and a simple truss. Let us add a member across AC.




                                                        26
To find forces in each member we start by first finding the forces applied by the
external supports. The free-body diagram of the truss is as follows:



                                           B                      C
                                NAy



                   NAx                                            5000N
                                  A                           D
                                                      ND



The direction of the forces applied by the external supports has been anticipated as
shown. To find ND, we balance torque about A to get
                  2  N D  (2  1.5 cos60)  5000
                                                     N D  6875N
                           2.75  5000            
Now balancing the vertical and horizontal forces on the truss gives
                          NAx = 0      and     NAy = 1875N
The negative sign again tells us that the direction of the force is opposite to that
shown.
We begin to apply the method of joints from point D since at this point there are two
unknown forces FAD and FCD. Assuming these forces to be tensile gives the free body
diagram of joint D as follows




                                      27
                                                          FCD




                                                    60
                     FAD


                                              6875N




Balancing the vertical forces on point D gives
                           FCD sin 60   6875  0  FCD  7939 N

The negative sign shows that the force FCD is compressive and not tensile as assumed.
Balancing the horizontal forces on point D gives
                           FCD cos 60   FAD  0  FAD  3969 N

The negative sign again shows that the force FAD is compressive and not tensile as assumed.
Next we go to point A and balance the forces there. The free body diagram of point A is


                                              FAB
                                                                    FAC


                                        60
                                                    
         3969N


                                      1875N




In drawing the figure above, we have shown the direction of FAD according to it being a
compressive force.

The length of rod AC is =    2  1.5 cos602  1.5 sin 602    3.04 m



                                              28
So
                         1.5 sin 60                    2  1.5 cos60
               sin                  0.427 and cos                  0.904
                            3.04                             3.04
Now balancing the horizontal forces at A gives
                               FAB sin 60   FAC sin   1875

and balancing the vertical forces at A gives
                                  FAB cos 60   FAC cos  3969

Solving these two equations gives FAB = 0 and FAC = 4390N
Since the sign of FAC is positive it is in the same direction as assumed and therefore tensile.
Now we can easily see that force FBC will be zero because point B is under equilibrium under
only two forces FAB and FBC and FAB has already been determined to be zero. Thus FBC = 0.
Thus all the forces are now determined. They are
FCD  7939 N (compressive)        FAD  3969 N (compressive)
FAB = 0   FAC = 4390N (tensile) and FBC = 0.


Finally to check our answer we make the forces at point C and see if they all balance. The
free body diagram of point C is




                                                                   7939N

                                                                   60

                                             

                                                            5000N
                                           4390N




Balancing the horizontal forces at C gives
                           7939cos60  4390cos  3969  3969  0
Balancing the vertical forces at C gives



                                               29
                   7939sin 60  5000  4390sin   6875  5000  1875  0
   This indicates that our answers are correct.
   Note: We see that FAB and FBC both vanish. This implies that members AB and BC may
   not bee needed for the truss. This is true because with just three members AD, AC and
   CD (m=3) there are only three joints (j=3) and the truss satisfies the condition 2 j  3  m
   for it to be a stable structure.


3.4 Free body diagram of the truss




                                  Ry


                            B                30cm                    C
        Rx
                                                      

                 20cm
                                                                    30N


                                                  N
                             A


Since point C is in equilibrium under one known and two unknown forces, both unknown
forces can be determined easily. The forces on C look as follows

                                                                           FAC


                                                                
                                       FBC




                                                          30N




                                             30
Balancing the vertical forces at C gives
FAC sin   30

                 2
With sin            this implies FAC = 54N (compressive)
                 13
Balancing the horizontal forces at C gives
                           3
FBC  FAC cos  54              45 N (tensile)
                            13
The only force left is at AB. We calculate this by balancing forces acting on pin A, which
look as follows.



                                                         FAB


                                                                  N
                                            

                            FAC


This gives
                                        FAB  FAC sin   30 N

Additionally we can also solve for the normal reaction N and the forces Rx and Ry. These
are
                                  N = 54N, Rx = 45N, and Ry = 30N


3.5 Rod AB provides a vertical force to hold pin A. However if it is removed and the
      vertical force is provided by a fixed pin joint, the structure will remain stable because we
      need 3j=6 forces for equilibrium of 3 joints; two of these are provided by the fixed
      supports and two by the two members. The forces in the members remain the same. So
      do the forces by the two support except that the fixed point at A also provides a vertical
      fore of 30N that was earlier provided by member AB.


3.6 Free-body diagram of the truss



                                                    31
                                           Ry

                                           B                C
                     Rx




                     N
                                       A                        D


                                                            100N


Since pin D has only two unknown forces acting on it, we can start our calculations from this
point onwards. The forces on D are




                                                     FCD




                               FAD


                                                     100N



It is immediately clear that
                                FAD = 0 and FCD = 100 N (tensile)


Next we go to pin C and balance the forces there. The forces acting on C are




                                                32
                                                                  FAC




                            FBC


                                                      100N



Balancing the vertical forces at C gives

                                  FAC  100 2N (compressive)
Balancing the horizontal forces at C then gives
                                     FBC  100 N (tensile)

Next we go to pin A. The forces there are as follows



                                                   FAB


                                                                   N
                            100 2 N




Balancing the vertical forces at A gives
                                     FAB  100 N (tensile)
Balancing the horizontal forces at A gives
                                             N=100N
Finally balancing forces at pin B will give the external forces
                                   Rx = 100N and Ry = 100N


3.7 Free body diagram of the truss is as follows



                                             33
                                         Ry
                                                          C                  D
                       Rx
                                         B

                                                                                 500N
                                Ny

                     Nx

                                     A                    E


(i)     There are 4 reaction forces at the supporting pins at A and B. In addition the forces
        generated by the members of the truss equal 6. This makes the total number of forces
        available = 10. The number of joints in the truss is 5 that require exactly 10 number
        of forces for equilibrium. Thus the truss is a stable one.
(ii)    It is also statically determinate since the number of forces available is equal to the
        number of equations to be satisfied for equilibrium.
(iii)   First we find Nx by balancing the torque about point B. This gives
                            0.75  Nx  1.5  500  Nx  1000N
        We now begin by balancing the forces at point D




                                                                     FED




                             FCD


                                                      500N


Balancing the vertical forces at D gives
                                   FED  500 2 N (compressive)



                                              34
Balancing the horizontal forces at D then gives
                                       FCD  500 N (tensile)

Nest we go to pin E because it has two unknown forces acting on it. The forces are as
follows




                                           FCE
                                                                  FAE




                                        500 2 N


Balancing forces at E gives
                                     FAE  500 N (compressive)
                                       FCE  500 N (tensile)

Next we go to pin at A. The forces there are




                                                 Ny


                                                          1000N
                              500N

                                          FAC



Balancing the forces gives
                                FAC  500 2N (compressive)
and
                                        Ny  500N (tensile)
Finally we go to point C where only one force FBC is unknown. The forces on C are


                                                35
                                                                      500 2 N
                                                                      FAC


                                 FBC                         500N


                                                      500N



Balancing the horizontal forces at C gives
                                               FBC=1000N
As a final check, the value for FBC gives the horizontal force Rx by pin B on the truss to be
1000N and vertical force Ry to be zero. This is consistent with the overall equilibrium of the
truss when it is treated as a system by itself.


3.8 Since each member of the truss weighs 50N, at each pin we take the load by each pin at
    that point to be 25N. The free body diagram of the truss is as follows; here each small
    arrow pointing down indicates the weight of the truss member, acting at its centre.



                                           NBy
                           NBx
                                                                  C
                                           B




                                                                                D
                                       A
                                                              E
                                                                                1000N
                                                       NE



We firs find NE. To do this we balance the torque about B. This gives




                                                 36
                         l                 3l
              l  NE       150  l  50   100  2l  1000  N E  2275N
                         2                 2
This immediately gives, by balancing forces on the entire truss
                                  NBx = 0 and NBy = 925N
The negative sign showing that the force I opposite to the direction assumed in the figure
above.
We begin at pin D as there are two unknown forces there. The force diagram on pin D is as
follows (there are two members meeting at pin D that give a load of 225=50N there)


                                  FCD



                                                            FDE


                                                  1050N



Balancing the forces gives

                              FCD  1050 2N  1485N (tensile)
and
                                 FDE  1050 N (compressive)
Next we go to pin E. The forces acting there are (including 325=75N from 3 members)


                                                  2275N



                                1050N                       FAE


                                                  FCE +75N


Balancing the forces gives



                                           37
                                  FCE  2200 N (compressive)

and
                                  FAE  1050 N (compressive)


Next we go to pin A. The forces acting there are (including 325=75N from 3 members)



                                           FAB             FAC




                             1050N
                                                   75N


Balancing the forces gives

                               FAC  1050 2N  1485N (tensile)
and
                FAB  75  1050 N  975 N  FAB  975 N (compressive)


Net we move to point B where the forces are (including 225N=50N from two members)

                                           FAB




                                FBC
                                                   50N

                                                   925N

This immediately gives
                             FAB = 975N (compressive) and FBC = 0
As a final check, one balances all the forces at C and sees that they all balance properly. This
implies that the answers obtained by us are correct.



                                            38
3.9 Free body diagram of the truss with the weight of each member included. The free body
    diagram is then as follows.


                                       B               C
                             NA                             ND




                   A
                                       F          E              D
                        Nx

                                                  5000N


By balancing the torque about A we get
12  N D  2  1000  4  500  6  1500  8  500  5000  10  1000        16750
                                                                         ND 
          67000                                                                 3

Balancing the vertical and horizontal forces on the truss, this gives
                                              11750
                                       NA          and Nx  0
                                                3
For calculating force in members, we take the weight of each member shared equally at each
joint. The forces on A are (including the weight of two members)

                             11750
                                   N
                               3

                                                           FAF




                                           500N
                       FAB


Balancing the forces at point A
                             FAB
                            11750
       F   y    0 gives
                        2      3
                                   500 or FAB  4832N (compressive)
                                   

                             F    10250
         Fx  0 gives FAF  AB  3 N (tensile)
                               2



                                                  39
Next we go to point F. The forces at point F are (including the weight of three members)



                                              FBF

              10250
                    N
                3                                              FFE



                                          750N

   Balancing the forces here gives
                                        FBF  750 N ( tensile)
                                                10250
                                        FFE          N ( tensile)
                                                  3
Next we go to point B since now there are only two unknown forces there. At point B the forces

look as follows (including the weight of four members)


                                                         4832N


                                                         FBC


                           750N


                                                          FBE
                                  1000N


Balancing the forces

                                FBE                      4832
           F   y    0 gives
                                  2
                                       1000  750 
                                                           2
                                                                  FBE  2357 N (tensile)

                                       4832       2357
           F   x    0 gives FBC 
                                          2
                                              
                                                    2
                                                          0  FBC  5083N


Negative sign above means that the direction of the force is opposite to the one assumed. So

                                  FBC = 5083N (compressive)


                                                    40
We next consider point C and balance the forces there. The forces at point C are (including

750N form the weight of three members)



                                   FCD
                                                               5083N


                                   750N

                                                 FCE


Balancing the forces

                                  FCD
            F   x     0 gives
                                    2
                                         5083  FCD  7188 N (compressive)

                                                FCD
             F   y    0 gives FCE  750 
                                                  2
                                                       5083  FCE  4333N (tensile)


                                                        16750
Next we go to pin D where the normal reaction is              N and balance the forces there. The
                                                          3

force diagram there is (including 500N form the weight of two members)



                                                  16750
                                                        N
                                                    3
                          FED


                                        500N

                                                           7188N


It is easily seen that the vertical forces balance at this point. This points to the correctness of our

calculations so far. Balancing horizontal forces then gives

                                         FED = 5083 N (tensile)



                                                  41
As a final check we should check whether all the calculated forces balance at pin E. The forces

at pin E are (including the weight of four members)



                             2357N                  4333N


                      10250
                            N                                      5083N
                        3
                                     1000N
                                                         5000N



       All these forces balance as can be seen by calculating the net x and y components of the

       forces. Thus our calculations are correct.

3.10     Free body diagram of the truss

                                                             2000N
                             NBy
                                                C
                                                                             1000N
                     NBx         B                                   D


                                          A              E

                                                              NE


        Taking torque about B we get

                                                                     8000
                                3  N E  4  2000  N E 
                                                                       3

        Balancing the horizontal and vertical forces now gives

                                                                    2000
                              NBx = 1000 N     and        NBy =         N
                                                                     3

        Negative sign above means that the direction of the force is opposite to the one assumed.



                                                    42
We begin at pin D. The forces there are

                                                             FED


                                                       
                          FCD
                                                                   1000N




                                               2000N


                                 1                      1.5 2  1
In the diagram above cos           0.667 and sin              0.745
                                1.5                      1.5
Balancing the vertical forces gives
                     FED sin   2000       FED  2683 N (compressive)
Balancing the horizontal forces gives
                                 FCD  FED cos  1000  2789N

Next we go to pin E. The forces there are

                                                    8000
                                                         N
                                                      3


                                                                   FAE
                                                       

                                                             FCE

                                      2683N


Balancing the vertical forces gives

                 2683  FCE sin   8000       FCE  895N (compressive)
                                           3
Balancing the horizontal forces gives
                     FAE  2683 cos  FCE cos  1192 N (compressive)


                                               43
Next we go to pin A. The forces there are


                                                         FAC


                                                     

                    1192N
                                                     

                                                          FAB


Balancing the vertical forces gives FAC = FAB.
Balancing the horizontal forces gives
           FAB  FAC cos  2 FAB cos  1192      FAB  895 N(compress ive)

This also means that
                                     FAC = 895N (tensile)
Now we go to pin C. The forces there are


                                895N


                                       
                                                                2789N
                       FBC
                                       

                                895N


The vertical forces are already balanced here. Balancing the horizontal forces gives
FBC  2  895  cos  2789     FBC  1596 N(tensile)

To check our answers, we finally balance the forces at pin B and see if they all balance there.
At pin B the forces are as follows




                                            44
                                      895N


                                             
                                                                        1596N
                        1000N

                                                       2000
                                                            N
                                                        3



   It is easily seen that all the forces above balance. So our answers are all consistent.


3.11    The weight of the road = volume of the roaddensityg
                                  = 128.3200010
                                  = 576000N
        This weight is divided equally between the two trusses on the sides.             Thus only
        288000N is supported by each truss.
        Weight of the members of the truss = 135000
                                                 = 65000N
        Total weight supported by each truss therefore is = 353000N


        Free body diagram of the truss is

                                                 B       C          D
                                NAy                                         NE


                        NAx
                                      A          H           G      F       E


                                                                 353000N

        From the balance of forces, is clear that
                                      353000
         N Ax  0     N Ay  N E             176500N
                                        2



                                                  45
Let us now consider forces at each pin one by one. Each pin has the following forces
                                                                     288000
acting on it: The weight of the road divided over 5 pins, which is           57600N ;
                                                                       5
the weight of the members at that pin; and the forces applied by the members.
Let us now balance forces at point E. The forces on pin E are (including the weight of
the members)

                                                     176500N



                            FEF


                                                 
                                  57600N                       FDE
                                                 5000N


In the figure above
                                     3                    4
                           sin        0.6   cos         0.8
                                     5                    5
Balancing the vertical forces in the figure above gives
       FDE cos  57600  5000  176500        FDE  142375 N (compressive)
Balancing horizontal forces then leads to
                           FFE  FDE sin   85425 N (tensile)
Next we consider pin D.       The forces on pin D are (including the weight of the
members)




                                        46
                        142375N

                                             

                                                                 FCD


                                       FDF


                                                  7500N



Balancing horizontal forces then leads to
                      FCD  142375 sin   85425 N (compressive)

Balancing the vertical forces gives
                     FDF  142375  0.8  7500  106400 N (tensile)
Next we look at pin F. The forces there are (including the weight of the members)

                                                      106400N



                            FGF                                       85425N


                                                  
                                  57600N                        FCF
                                                  10000N


Balancing the vertical forces in the figure above gives
       FCF cos  57600  10000  106400         FCF  48500 N (compressive)

Balancing horizontal forces then leads to
                     FGF  FCF sin   85425 N  114525N (tensile)

By symmetry of the problem, forces on the members to the left of member CG will be
exactly the same as on the corresponding members to its right. The only force that we
now have to calculate is on member CG. For this we consider point G. Two horizontal



                                      47
        forces at G are by HG and GF and are equal to 114525N each. The forces at point G are
        then given as

                                                          FCG



                     114525N                                                114525N


                                                      
                                         57600N
                                                      7500N




        This gives FCG = 65100N


        Finally we check our answer at point C. The forces there (including the weight of 5
        members meeting there) are

                                  48500N                         48500N



                            85425N                                     85425N




                                         65100N                  FCF
                                                      12500N



        As is easily seen, the forces at C balance and therefore our calculations have been
        consistent throughout.


3.12   Free body diagram of the truss is as follows




                                              48
                                   2000N               1000N

                  NA                                                             NE
                                             C
                               B                                   D

     RA
                           
                   A                H                 G            F              E


Here the distances and the angles are
AH = HG = GF = FE = 2.5m             BH = DF = 1.25m (similarity of triangles)
          1.25
tan           0.5    26.6  sin   0.447 and cos  0.894
           2.5
Balancing torque about point A gives
10  N E  5  1000  2.5  2000      N E  1000 N
Thus NA = 2000N and RA = 0
Now that the external reactions have been determined, we can go about calculating the forces
in the members. We start with pin E because there are only two unknown forces there. The
forces at E are

                                                      1000N




                               FEF
                                                          E    
                                                                       FDE


     Balancing the vertical forces in the figure above gives
                        FDE sin   1000      FDE  2236 N (compressive)
     Balancing the horizontal forces gives
                         FEF  FDE cos  2236 * 0.894  2000 N (tensile)



                                                 49
Nest we go to pin D. The forces there are

                                                         FDF
                       2236N




                                                     D
                                                                       FCD


Balancing the forces gives
                                     FCD  2236 N (compressive)

                                              FDF = 0


Next we go to point F where the forces are as shown below.

                         FCF

                                                 0N


                         FGF
                                                 F                 2000N


Balancing the forces gives
                             FGF = 2000N (tensile)       FCF = 0
Pin G is considered next. Since the forces there are only vertical and horizontal, even
without making the forces there, we immediately can write
                               FCG = 0 and FGH = 2000N (tensile)
Next we consider point A where two members AB and AH meet and therefore there are two
unknown forces. The forces there are




                                            50
                                          2000N




                                                               FAH
                                      
                                             A
                            FAB


Balancing the vertical forces gives
                                                  2000
                FAB sin   2000  FAB                  4472N (compressive)
                                                 0.4472
Balancing horizontal forces then gives
                             FAH  4472 cos  4000 N (tensile)
Next we go to point B. Here there are four forces acting and each pair (FBH and 2000N; and
FBC and 4472N) has two forces in opposite directions. Thus without solving the detailed
force balance equations, we can directly write
                FBH = 2000N (compressive) and FBC = 4472N (compressive)
Next we go to point H and balance the forces there. The forces there are as follows

                                                         FCH




                                                       45
                                                                2000N
                            4000N                  H



                                            2000N


By balancing the forces at H, it becomes clear that FCH  2000 2N (tensile)
Finally the answers are checked at C. The forces at C are as shown below




                                            51
                                 2236N
                                                                4472N


                                                                D



                      2000 2 N                 1000N


As is easily seen, the horizontal a n vertical forces all balance at C. Thus our answers are all
correct.
To calculate the forces by method of sections, we make a cut through the truss so that it
passes the concerned members. In the present case we take the following section of the truss
and show various forces on that section.

                                            2000N

                        2000N
                                                        FCB
                                           B                  FCH


                                       
                                                                FGH
                                 A             H


In the figure above, FCH is determined easily by taking torque about A since the torque due to
FCB and FGH both vanish about A. This gives
       FCH
AH             AH  2000    FCH  2000 2 N (tensile)
           2
To find FCB, we balance the vertical component of the forces to get
FCH                                  2000
       FCB sin   0  FCB                4472 N
  2                                  sin 
Negative sign here means that the force is opposite to the direction assumed and therefore is
compressive in nature.
Finally, balancing the horizontal forces leads to


                                               52
                  FCH
          FGH           FCB cos  0  FGH  2000  4472  0.894  2000 N (tensile)
                    2


3.13   The free-body diagram of the truss on one side is as follows (Notice that the weight of
       the truck is equally divided between the two trusses)


                                         B              C          D
                                                                       NE
                            NA

                              A                                        E
               RA
                                              H         G    F



                                                            50kN


   We first calculate NE by balancing torque about A. This gives
   16  N E  12  50 kN  N E  37 .5 kN
   This gives NA = 12.5 kN and RA = 0
   To find forces in members CD and DG, we make a cut through CD, DG and GF. This looks
   like the following


                                     FCD                           D
                                                                       37.5 kN


                                                                       E
                                   FGF                       F

                                             FGD

                                                            50kN


   To find FGF, we balance torque about point D about which the torques due to FCD and FGD
   vanish. This gives


                                                   53
                         4  37 .5  5  FGF    FGF  30 kN (tensile)

To obtain FCD, we take torque about point where FDG and FGF intersect, which is point G.
This leads to
                 5  FCD  8  37 .5  4  50  FCD  20 kN (compressive)

Now we balance the horizontal and vertical forces on the truss. Balancing horizontal forces
gives
                                  4
                         FGD            20  30  0  FGD  16 kN
                                  41
Negative sign here means that the force is opposite to the direction assumed and therefore is
compressive in nature.
To find the forces in the members BC and BG, we make a cut through the members BC, BG
and HG as follows and then calculate the forces.


                                       B               FBC
                    12.5 kN

                           A
                                                             FGH
                                           H
                                                         FBG


To obtain FBC, we take torque about point where FGH and FBG intersect, which is point G.
This leads to
                     5  FBC  8  12 .5  FBC  20 kN (compressive)

Next we find FGH by taking torque about B. We get
                         5  FGH  4  12 .5  FGH  10 kN (tensile)

Finally we get FBG by balancing vertical and horizontal forces. Horizontal force balance
gives
                                  4
                          FBG           10  FBG  16 kN (tensile)
                                   41
Finally to find FCG, we make the following cut through the truss




                                               54
                                    B        FCG               D
                                                                   37.5kN
                     12.5kN

                           A                                       E
                                        H           G    F



                                                        50kN


Since the vertical forces all balance, this implies FCG = 0.
Further, the horizontal forces are also balanced.




                                             55
                                                Chapter 4


      4.1




                       sN



                 Frictional
                 force




                                                       Fmax        Applied force



4.2         Since the block is in equilibrium under three forces, the three forces must pass through
            the same point. Thus the normal reaction will be at the point where the arrow showing
            the weight meets the inclined plane. This is shown below.

                               f
                                                       N




                                      mg

                                                               θ




4.3         The free body diagram of the block is as follows




                                                  56
                                                              N

                                                F


                                            
                             f

                                                           mg


Balancing the horizontal forces gives
                                                f  F sin 
Balancing the vertical forces gives
                                        N  mg  F cos

Since the maximum frictional force f m ax  N , for equilibrium we should have

                                                                        mg
                       F sin    mg  F cos   F 
                                                                  sin    cos


4.4     We consider two different situations when the weight on the table is about to move to
        the left or to the right. When it is about to move to the left, its free body diagram will
        look as follows

                                                          N




                            10g
                                                                            mg

                                                                   f



                                                     50g

By equilibrium conditions, we have
                                                 N = 50g



                                                     57
                                            f + mg = 10g
Since
                                               f  N
We have
                               f  10g  mg  0.1 50g  5  m
Thus the minimum value of m is 5kg when the frictional force is at its maximum pointing to the
right. As m is increased above 5kg, frictional force becomes less and less, eventually changing
direction and attaining its maximum value pointing left. In that situation, the free body diagram
of the block on the table is as follows.

                                                       N




                             10g
                                                                      mg


                                   f


                                                 50g

In this situation, the equation for horizontal force balance is
                                            f + 10g = mg
This coupled with f  N leads to
                               f  mg  10g  0.1 50g  m  15
Thus
                                             5  m  15


4.5 Taking the x axis along the plane and the y axis perpendicular to the plane, the free-body
diagram of the block looks as follows.




                                                  58
                                                               Y
                                    N
                                                                          X


                F         30º


                                                    30º
                    F

                                                 100g


The equations describing equilibrium in the X and the Y directions are

                          F    x    0  F cos30  F   100g sin 30  0

                          F    y    0  N  F sin 30  100g cos30  0
The first equation implies that
                                            F    F cos30  100g sin 30
                  2
Taking g = 9.8ms , the value of F for different values of F is
                                            F  600N  F   29.6 N
                                            F  500N  F   57.0 N
                                            F  100N  F   403.4 N


4.6   Free body diagram of the box when it is about to move (i.e. the frictional force is at its
      maximum) is shown below


                                                           N
                    F


                                        h                          b

                                                          a

                         N
                                                   mg




                                                              59
When the box is about to move, the friction is at its maximum and is equal to N. The force F
also equals N at this point. This creates a couple that is counterbalanced by the couple formed
by the weight of the box mg and N (=mg). This is the reason that N shifts towards the direction
                                                                                                      a
of the push. However, the maximum couple moment that can be created by mg and N is                      mg .
                                                                                                      2
Thus for the box not to topple, the couple created by F and the friction should remain less than
a
  mg . Thus implies
2
                                                           a                       a
                                          h  mg           mg        h
                                                           2                      2


4.7 suppose each break show makes an angle  at the centre as shown below



                                                            



The force F is assumed distributed uniformly over the shoe. Then the torque due to the
frictional force will be
                      b
                   
                                Fr
                                                r dr 
                                                                  
                                                           2 F b 3  a 3
                                                                                       
                                                                            2 F a 2  ab  b 2   
                      a
                           
                               b   2
                                         a2                     
                                                             3 b2  a2       3      a  b 
                           2
With two shoes therefore, the torque would be

                                                   
                                                             
                                                        4 F a 2  ab  b 2   
                                                          3      a  b 

4.8 It is given that mass M is balanced by mass m. The contact angle is π. Since each time the
    string is wound once more around the rod, the mass M that can be balanced by m becomes
    twice as large, we have
                                M  m exp( )    
                                                  
                                2 M  m exp(3 )   2  exp(2  )
                                4 M  m exp(5 )



                                                                 60
   This gives  = 0.11


                                                                                           L1
4.9 Neglecting the length of the rope passing over the pulley, we have mass                     M on one
                                                                                        L1  L2

                                                     L2
    side of the pulley that is balanced by mass           M on the other side. Thus we have
                                                  L1  L2

                    L1                     L2
                         Mg  exp(  )         Mg                 L1  exp(  ) L2
                 L1  L2                L1  L2


4.10 As the weight is put, it has a tendency to move down. Hence the frictional force will be in
    the counterclockwise direction. Thus if the tension in the rope on the spring balance side is
    T1 and that on the weight side is T2 then
                                               
                                  T2  T1 exp   
                                               2


     Now it is given that T1 = 5g and T2 = mg. Thus we get
                                  m  5 exp(0.2   / 2)  6.85kg
     An interesting possibility exists if a person had pulled the weight down and then slowly
     brought it to equilibrium. In that case the tension will work in the other direction and
                                 m  5 exp(0.2   / 2)  3.65kg
     However we have not considered this possibility.


4.11 There is a range of M2 that exists because frictional force can act with its maximum value
    in one direction to the maximum in the other direction.         Largest value of M2 is when the
    mass M1 is about to slide up the plane. The free body diagram of M1 in that case is as
    follows




                                                  61
                            N
                                                          T




                                            f

                                
                                         M1g


When the mass M1 is about to slide up, we have
                  T  M 1 g sin   1 M1 g cos  M1 g (sin   1 cos )
                                                         
The contact angle between the rope and the pulley is    
                                                     2    
Since the rope has a tendency to move clockwise, the frictional force due to the pulley will be
acting counterclockwise. Thus we have
                                                                                 
       M 2 g  T exp 2            M 2 g  M 1 g sin   1 cos  exp 2    
                     2                                                     2       
       Thus
                                               
       M 2  M 1 sin   1 cos  exp 2    
                                        2       
The other extreme is when the mass M1 is about to slide down the plane. In that case the free
body diagram of M1 is


                            N
                                                          T




                                            f

                                
                                         M1g


Thus we have
                      T  M 1 g sin   1 M 1 g cos  M 1 g (sin   1 cos )


                                                 62
Now the rope has a tendency to move counterclockwise, the frictional force due to the pulley
will be acting clockwise. Thus we have
                                                                                     
      M 2 g exp 2      T           M 2 g  M 1 g sin   1 cos  exp  2    
                2                                                                 2    
      Thus
                                                
      M 2  M 1 sin   1 cos  exp  2    
                                            2    


4.12Free body diagram of the tire when it is loaded and is about to roll is as follows




                                                                       F


                                      f
                                                   W      N


Balancing the vertical forces gives
                                                                W
                                      N  abP  W        a
                                                                bP


Balancing the horizontal forces gives F = f


Balancing torque about the centre of the wheel gives
                                             a                 W2
                                      FR      W    F
                                             2                2bPR




                                                   63
                                                  Chapter 5


5.1   Consider a composite surface of total area A made up of N different surfaces. Then the
      coordinates  X C , YC  of the centroid satisfy

                                                 AX C   xdA
                                                 AYC   ydA

 If the area of each surface is Ai (i  1 N ) then A   Ai
                                                                                      i

Now in the definition of the centroid, the integrals can be performed separately over each surface
so that we can write

                                         AX C    xdA   Ai X Ci
                                                      i       i                   i

                                         AYC    ydA   Ai YCi
                                                  i       i                   i


This immediately gives
                          Ai X C i      Ai X C i                  Ai X C i   Ai YC i                      
                  XC                 
                                        A                and YC 
                                                                              
                                                                                 A                             
                                                                                                                 
                             A                i                       A              i                        


5.2   By symmetry it is clear that XC = 2. We are nevertheless going to prove it below. We first
      calculate the area of the surface. It is

                                                                                             
                                             4                    4
                                        A   ydx   4  x  2 dx
                                                                                          2

                                             0                    0
                                                                          4
                                                           16   x  2 dx
                                                                                          2

                                                                          0

Substituting z  x  2 we get
                                                 2
                                                                                      16 32
                                       A  16   z 2 dz  16                           
                                                 2
                                                                                       3   3
 Y                                          To calculate Xc, we take vertical strips of width dx on the
                                            surface at distance x from the origin and then calculate

                                                                                                  XC 
                                                                                                          xdA
                                                                                                          A

 O                               X
       x
                                                                  64
Thus


                                               x 4  x  2 dx
                                              4
                                                               2


                                       XC    0
                                                                     2
                                                      32
                                                           3
Similarly to calculate YC, we take horizontal strips of width dy at height y and calculate

                                                                             YC 
                                                                                     ydA
 Y                                                                                   A
                                                  At height y, the strip extends from x1 to x2. These
                                                  points are given by the equation

                                                           y  4  x  2              x1  2  4  y
                                                                             2

 y
                                                                                      x2  2  4  y

        x1           x2            X              Therefore dA  ( x2  x1 )dy  2 4  ydy
 O
We thus have
                                                      4
                                         32
                                            YC  2 y 4  y dy
                                          3       0


Substituting y  4 sin 2  so that dy  8 sin  cos d , we get
                        2                                               1
                YC  2  4 sin 2   2 cos  8 sin cos d  128 sin 2  cos2  d cos 
             32
              3        0                                          0


                                                                                           
                                                                     1
                                                                128 cos2   cos4  d cos 
                                                                     0

                                                                   256
                                                               
                                                                   15
This gives
                                                           8
                                                    YC 
                                                           5
                         8
Thus the centroid is at  2, 
                         5




                                                      65
5.3   One curve (call it curve 1) y  4  ( x  2) 2 in this problem is the same as that in the
      problem above. The other curve (curve 2) is
                                                                   
                                   y  16  4( x  2) 2  4 4   x  2 
                                                                                   2
                                                                                       
                                                                                                             128
The y-axis of curve 2 is thus 4 times curve 1. The area of curve 2 is therefore                                  . Similarly
                                                                                                              3
                                                                                                                    8 32
the x coordinate the centroid of curve will remain at 2 but the y coordinate will be 4                                 .
                                                                                                                    5 5
Thus we have

                A1 
                       32
                             X C1 , YC1    2, 8 
                                                      ;        A2 
                                                                        128
                                                                               X C 2 , YC 2    2, 32 
                                                                                                        
                        3                       5                      3                          5 

The area A for which we wish to obtain the centroid  X C , YC  is obtained by removing surface
formed by curve 1 from the surface formed by curve 2. We thus have


                 128 32
             A             32
                  3     3
             AX C  A2 X C 2  A1 X C1  32 X C   A2  A1   2                                  XC  2
                                                            128 32 32 8
             AYC  A2YC 2  A1YC1  32YC                                                        YC  8
                                                             3  5   3 5


5.4   Trapezoidal loading is shown in the figure below




                                                                                       w2
                    f(x)


                                w1


                                                                                                        X
                                     X1                                       X2




                                                            66
                 The total force on the beam will be equal to the area under the curve. Thus the total force is
                 equal to
                                                                           w1  w2  
                                                                                          X 2  X1 
                                                                                2
                 This load will be acting at the centroid of the area. Thus it acts at
                       w2  w1            
       X2

        x w
                               x  X 1 dx
                                           
                        X 2  X1
                1
                                          
XC 
       X1

                w1  w2  X 2  X 1 
                                  2

       w1
            X      2
                    2   X   1
                              2
                                    w
                                w1  X 12  X 2  X 1 X 2 w2  w1  X 1  X 2 X 1
                                           2
                                               2
                                                           
                                                                          
                       2                  3                             2
                                    w1  w2  X 2  X 1 
                                               2
  3w1 X 2  3w1 X 1  2 w2 X 1  2w2 X 2  2w2 X 1 X 2  2 w1 X 12  2 w1 X 2  2w1 X 1 X 2  3w2 X 12  3w2 X 1 X 2  3w1 X 12  3w1 X 1 X 2
        2          2          2           2                                  2

                                                            3w1  w2  X 2  X 1 




                                                        w1 X 2  2w1 X 12  w2 X 12  2w2 X 2  w2 X 1 X 2  w1 X 1 X 2
                                                             2                              2
                                                    
                                                                           3w1  w2  X 2  X 1 


                                                        X 2 w1  2w2   X 12 2w1  w2   w2 X 1 X 2  w1 X 1 X 2
                                                          2
                                                    
                                                                         3w1  w2  X 2  X 1 


                 Adding and subtracting w2 X 1 X 2 and w1 X 1 X 2 in the numerator we get


                                          X 2 w1  2 w2   X 12 2 w1  w2   2 w2 X 1 X 2  w1 X 1 X 2  2 w1 X 1 X 2  w2 X 1 X 2
                                            2
                                      
                                                                           3w1  w2  X 2  X 1 


                                          2w1  w2 X 1  X 2  X 1   w1  2w2 X 2  X 2  X 1 
                                      
                                                           3w1  w2  X 2  X 1 


                                          1 2 w1  w2 X 1  w1  2 w2 X 2
                                      
                                          3            w1  w2 



                                                                                     67
5.5   From figure 5.14, for a plate of width w
                                     w1  gh1 w             w2  gh2 w
Similarly, from figure 5.15
                                              h1                           h2
                                X 1  Y1                    X 2  Y2 
                                             cos                         cos
This gives from formula 5.9
                                             1 (2h1  h2 )h1  (h1  2h2 )h2
                              X C  YC 
                                             3       (h1  h2 ) cos


                                         
                                               
                                             2 h12  h1 h2  h2
                                                              2
                                                                   
                                             3 h1  h2  cos
which is equivalent to the depth given by formula (5.17)


5.6   Loading on the tank door is triangular as shown below



                                                   N1

                                                                   NA

                               0.25m

                      0.5m

                                                                 153.125N

                                                                 NB



                                         19.6N


The average pressure is the pressure of water at the centroid of the submerged part. Thus the
average pressure will be
                           ghcentroid( plate )  1000  9.8  0.125  1225Nm2
Thus the total force due to the water pressure is



                                                        68
                                  F  0.5  0.25 1225  153.125N
This force acts at the centroid of the loading that is triangular in this case. Thus it is at a distance
                                     1
                                       0.25  2  0.50  0.417m
                                     3
below point A.
We now apply equilibrium conditions to the door. This leads to
                           N1  19.6 N
                          0.5  N B  0.417  153.125  N B  127.7 N
                           N A  N B  153.125  25.4 N


5.7   The rectangular surface looks as follows


                                                              Y




                                                                                   b

                                                                                            X



                                                              a




To find Ixx, we take a strip (see figure above) of width dy parallel to the x-axis and calculate
                                                  b2
                                                              ab 3
                                         I xx     y ady 
                                                         2

                                                 b 2
                                                              12

Similarly to find Iyy, we take a strip (see figure above) of width dx parallel to the x-axis and
calculate
                                                  a2
                                                                  a 3b
                                         I yy      x bdx 
                                                      2

                                                  a 2
                                                                  12



                                                         69
To find Ixy, we take a small square (see figure above) of size dxdy parallel and calculate
                                                     a2 b2

                                            I xy       xydxdy  0
                                                      a 2 b 2


by symmetry of the inegrand.


5.8


                                                            Y’
                                                                                         X’




                                   b                                       
                                                                       O


                                                                       a



From the figure
                                                 b                           a
                             sin                                cos 
                                               a2  b2                     a2  b2
                                                      2ab
                             sin 2  2 sin  cos 
                                                    a  b2             2


                                                     a2  b2
                             cos 2  1  sin 2 2  2
                                                     a  b2
From the formula for transformation of area moments (taking X and Y axis as in the problem
above) we get
                                       I xx  I yy       I xx  I yy
                           I x'x'                    cos 2  I xy sin 2
                                           2     2
                                 aba 2  b 2  abb 2  a 2  a 2  b 2 
                                              
                                     24             24         a 2  b 2 
                                      a 3b 3
                               
                                   6a 2  b 2 
Similary


                                                             70
                                          I xx  I yy          I xx  I yy
                             I y'y '                     cos 2  I xy sin 2
                                                           
                                                 2  2
                                    aba 2  b 2  abb 2  a 2  a 2  b 2 
                                                 
                                        24             24         a 2  b 2 
                                      aba 4  b 4 
                                  
                                      12a 2  b 2 
and
                                                     I xx  I yy
                                         I x' y'                  sin 2  I xy cos 2
                                                           2
                                                  abb 2  a 2  2ab
                                                
                                                      24        a 2  b 2 
                                                     a 2 b 2 a 2  b 2 
                                                
                                                      12a 2  b 2 


5.9

                                                                              Y




                        b                                                                   X




                                                                         a



To calculate IXX we take a horizontal strip of width dy at y (see figure) for dA and calculate
                                                                    b
                                                                             a 2
                                      I XX   y dA 2  y 2
                                                       2
                                                                               b  y 2 dy
                                                                   b
                                                                             b

Taking y  b sin  , we get
                             2                                                    2
                                     a 2                    sin 2 2      ab 3
                 I XX    2  b sin  b cos d  2ab 
                                  2        22        3
                                                                     d 
                            2
                                     b                  2
                                                                4           4



                                                                   71
Similarly for IYY, we take a vertical strip at x for dA and calculate
                                                                     a
                                                                             b 2
                                      I YY   x 2 dA  2  x 2                a  x 2 dx
                                                                    a
                                                                             a

Taking x  a sin  , we get
                                                                     a 3b
                                                            I YY 
                                                                         4
And by symmetry
                                                                I XY  0
                                                                                                   
We now calculate the moments and product of inertia about a set of axes rotated by an angle
                                                                                                   3
with respect to the original one and with the same origin. Thus
                                      I xx  I yy       I xx  I yy
                          I x'x'                                       cos120  I xy sin 120
                                     2          2
                                 ab(b  a ) 1 ab(b 2  a 2 )
                                       2  2
                                             
                                      8         2   8
                                 ab 2
                               
                                 16
                                          
                                     3a  b 2               
Similarly
                                      I xx  I yy       I xx  I yy
                          I y'y '                  cos120  I xy sin 120
                                                    
                                     2            2

                               
                                               
                                                        
                                 ab b 2  a 2 1 ab b 2  a 2                    
                                      8           2   8

                               
                                         
                                 ab a  3b
                                       2      2
                                                            
                                      16
Product of inertia is calculated using the formula
                                              I xx  I yy
                                  I x' y'          sin 120  I xy cos120
                                               2

                                         
                                            3 ab b 2  a 2             
                                           2       8
                                            ab
                                         
                                             16
                                                 3 a2  b2                  




                                                                    72
5.10


                         Y




                                                            y

                                                                                       X
                                 x1                                          x2
                          O


To calculate IXX we take a horizontal of width dy strip at y (see figure) for dA and calculate
                                                                R
                                I XX   y dA  2 y 2 R 2  y 2 dy
                                               2

                                                                 0

To evaluate the integral, we substitute y  R sin  so the integral is transformed to
                                                        2
                                      I XX  2 R        sin              cos2  d
                                                   4                 2

                                                        0
                                                   4  2
                                               R
                                                        sin            2 d
                                                                     2

                                               2        0

Now substituting z  2 , we get
                                                                
                                                     R4
                                                        sin z dz
                                                             2
                                          I XX
                                                     4 0
                                                         R4
                                                   
                                                             8




                                                             73
                                 Y




                                                                        x

                                                                                                                  X
                                 O


Similarly for IYY, we take a vertical strip of width dx at x for dA (see figure) and calculate

                                          I YY   x 2 dA
                                                                2R
                                                               x           R 2  x  R  dx
                                                                         2                     2

                                                                0


Taking x  R   R cos we get
                                                    0
                                     I YY  R        1  cos                     sin    sin  d
                                                4                                2

                                                    
                                                        
                                                            
                                           R 4  1  2 cos  cos2  sin 2  d               
                                                        0

                                                                               
                                                                                                         
Now  sin  d 
           2
                         ,    cos sin
                                          2
                                               d  0 and  sin 2  cos2  d                               . This gives
      0
                     2       0                                                   0
                                                                                                          8

                                                                                 5 4
                                                                        I YY       R
                                                                                  8


5.11 Product of area about the origin O is given as
                                                I xy   xi yi Ai   xy dA
                                                                    i

 If the centroid is at O’ which has the coordinates  x0 , y 0  and the coordinates of a point with

respect to O’ are x, y  then
                                              x  x0  x                             y  y0  y 




                                                                             74
                                 Y                          Y’




                                                  (x0,y0)                      X’




                                                                                     X
                             O


Therefore
I xy   xy dA   x0  x y0  y dA  x0 y0 A  x0  y  dA  y0  xdA   xy  dA

However, by definition of the centroid

                                          xdA  0          ydA  0
Thus
                                           I xy  x0 y0 A   xy  dA


5.12 Consider the moments and product of inertia of a square about a set of axes parallel to its
       sides and passing through its centre.



                                                   Y




                                                                           X



                                                       a


For this set of axes



                                                       75
                                                        a4 
                                         I xx  I yy   
                                                        12            I xy  0
                                                        
Now by the formula
                                               I xx  I yy
                                 I x' y'                    sin 2  I xy cos 2
                                                   2
Ix’y’ will always remain zero because of the equality of Ixx and Iyy irrespective of the angle of
rotation of the new set of axes x’y’. Thus any set of axes passing through the centre is the
principal set of axes.


5.13 The formulae for the moments of inertia in rotated frames are


                                     I xx  I yy       I xx  I yy
                          I x'x'                                   cos 2  I xy sin 2
                                           2                 2
                                     I xx  I yy       I xx  I yy
                          I y'y'                                   cos 2  I xy sin 2
                                           2                 2


Taking the second derivative of these expressions with respect to  , we get


                             d 2 I x'x'      I xx  I yy
                                         4             cos 2  4 I xy sin 2
                              d 2                2
                             d 2 I y'y'          I xx  I yy
                                           4                    cos 2  4 I xy sin 2
                               d    2
                                                       2


Thus the two derivatives have opposite signs. This implies if one of them is a maximum, the
other one will be a minimum.




                                                             76
                                              Chapter 6


6.1 (i) (a) and (d) are the virtual displacements because these are the only ones consistent with
    the constraint that the block can move only in the vertical direction.
   (ii) Suppose the strech is y0. In that situation, the forces on the block are as shown



                                        ky0




                                        mg



     Now a virtual displacement gives a displacement of y in the vertical direction. Taking it
     in the vertically up direction gives the virtual work to be
                                          W  ky0  mg y
     Equating this to zero gives
                                                        mg
                                                 y0 
                                                         k
6.2 Figure below shows the students and the plank on a wedge and a possible virtual
    displacement of the system.




                              x                                      (3-x)

                                                                        

                                         30g
         40g
                                                                                            50g


It is clear that as long as the point on the wedge does not move, the only possible displacement is
the rotation of the plank about this point and the system has only one degree of freedom. For the


                                                 77
virtual displacement shown, the displacement and the virtual work done by various forces is as
follows:
40kg : virtual displacement  x ; virtual work  40xg
Center of gravity of plank : virtual displacement  x  1.5 ; virtual work  30x  1.5g
50kg : virtual displacement  3  x  ; virtual work  503  x g


Since the net virtual work must vanish for equilibrium, we have

       503  x   40x  30x  1.5g  0      120x  195  0  x 
                                                                                     195
                                                                                          1.625
                                                                                     120


6.3 (i) Since the number of parameters required to describe the system is 1, the number of
    degrees of freedom is 1. We choose it to be the angle , the rod makes from the vertical. A
    virtual displacement will be to change  by .




                                                1.5
                                                 m



                                           2kg
                                                                     T

                                                              20kg


(ii) To apply the principle of virtual work, we need to calculate the virtual work done by various
force when  is changed to  +.      For this we first write the coordinates (x, y) of the tip of the
rod and yCG of the centre of gravity of the rod with respect to the pivot point. These are
                           x  1.5 sin    y  1.5 cos     y CG  0.75 cos

As  is changed to  +, these coordinates change and the changes are given by
                   x  1.5 cos     y  1.5 sin      yCG  0.75 sin  




                                                   78
Thus the total virtual work done by the external forces – 2g at the CG, 20g at the tip, both in the
positive y direction, and T at the tip in the positive x direction – is
                       W  1.5 cos  T  1.5 sin   20 g  0.75 sin   2 g 
Equating this to zero gives
                                                T  21g tan 


6.4 To find the forces applied by the bricks, we treat these forces as external. For only vertical
     motion, there are two degrees of freedom. One is the vertical displacement of the centre of
     gravity and the other the rotation of the plank about the CG. Equivalently, we can take the
     vertical displacements of the ends A and B as the virtual displacements. We choose the
     second option because this is related directly to the forces applied by the bricks. The plank
     in equilibrium and virtually displaced is shown below




                                                1.5m                                                  y2
            y1                  1m



                  NA                                   100N                                      NB
                                                         2m
                                                                              500N

Now the vertical virtual displacement of different points is
Point A :                                y1
Point B :                                y 2
                                         y1  y 2
Centre of gravity of the plank :
                                                2
                                                         y 2   y1 
Athlete :                                y1  1.5                     0.25 y1  0.75 y 2
                                                              2      
Thus the total virtual work done is



                                                        79
                                             y1  y 2 
                W  N A y1  N By 2  100              5000.25y1  0.75y 2
                                                 2      
                      N A  50  125y1  N B  50  375y 2
Equating the virtual work to zero and therefore the coefficients of each independent
displacement (y1 and y2) to zero gives
                                    NA=175N and NB = 425N


6.5 (i) Constraints on the system: length of the strings holding the masses is fixed
    (ii)    There is only one degree of freedom. This can be understood as follows. There are
            three variables needed to describe the system: The distance of two masses and one
            pulley from the ground. However there are two constraints: To of the strings have
            fixed lengths. Thus only one variable is left to change freely.
    (iii)   In terms of the lengths shown in the figure




                           h1
                                 M1

                                                                 h2
                                       y1                 M2

                                                     y2




    The constraint that the longer string has fixed length is expressed as
                            h1  y1   2h1  h2   3h1  2h2  y1  const.
    The constraint that the shorter string has fixed length is expressed as
                                            h2  y 2  const.




                                                80
    (iv)   The constraint forces are the tension in the two strings. It is by these tensions that
           the constraints are maintained.
    (v)    To apply the method of virtual work, we make a virtual displacement y1 of mass 1.
           The corresponding displacement of mass 2 is then y 2 . However, since there is only
           1 degree of freedom, we will finally express y 2 in terms of y1 to apply the method
           of virtual work. The virtual work done in these processes is
                                       W  M 1 gy1  M 2 gy 2
    Notice that the two virtual displacements are not independent. Therefore we should not
    conclude that M1=M2=0 for equilibrium. To apply the method, we first express the virtual
    work in terms of only y1 .
    From the first constraint equation, since h1 if constant,
                                                                    y1
                                   2h2  y1  0  h2  
                                                                     2
    From the second constraint we have
                                                                    y1
                                  h2  y 2  0  y 2  h2  
                                                                      2
    Substituting these in the expression for the virtual work gives
                                                     y1         M 
                           W  M 1 gy1  M 2 g         M 1  2  gy1
                                                      2           2 
    Equating this to zero then gives, for equilibrium
                                               M 2  2M 1


6.6 (i) Since the two blocks are free to move in one dimension without any constraint, the
    number of degrees of freedom for the system is 2.
    (ii)   The system in equilibrium is shown below. At equilibrium spring on the left is
           stretched by x1 and the total stretch of the two springs together is by x2. Thus spring
           on the right is stretched by (x2x1).      Thus force on mass m1 is k1 x1 to the left and

            k 2 ( x2  x1 ) to the right. These two forces must be equal for equilibrium but we wish
           to obtain this by applying the principle of virtual work. Similarly the two force




                                                81
            acting on m2 are F and k 2 ( x2  x1 ) to the left. Again these two forces must be equal
            for equilibrium and we will obtain this by applying the principle of virtual work.

                          k1                   m1                k2                   m2

                                                                                            F



                               x1                                     x2


    As the mass m2 is moved by a virtual displacement  x 2 , let us assume that mass m1 moves
    by  x1 . Then the virtual work done will be
                       W  k1 x1x1  k 2 ( x2  x1 )x1  Fx2  k 2 ( x2  x1 )x2
    Equating this to zero and therefore the coefficients of  x1 and  x 2 to zero gives
                                    k1 x1  k 2 ( x2  x1 ) and F  k 2 ( x2  x1 )

                      F             1 1            
    This gives x1       and x2  F  
                                    k              .
                                                    
                      k1             1 k2          


6.7 (i) There is only one degree of freedom. Although the piston moves in the vertical direction
    and the wedge in the horizontal direction, their movements are connected because the piston
    moves on the surface of the wedge.
    (ii) Normal reactions on all the surfaces are the constraint forces. In the present context, the
    constraint force specific to the piston’s movement on the surface of the wedge is the normal
    reaction of the wedge surface on the piston.
    (iii)   if the distance of the middle line of the piston is at a distance a from the origin, its
            height is y and the distance of the left edge of the wedge is x (see figure) then the
            constraint is expressed as




                                                       82
                                            m



                                                 y                         F
                            x            
                                  a




                                y  a  x  tan         y   x tan 
Now the method of virtual work is applied by considering x as the free variable, varying it by x,
and equating the total virtual work to zero. The virtual work is
                                        W   F x  mg  y
                                             F  mg tan   x

Equating the coefficient of x to zero gives F  mg tan 


6.8 When the equilibrium angle is , the distances of various points (see figure) , taking A as the
    origin are as follows
                                                                                 
                         x B  l sin             x D  l sin        yC  2l cos
                                        2                       2                  2
     The force on the two points B and D due to the spring is
                                                               
                                                k  l 2  2l sin 
                                                               2
     to the left on B and to the right on D




                                                       83
                                                          A



                                                                                                   x

                                 B                                     D


                                                                                 y


                                                              C

                                                                   W


       It is clear that we need only one parameter  to specify the system. Thus there is only one
       degree of freedom. Now let us make a virtual displacement by changing  by  . In that
       situation
                             l                         l                                 
                    x B    cos                x D   cos            yC  l sin
                             2            2             2       2                          2


       Thus the total virtual work done as  is changed by  is
                                l                    l            
            W  k  l 2  2l sin  cos  k  l 2  2l sin  cos  mgl sin 
                                2 2  2                 2 2   2         
                                                                           2                   
       Vanishing of the virtual work then implies
                               l                          l                  
               k  l 2  2l sin 2  2 cos 2  k  l 2  2l sin 2  2 cos 2  mgl sin 2   0
                                                                                   
Or equivalently
                                                            mg      
                                              2  2 sin         tan
                                                          2  kl      2
     mg                                                       mg                      
If       0 then the solution is   . For very small value of    therefore we have    x ,
     kl                             2                          kl                      2
where x is very small.     Substituting in the equation above, we get



                                                              84
                                                x   mg    x 
                               2  2 sin  45         tan 45  
                                                2   kl          2
        x
Since     is small, we get by Taylor series expansion
        2
                                    x                      x  1     x
                           sin  45    sin 45   cos 45      1  
                                    2                      2   2  2
                                     x
                             tan 45    tan 45   sec2 45   1  x 
                                                               x
                                     2                       2
Thus the equation for equilibrium is
                          1  x   mg                            1 mg  mg
                   2 2      1     (1  x)                x     
                           2  2   kl                           2 kl  kl
Or to a good approximation
                                                    mg
                                              x       2
                                                    kl
Thus
                                                       mg
                                                        2
                                                2       kl


6.9 (i) Even if we consider only one dimensional motion, we would require two variables, one
    the angle that the bar makes with the vertical and the other describing the position of the
    mass. However, the two are connected by a rope. Hence the two variables cannot vary
    independently since the length of the rope is foxed (constraint). Therefore there is only one
    degree of freedom in the system.
    (ii)     The constraint is the length of the rope remaining constant. It is enforced by the
             tension that develops in the rope. This tension makes the movement of the bar and
             the mass restricted. Thus it is the tension in the rope that is the force of constraint.
    (iii)    A virtual displacement would be to displace the bar by an angle  from its
             equilibrium position. Since the length of the rope is fixed, the midpoint of the bar
             moves by the same distance as the mass connected to the rope. Therefore the tension
             does positive work at one end of the rope and exactly equal but negative work at the
             other end. The sum of the work done by the tension then vanishes.



                                                    85
    (iv)    Let us say we make a virtual displacement of the bar by turning it counterclockwise
            by an angle from the vertical. Then the end of the rod moves by l in the
            direction of the force. At the same time, the virtual displacement of the mass is
                                                                                              l
            equal to the movement of the midpoint of the bar. Thus the mass moves by            
                                                                                              2
            opposite to the gravitational force (see figure)

                           F
                                  




                                                         m




                 The net virtual wok therefore is
                                                              l
                                          W  Fl  mg 
                                                              2
                 Equating this to zero then gives
                                                         mg
                                                    F
                                                          2


6.10 Initially the bars are at 90. The weight has a tendency to fall down so the torque applied is
    such that it tends to pull the weight up. Thus is a virtual displacement of angle  is made
    in the direction of the torque, the weight will be lifted up by the corresponding distance y.
    The corresponding virtual work done by the torque  is  and the corresponding work by
    the gravity is Wy. By the method of virtual work then we have
                                             Wy  0
    This gives
            y
      W
            


                                                    86
Next we calculate the relationship betweeny and . As the rod is rotated by angle , the
                                                         p
length of the horizontal diagonal decreases by              . If the corresponding angle at the
                                                        2
vertical corner changes from 90 to 90+α, then we have (see figure)



                                                      (90+α)

                            




                                                                 W



Change in the length of the horizontal diagonal
                       90     l                               l   l
             = 2 l sin
                                     2 l sin 45  l
                                                            cos 45        
                           2       2                    2            2   2
                     p
This should equal       so we have
                    2
                                            l           p
                                                       
                                            2          2
As the angle changes the length of the vertical diagonal changes by
                     90     l                               l    l
               l cos
             2                     2 l cos 45  l
                                                          sin 45        
                         2       2                    2            2    2
Thus we have
                                 l           p                   y   p
                          y                                 
                                 2          2                    2
This gives
                                                      pW
                                                 
                                                      2




                                                87
6.11 (a)    For the motion in a plane, the orientation of the rod can be described by the
     displacement of its centre of mass and the angle it has rotated by about its CM. Or
     equivalently the displacement of its two ends is sufficient to describe its orientation. Thus
     the degrees of freedom is 2.
     (b) We take the vertically down displacement of the two ends as the virtual displacement as
     shown in the figure.




                                                 yCM
                            y1

                                                                     y2




                                                      2L
        Let the centre of mass be at a distance          from the left hand end of the rod. In that case,
                                                       3

the centre of mass moves down by yCM  y1 
                                                             2
                                                               y 2  y1   1 y1  2 y 2 . The virtual work
                                                             3                 3       3
done by the springs in such a virtual displacement is  ky1y1 and  ky2y 2 respectively while that
                                         W       2W
by the weight of the rod is WyCM         y1     y 2 . Thus equating the net virtual work to zero
                                         3        3
gives
                                  W       2W
                                    y1     y 2  ky1y1  ky2y 2  0
                                  3        3
Now equating the coefficient of each independent displacement to zero gives
                                                 W                  2W
                                          y1                y2 
                                                 3k                 3k




                                                        88
                                                       Chapter 7


   7.1     For Cartesian coordinates (x,y), the planar polar coordinates are r  x 2  y 2 and

                        y
             tan 1   . If y<0 and x<0, 270>>180 and if y<0 and x>0, >270.
                       x
      Unit vectors are given by
             ˆ j
            xi  yˆ           ˆ ˆ
                         ˆ  yi  xj
      r
      ˆ             and  
               r               r


   7.2
                                                    
                                                        ˆ ˆ
                                                                 ˆ ˆ
                                                                                     
           At each point the velocity v is given as v .r  r  v .  . Thus for (iv) v  2i  3 ˆ for
                                                                                             ˆ     j
           a particle at (-2,-3), the velocity is

       2iˆ  3 ˆj   2i  3 j  rˆ   2iˆ  3 ˆj  3i  2 j ˆ   5rˆ  12
                          ˆ     ˆ                         ˆ        ˆ                    ˆ
                           13                                 13                  13
   7.3     The two points with polar coordinates                        r1 , 1  and r2 ,2       and the corresponding
           vectors are shown in the figure below

                                                                                   
                                                                                   r2


                                                                                             
                                                                                            r1  r2
                                          ˆ1
                                ˆ2                                                           
                                                                       (21)                r1
                                                          2
                                                               1




                                                                     
(i) As is clear from the figure, the angle between the vectors r1 and r2 is 2  1  . This also the

angle between unit vectors r1 and r2 and unit vectors ˆ1 and ˆ2 . Therefore
                           ˆ      ˆ

                                            r1  r2  1   2  cos 2  1 
                                            ˆ ˆ        ˆ ˆ




                                                              89
                                                                                
Similarly, angle between r1 and ˆ2 is   2  1  and that between r2 and ˆ1 is   2  1  .
                                      2                                            2
Therefore
                                                        
                               r1   2  cos   2  1    sin  2  1 
                               ˆ ˆ
                                             2           
                                                          
                               r2  1  cos  ( 2  1 )   sin  2  1 
                               ˆ ˆ
                                             2             
                                            
   (ii)      Similarly, from the figure r1  r2  sin 2  1 z
                                                                ˆ
               
   (iii)      r1  r2  r12  r22  2r1r2 cos2  1 


   7.4      The trajectory of the projectile is shown schematically in the figure below. The
            horizontal distance from the origin to the point of highest elevation (height 1.25m)
            is 2.5 3m . The vertical component of the velocity at this point vanishes while the

            horizontal component is 5 3ms -1 .               Similarly, the vertical component of the
            acceleration is 10ms-2 vertically down. Also shown in the figure are unit vectors in the
            radial and the tangential directions at the highest point and on the ground.


                                                                                  ˆ
                                                                                             ˆ
                                                                                             r

                                                                                  5 3ms -1       ˆ

                                                                  1.25m
                                            
                                                                                                      ˆ
                                                                                                      r
                                        2.5 3m


     From the figure it is clear that for the point of highest elevation

                                1.25            1                 1                     3
                      tan                        ;   sin          ;   cos  2
                                2.5 3       2 3                   13                   13


     Therefore the radial and tangential components of the velocity at the highest point are




                                                        90
                                       3   30                                                3 1
          vr  5 3 cos  5 3  2             ms1                v  5 3 sin   5        ms
                                      13    13                                              13
       Similarly the radial and tangential components of the acceleration are

                                    10                                                  3 2
               ar  10 sin            ms2               a  10 cos  20          ms
                                     13                                                13
       On the ground, the radial unit vector points towards positive x direction and the
       tangential unit vector is in the negative y direction.               Thus the radial component of the
       velocity is its x component and the tangential component is its y component. Therefore

                                     vr  5 3ms1                  v  5ms1

       Similarly, since the gravitational acceleration on the ground is in the negative y direction,
       its radial and tangential components are
                                          ar  0              a  10ms2


7.5   The position of the particle at time t is shown in the figure below.




                                               2ms-1



                                                                            1

                                               

                                                       2t


      The polar coordinates of the particle at time t are given as
                                                                       1
                                r  4t 2  1                   tan 1    
                                                                        2t 
      Therefore
                                          4t                          2
                                r
                                                            
                                         4t  1
                                           2                         4t  1
                                                                       2




                                                        91
This gives the velocity in polar coordinates as
                                         ˆ                   4t               2             ˆ
                               v  rr  r 
                                   ˆ                                   r
                                                                        ˆ                    
                                                           4t  1
                                                               2
                                                                              4t 2  1
Differentiating the velocity vector gives


               
              dv       4                 16t 2                     4t                  8t         ˆ     2       ˆ
                             r
                              ˆ                       r
                                                      ˆ                  r
                                                                         ˆ                                     
              dt     4t  1
                         2
                                   4t   2
                                              1
                                                 32
                                                               4t  1
                                                                    2
                                                                              4t   2
                                                                                         1
                                                                                              32
                                                                                                        4t  1
                                                                                                          2




                  ˆ            2 ˆ         
                                             ˆ    ˆ    2
Now substituting r    
                 ˆ                     and    r  2     ˆ
                                                             r , we get the acceleration above to
                               4t  1
                                   2
                                                      4t  1
be zero.


7.6                                                                      
      (i) Kepler’s second law states that the rate of the area swept (  A) by the radius vector is
      constant. This can be expressed as (see figure for the symbols used)

                                                1 
                                               A  r 2  constant
                                                  2




                                                      
                                                      r

                                                           

                                             Sun




Now differentiating the equation above with respect to time gives
1 2 
         
  r   rr  0                       
                              r 2  2rr  0
2
The expression on the right is the tangential acceleration. Thus Kepler’s second law gives
tangential acceleration to be zero.




                                                           92
      (ii)    Since the tangential acceleration is zero, the force is only in the radial direction.
              Acceleration in the radial direction is
                                                          
                                                          r      
                                                    a r    r 2   
                                                       r0
      We differentiate the orbit equation r                  with respect to time to get
                                                  1  e cos 

                                        r0 e sin                    
                                                          r 2e sin  2 Ae sin 
                                  r
                                                                   
                                       1  e cos 2           r0        r0
            
      Since A is constant, differentiating the equation above once more with respect to time, we get

                                                          r    
                                                       2 A 0  1
                                       
                                     2 Ae cos          r            2
                                                                   4 A  r0  1
                                 
                                r                                              
                                          r0                 r0      r0 r 2  r  
      This gives
                                                    
                                                 4 A 2  r0   
                                           ar                      
                                                          1   r 2
                                                  r r
                                                    2
                                                              
                                                4A  2  r0          
                                                                   4 A2
                                                        1  r 4
                                                r0 r 2  r         r
                                                       
                                                   4A  1
                                                         2
                                                r  r2 
                                                   0 
      This shows that the force is proportional to r2.


7.7     It is given that
                                                  m   F sin  t
                                                   x
                                                  m  F cos  t
                                                   y
      which gives
                                                    F  m sect
                                                         y
      Now write y  x tan t and differentiate it twice with respect to t to get

                                  tan t  2 x sec2 t  2 2 x sec2 t tan t
                               y x                

      This gives
                                                                                      
                       F  m  m  tan t  2 x sec2 t  2 2 x sec2 t tan t sect
                            y      x             

      Substitute this in m   F sin  t to get
                          x



                                                        93
                                                                                        
                             tan t  2 x sec2 t  2 2 x sec2 t tan t sin t sect
                         x      x             
Multiplying and rearranging terms gives
                                            2 x tan t  2 2 x tan 2 t
                                          x        
The equation does not appear to be easily ingrable.


                                 
7.8 It is given that r  2t and   1rad s 1
                     
(i)              Since r (0)  1 , integrating the equation for r gives
                                                                
        r (t )       t

          dr   2t ' dt'  r (t )  1  t or equivalently r (t )  t  1
                                           2                          2

          1         0

               ˆ                  ˆ
(ii) v  rr  r  2tr  (t 2  1)
         ˆ           ˆ

If the velocity vector is at 45 to the radius vector, we have
                         
                            ˆ
                                
                         v .r  v cos 45            2trˆ  (t   2        ˆ ˆ
                                                                        1) .r 
                                                                                    1
                                                                                    2
                                                                                       4t   2
                                                                                                  (t 2  1) 2   
This gives

                                          4t 2  t 4  1  2t 2   t 4  6t 2  1
                                     2t                        
                                                    2                   2
Squaring both sides gives
                                      8t 2  t 4  6t 2  1 OR t 4  2t 2  1  0
This is equivalent to

                                             t   2
                                                         2
                                                       1  0  t  1s
(iii)            Radial acceleration zero implies
                                                            r     
                                                      a r    r 2  0  
 r  2t    2 which leads to
         r
                                                             
                                            2  t 2  1  0 giving t  1s
This also gives the distance from the origin to be r (t  1)  1  1  2m


7.9 Free body diagram of the bead when its radius vector is making an angle  (increasing as
      the particle slides down) from the vertical is shown below




                                                                  94
                                                                           N

                                                              R
                                                                   
                                                      

                                                                  mg




Taking the components of the forces in the radial and tangential directions and equating these
to mass times the radial and tangential components of the acceleration, respectively, gives
In the radial direction                                  
                                  N  mg cos  m   r 2
                                                  r                    
In the tangential direction                         
                                  mg sin   m r  2r           
Since the particle moves on a path of constant radius r  R , we have r    0
                                                                       r
When substituted in the equations above, this gives
                                                      
                                   N  mg cos  mR  2
                                                    
                                         g sin   R
                  2
           1 d , we get from the second equation above
Now using  
               2 d
                                                
                                            R d 2
                                     
                                    R            g sin 
                                            2 d
which gives upon integration
                              
                      g  sin  d  g 1  cos   R 2  2 g 1  cos 
                  R 2                                   
                  2       0

                                                    
When substitutes in the equation N  mg cos  mR  2 , this gives
                                    N  m3g cos  2 g 


7.10       The free body diagram of the bead at equilibrium is shown in the figure below.




                                             95
                                                        N
                                                                


                                                                mg


                                               




The horizontal components of the normal reaction N provides the centripetal force while the
vertical component balances the weight of the bead. Thus
                                             N sin   mx 2
                                             N cos  mg
Dividing the first equation by the second one gives
                                                       x 2
                                               tan  
                                                        g
The slope of the curve is also equal to tan . Thus
                                             x 2 dy
                                                      4cx 3
                                              g    dx
This gives

                                   2                                         4
                             x                    and y  cx       4

                                   4cg       2 cg                             16cg 2


                         1 2
7.11         Since       t , we have
                         2
             r  2  t 2 , r  2t
                                     and        2 ;   t
                                               r                    and   1
                                                                         

                                                                         
             Thus a r    r 2   2  t 4 and a  r  2r  t 2  4t 2  5t 2
                         r                                   


7.12         We know that
                                              ˆ                        ˆ
                               r  sin  cos i  sin  sin  ˆ  cos k
                               ˆ                              j



                                                   96
                              ˆ            ˆ                        ˆ
                               cos cos i  cos sin  ˆ  sin  k
                                                          j
                                          ˆ          ˆ
                                            sin  i  cos  ˆ
                                                               j
Therefore
         
         ˆ                                                                              ˆ
                          ˆ               ˆ 
           sin  cos i   cos sin  i   sin  sin  ˆ   cos cos ˆ   cos k
                                                            j              j 
                                                    ˆ               ˆ
              sin  cos i  sin  sin  ˆ  cos k   cos  sin  i  cos ˆ
                            ˆ
                                           j                                    j   
                ˆ 
              r   cos ˆ

Similarly

                                         cos  i  sin  ˆ 
                                       
                                       ˆ           ˆ         j
                                               
                                            sin  r  cos 
                                                     ˆ         ˆ   

7.13        Since m3 3  T2  m3 g
                     y

                                 4m1m2   m1  m2  m3
                                         
                                         
                                                   
                                                   
We get from the solution 3 
                         y                                   g
                                 4m1m2   m1  m2  m3
                                         
                                         
                                                   
                                                   




                                         4m1 m2  m1  m2 m3
                             T2  m3                           g  m3 g
                                         4m1 m2  m1  m2 m3


                                          8m1 m2 m3 g
                                 
                                     4m1 m2  m1  m2 m3

From T2  2T1  0 , we get
                                                 4m1 m 2 m3 g
                                     T1 
                                            4m1 m 2  m1  m 2 m3

Now from m1 1  T1  m1 g we get
            y
                                         3m 2 m3  4m1 m 2  m1 m3
                                 1 
                                 y                                 g
                                          4m1 m 2  m1  m 2 m3

And from m2 2  T1  m2 g we get
            y
                                         3m1 m3  4m1 m 2  m 2 m3
                                 2 
                                 y                                 g
                                          4m1 m 2  m1  m 2 m3




                                                    97
7.14


                                                                
                                         l             m
                                                                      T2       T1


(i) The tension in the outermost string provides the centripetal force for the outermost bead.
Let the tension in this string be T1. Then
                                             T1  Nml  2
Similarly, centripetal force for the second bead from outer end is provided by the difference
in the tension T2 in the second string and tension T1 in the first string. Thus
                         T2  T1  ( N  1)ml  2       T2  2 N  1 ml  2
Extending further
                         T3  T2  ( N  2)ml  2       T3  3N  3 ml  2

                        T4  T3  ( N  3)ml  2        T4  4 N  6 ml  2

and so on. In general we can write for the ith string from the outside,
                     Ti  Ti 1  N  (i  1)ml 2
                         Ti 2  N  (i  2)ml 2  N  (i  1)ml 2  
This is then easily seen to be
                    Ti  iN  (i  1)  (i  2)  (i  3)    2  1  0ml 2
                             i (i  1) 
                       iN             ml
                                              2

                                  2 
                        iml 2
                               2 N  i  1
                           2


(ii)    Now we generalize the result of part (i) to a rope of length L and mass per unit length
        λ. To make the transformation from the problem in part (i) to this problem, we
        consider the mass of each bead to be distributed over the connecting string whose
        length we take to be vanishingly small i.e. l  0 . Thus we have
                           L  Nl , x  ( N  i)l , L  x   il and m  l
Substituting this in the expression for Ti above, we get


                                                  98
                                         ( L  x) 2
                           T ( x)  Ti               (2 Nl  il  l )
                                               2
                                             2
                                                  ( L  x)(2 L  L  x  l )
                                             2
                                             2
                                                  ( L  x)(L  x  l )
                                             2
On taking limit l  0 , we then get
                                                    2
                                       T ( x)            ( L2  x 2 )
                                                    2
To get this answer by considering the rope directly, we take a small portion of the rope of
length x at a distance x from the centre O. The centripetal force to it is provided by the
difference in the tension at its two ends, as shown in the diagram below

                        T(x)                                             T(x+x)


Then
                                                             T ( x)  T ( x  x)
                T ( x)  T ( x  x)  x 2 x                                    2 x
                                                                      x
This gives the differential equation
                                                dT
                                                    2 x
                                                dx
The solution of this equation is
                                                           2 x 2
                                         T ( x)  C 
                                                              2
where C is the integration constant. With the condition that the tension vanishes at x=L, we
then get
                                2 L2                             2
                          C
                                   2
                                            and T ( x) 
                                                                   2
                                                                         L
                                                                          2
                                                                               x2   

7.15 Consider a small section of the rope making a small angle  at the centre of the loop.
    The force at its two ends due to the tension in the rope is shown in the diagram below




                                                  99
                                                                  /2

                                                  
                        T                                                 T
                                                          Tsin(/2)



For small angle the forces at the ends give a net force towards the centre which is equal to
                                                
                                    2T sin    2T  T
                                           2     2
This provides the required centripetal force for the segment. Therefore
                                T  R  R 2    T  R 2  2


7.16 The relevant coordinates and the free body diagrams of the two masses are shown below




                                                             r
                            T
                                                                         m


                                            y                         T




                                                                              mg
                                      M
                            Mg


We treat the mass m using the polar coordinates as shown in the figure. The total number of
unknowns in the problem are : y coordinate of mass M, polar coordinates of mass m and
tension T in the string. The corresponding equations are




                                            100
                                    M   Mg  T
                                       y
                                            
                                    m   r 2  mg cos  T
                                      r
                                    mr  2r   mg sin 
                                            
And the constraint equation
                       r  y  constant  r  y  0 and     0
                                                      r y


                                                                                           1 2
7.17 (i) After time t, the end of the rod that was at the origin has moves by a distance     At .
                                                                                           2
     Thus the relationship between the x and y coordinates will be
                             y                                        1 2
                                  tan               y cot  x      At
                             1 2                                      2
                          x  At
                             2


(ii) Free body diagram of the bead is shown below




                                                        θ
                                        θ                      N


   Thus the equations of motion are
                                         m   N sin 
                                           x
                                         m    N cos 
                                           y
     From the constraint equation
                                            A   cot  
                                                y         x
     Thus
              N sin   m A   cot    N  mA cos ec  m cot  cos ec
                               y                              y
     which gives
                                 N cos  mA cot   m cot 2 
                                                      y
     Thus
                   m  mA cot   m cot 2 
                    y                 y               m cos ec 2  mA cot 
                                                        y


                                               101
    Or
                                  A
           A cos sin   
        y                           sin 2
                                  2
(iii)      Integrating the equation above with the condition y(0)  0 and y(0)  d gives
                                                             

                                                           A
                                             y(t )  d      sin 2 t 2
                                                           4
                                                      4d
           Thus the bead will take time t                  to reach the lower end.
                                                   A sin 2


7.18       Suppose the lower corner with angle  is at the origin at t=0. Then if the position of
    the mass is (x, y) at time t, the relationship between these coordinates is
                                 y                                           1 2
                                        tan                y cot  x      At
                                 1                                           2
                              x  At 2
                                 2
if the acceleration of the wedge is A. This implies
                                                A   cot  
                                                    y         x
    (i)       Now if the mass falls vertically down,    g which gives A  g cot i.e. the
                                                     y
              wedge accelerates to the right with acceleration g cot .             The interpretation is
                                                                                       1 2
              quite simple: when the wedge moves by horizontal distance                  At , the mass
                                                                                       2
                                                           1 2
              should move vertically down by                 gt and the relationship between the two at
                                                           2
                                      1 gt 2
              minimum acceleration is 2 2  tan   A  g cot .
                                      1 At
                                       2
    (ii)      For the particle not to move with respect to the wedge, we have
                                                       0
                                                     y
           which implies
                                                      A .
                                                    x
           The free body diagram of the mass is as follows




                                                   102
                                    Y
                                                      N




                                         θ
                                                                       X



   Thus we have by the conditions above
                                                 N cos  mg
                                                  N sin   mA
   which gives
                                                  A   g tan 


7.19      From example 7.9
                                          m2  N1 sin 
                                           x
                                          M 1   N1 sin 
                                             x
                                          2  ( 1  2 ) tan 
                                          y       x x
                                                       mg cos
                                             N1 
                                                        m         
                                                    1    sin 2  
                                                     M            
         This gives (assuming x1 (0)  x1 (0)  0 )
                              
                                                m
                                                    g cos sin 
                                        1  
                                        x       M
                                                     m        
                                                1  sin  
                                                            2

                                                 M            
        Similarly
                                                   g cos sin 
                                         2 
                                         x
                                                     m          
                                                 1     sin 2  
                                                  M             
       This gives




                                                    103
                                                   m
                                               1   g sin 
                                                           2


                                       2   
                                                    M
                                       y
                                                   m        
                                                1  sin  
                                                         2

                                                 M          
Thus if mass m starts from height h, it will take time

                                       2h         2h (m  M cosec 2 )
                                  t       
                                       2
                                       y               (m  M ) g

And at this time the speed of the wedge will be

                                                         2 gh
                             1t  m cot
                             x
                                               (m  M )(m  M cosec 2 )


7.20 The coordinates used in solving the problem are shown in the figure below


                                  P1
                                                                 P2

                                  T
                                                  T
                                                        T
                      h1               m                         T         h2
                                             P3
                             y1
                                                                 m
                                                            y2
                                                                      y3




Let the tension in the string be T. The equations of motion for the two masses are
                                             m1  T  mg
                                              y
                                             m2  T  mg
                                              y
And since the pulley P3 is massless, we have
                                           2T  k  y 3  h   0




                                                      104
Thus we have four unknowns y1, y2, y3 and T but only three equations. One more equation is
provided by the constraint equation. The constraint is that the length of the string is a constant.
If the heights of the two fixed pulley are h1 (for P1) and h2 (for P2), the constraint is expressed as
                          h1  y1   h1  y3   h2  y3   h2  y 2   const.
or equivalently as
                             y1  2 y3  y 2  const.              1  2  2 3
                                                                    y y            y

Now adding the equations of motion for the two masses gives
                                         m 1  2   2T  2mg
                                            y y

On substituting 1  2  2 3 and 2T  k  y3  h  from the equations above, we get
                y y            y

                                         2m3  ky3  kh  2mg
                                            y

This gives
                                                  k           kh 
                                         3 
                                         y          y3   g     
                                                 2m           2m 
The general solution of the equation above is the sum of its homogeneous solution yh(t) and the
particular solution yp(t). We have
                                        y h (t )  A cos t  B sin t
                                                         2mg
                                        y p (t )  h 
                                                          k

                                                                                             k
Here A and B are two constants to be determined by the initial conditions and                . Thus
                                                                                            2m
the general solution is
                                                                      2mg 
                                y 3 (t )  A cos t  B sin t   h      
                                                                       k 
Now the initial condition is
                                     y3 (t  0)  h            y3 (t  0)  0
                                                               

These give
                                                   2mg
                                           A                   B0
                                                    k
Thus




                                                         105
                                                    2mg          k 
                                   y3 (t )  h         1  cos    t
                                                     k         2m 
This immediately gives through the equation 2T  k  y 3  h   0

                                                        k 
                                        T  mg 1  cos
                                                          t
                                                       2m 
This gives

                                                             k
                                            1   g cos
                                            y                  t
                                                            2m
This is easily integrated. Since it is gives that y1 (t  0)  0 and y1 (t  0)  h , we get
                                                  

                                                    2mg          k 
                                   y1 (t )  h         1  cos    t
                                                     k         2m 
In exactly tha same manner we get

                                                    2mg          k 
                                   y 2 (t )  h        1  cos    t
                                                     k         2m 


7.21 (i) Wedge m3 is free to move only in the horizontal direction; m1 and m2 move both
        horizontally as well as vertically. Thus we would have had 5 degree of freedom.
        However, there are three constraints: The length of the string is fixed, mass m1 moves
        only in the vertical shaft and mass m2 moves on the plane of the wedge.                These
        constraints reduce the degrees of freedom to 2. Thus there are only two degrees of
        freedom.
(ii) The origin and the coordinate axes chosen to describe the motion are given in the figure
below. Also shown are the free-body diagrams of the three masses




                                                      106
                                          m1
                                                                        m2
                                          y1             m3
                                                                  y2
                                     x1                                 
                                                x2

                                                             x3



                                                         T
                    T                                                                    T
                                            T                               N2                 N2

                             N1           N1             m3

                    m1 g
                                                                                                  m2g
                                                      m2g          N3



We see that there are in total 8 unknowns: x1, y1, x2, y2, x3, N1, N2, N3 and T. The equations of
motion are:
                               m1  N 1
                                x              (i)
                               m1  T  m1 g
                                y                     (ii )
                               m3 3   N 1  N 2 sin   T cos
                                  x                                         (iii )
                               m2 2  N 2 sin   T sin 
                                  x                             (iv )
                               m 2  2  N 2 cos  T sin   m2 g
                                   y                                         ( v)
                               N 3  T 1  sin    N 2 cos  m3 g            ( vi)
These are the equations of motion. In addition there are three constraint equations.
          y 2  x3  x 2  tan          2  3  2  tan 
                                           y      x x                 ( vii )
         h  y1   x2  x1 sec  const.   1  2  1 sec  0
                                                               y    x x                      ( viii )
         x3  x1   const.  1  3x x            (ix )




                                                     107
In the above, h is the height of the wedge. There are a total of nine variables and nine equations.
Of these equation (vi) is not relevant for the motion since the wedge moves only in the horizontal
direction.
Equations (i), (iii), (iv) and (ix) give
                                        m3 1  m1 1  N 2 sin   T cos
                                           x         x
                                               m1 1  m2 2
                                                    x         x
which gives
                                                m1  m3 1  m2 2
                                                          x        x     0

This is the equation expressing momentum conservation in the horizontal direction.
Similarly equation (iv) and (v) along with (ii) give
                            m2 2 cos  m2 2 sin   T  m2 g sin 
                               x             y
                                                       m1 1  m1 g  m2 g sin 
                                                            y

Now substituting for 1 from (viii) and 2 from (vii) and using (ix), we get
                     y                   y
              m2 2 cos  m2 1  2  tan  sin   m1 2  1 sec  m1 g  m2 g sin 
                 x              x x                            x x
Which is equivalent to
               m2 2 cos2   m2 1  2 sin 2   m1 2  1   m1  m2 sin  g cos
                  x                x x                       x x
               
               m1  m2 2  m1  m2 sin 2  1  m1  m2 sin  g cos
                         x                       x

Now using m1  m3 1  m2 2  0 , we eliminate 2 from the equation above to get
                    x        x                      x

                    m1  m2 m1  m3  
                
                            m2
                                                                   
                                               x1  m1  m2 sin 2  1  m1  m2 sin  g cos
                                                                    x

This gives
                                          m2 m1  m 2 sin  g cos 
                             1 
                             x                                                   3
                                                                                  x
                                     m  2m1 m 2  m1 m3  m2 m 3  m 2 sin 2 
                                      2
                                      1
                                                                      2



And using m1  m3 1  m2 2  0 , we then get
                    x        x

                              2  
                                               m1  m3 m1  m2 sin  g cos
                              x
                                          m  2m1 m2  m1 m3  m2 m 3  m2 sin 2 
                                           2
                                           1
                                                                         2



Now using equation (viii) we get

                              1  
                                               m1  m2  m3 m1  m2 sin  g
                              y
                                          m  2m1 m2  m1 m3  m2 m 3  m2 sin 2 
                                           2
                                           1
                                                                         2




                                                           108
Using equation (vii), we get

                              2 
                                       m1  m2  m3 m1  m2 sin  g sin 
                              y
                                      m12  2m1 m2  m1 m3  m2 m 3  m2 sin 2 
                                                                       2



As m3   , we get

 1  3  0 2  
                         m1  m2 sin  g cos      1  
                                                               m1  m2 sin  g   2 
                                                                                           m1  m2 sin  g sin
 x x           x                                     y                             y
                               m1  m2                          m1  m2                     m1  m2 

7.22 Consider two parts of the rope x and (L-x) in length. The free body diagrams (showing
        only the horizontal forces) of the rope and these two parts are shown below


                                         x
                F
                                                                    Friction




                 F                                                        T(x)

                                       Friction


                                T(x)
                                                                     Friction



   Since the rope is moving with constant speed, there is no net force on the rope or any part of
   it. Thus from the free body diagram of the rope
                                                     F  Mg
   From the free body diagram of the left portion of the rope

                             F  T ( x)  
                                              M
                                                xg  0  T ( x)  
                                                                    L  x  Mg
                                              L                        L
   Or from the free body diagram of the right portion of the rope

                                               T ( x)  
                                                            L  x  Mg
                                                               L




                                                      109
   It is also instructive to solve the problem by considering a small portion of length dx at
   distance x from the left and balancing the forces there to get a differential equation for T(x).
   The free body diagram of such a portion is as given below

                                                       dx

                                   T(x)                                        T(x)+dT(x)

                                                     Friction


                    T ( x)  T ( x)  dT ( x)  
                                                       dx                     dT ( x)   M
                                                          Mg                          g
                                                       L                       dx       L
With the condition that T(L) = 0, the equation above can be integrated to get
                          T ( x)
                                          M
                                                x
                                                                              L  x Mg
                             dT  
                            0
                                          L 
                                            g dx 
                                             L
                                                                  T ( x) 
                                                                                 L


7.23 The force applied should be such that the frictional force on mass m is sufficient to balance
        its weight. Free body diagrams of the two blocks are shown below



                                          friction
                     F
                                                      N                                       M
                                          mg                           N
                                                                           friction
                                                                                      Mg    Normal reaction




If the entire system is moving with acceleration a then
                                                      N  Ma
                                                      F  N  ma
This gives
                                             F                          M
                                      a                  and     N        F
                                            M m                       M m
If mass m is not falling then


                                                            110
                     friction  N  
                                          M
                                              F  mg          F
                                                                       M  mmg
                                         M m                                M
Thus

                                         Fm in 
                                                   M  mmg
                                                        M
For m=16kg, M=88kg, =0.38 and g=9.8ms-2, we get
                                           104  16  9.8
                                  Fmin                    487.7 N
                                             0.38  88


7.24 The forces on the bead are: Its weight, normal reaction NV in the vertical direction and
        normal reaction NH in the horizontal direction. Thus the free body diagram of the bead is
        as follows (assuming the rotating arm is going into the page)




                                                NV
                              friction                                             ˆ
                                                        NH                         r

                                                   mg



These force provide the radial and the tangential acceleration given by
                                   a    r 2 r  r  2r 
                                   
                                        r       ˆ             ˆ

Since the rod is rotating with a constant angular speed , we have
                                       a    r 2 r  2r
                                       
                                            r          ˆ     ˆ

(i) If the bead is stationary at r=R, we have
                                             
                                            a   R 2 r
                                                       ˆ

This gives
                              N V  mg      NH  0       friction  mR  2

Since friction    N , we get for stationary bead
                                                                  g
                                  mR 2  mg   0 
                                                  2

                                                                   R


                                                   111
(ii)      If    0 and negligible weight of the bead, we have

            NV  mg            N H  2mr
                                                 N  m g 2  4r 2 2  2r
                                                                                        friction  N  2mr
                                                                                                               

The minus sign for the friction shows that since the bead slides outwards, the frictional force in
inwards. The equation describing the motion of the bead is then
                                       r 2  2r OR
                                     r                                       2 r   2 r  0
                                                                            r        

Assuming a solution of the form r (t )  e t and substituting it in the equation above we get

                                                        2  2   2  0
Solution of this equation gives

                                              
                                  1   1   2                                   
                                                                  and 2   1   2                 
Thus the general solution is


                                                                                
                                r (t )  A exp t 1   2    B exp  t 1   2                        
Here A and B are to be determined by the initial conditions that r (t  0)  R and r (t  0)  0 .
                                                                                   
Thus

                                  A B  R               1     2
                                                                             
                                                                        A  1  2   B  0     
The solution is

                                     R  1                                    R  1    
                                             2                                             2
                                  A              and                         B             
                                     2   1  2                                  2   1  2  
                                                                                             
Thus the distance of the bead from the center is given as




       r (t ) 
                     R
                  2 1    2
                                1     2
                                                    
                                                exp t 1   2               1      2
                                                                                                         
                                                                                                    exp  t 1   2     

7.25 The solution for the distance travelled by the particle is

                                                    x(t ) 
                                                              F
                                                              k
                                                                   m
                                                                  
                                                                            
                                                                  t  k 1  e
                                                                               k m  t 
                                                                                         
                                                                                         
                                                                                          
                                                                      112
   For k=0, we can’t substitute this directly in the formula since we are dividing by k in the
   expression above. Thus we take the limit k  0 . In this limit, the first nonzero term we get
   is

                            m       k    1 k2 2                      
                             1  1  t 
                          t                2
                                               t  higher order terms 
                                                                      
                            k       m    2m                          
                              1 k 2
                                t
                              2m
   This gives
                                                             1F 2
                                                   x(t )       t
                                                             2m


7.26 The equation of motion for the particle is
                                                 k          k
                                       g 
                                    y              y OR   y   g
                                                       y     
                                                 m          m
   The solution of the homogeneous equation
                                                          k
                                                    
                                                   y        y0
                                                            
                                                          m
   is
                                                            k 
                                                 y  A exp   t 
                                                 
                                                            m 
   Here A is a constant to be determined by the initial conditions. The particular solution is
                                                             mg
                                                     y
                                                     
                                                              k
   Therefore the full solution is
                                                     k  mg
                                          y  A exp   t  
                                          
                                                     m  k
   Now the initial condition is y (t  0)  v0 . This gives
                                

                                                              mg
                                                   A  v0 
                                                               k
   Thus
                                                   mg       k  mg
                                    y (t )   v0 
                                                       exp   t  
                                                    k       m  k



                                                      113
This is easily integrated to get y(t) also. Thus
                                         mg    m      mg         k 
                             y(t )        t   v0     1  exp  t 
                                          k    k       k         m 
When the ball reaches the highest point, its speed is zero. If this time is tup, then
                 mg      k       mg                    k           mg k        1
       0   v0      exp  t up                  exp  t up           
                  k      m  k                          m   v  mg   kv0 
                                                                        0     1    
                                                                           k   mg 
                                                                                      
       OR
                m  kv0 
       t up     ln 1  
                k  mg 
                        
This gives the height h to be
                                                   
           mg m  kv0  m         mg         1  mv0 m 2 g  kv0 
       h     ln 1     v0      1                    mg 
                                                          2 ln 1 
            k  k  mg  k 
                                  k         kv0   k   k    
                                                                     
                                                                     
                                            1     
                                               mg 

If the total time of flight is T then T=tup+tdn, where tdn is the time taken to come down from
the highest point. At time T, y(T)=0. Therefore

                 0
                       mg
                          tup  t dn   m  v0  mg 1  exp  k tup  exp  k t dn 
                                                                                    
                        k                 k        k        m            m 
Thus tdn will be given by solving
                                                                               
                                        m      mg                            
                  mg           mg                         mg k        k      
                     t up       t dn   v0      1          exp  t dn 
                   k            k       k       k        mg      m 
                                                       v0  k                
                                                                             
                  OR
                                   1      mg         k       
                  t up  t dn     v 0     1  exp  t dn 
                                   g       k         m  


                                                  v0 m        k      
                                       t dn       1  exp  t dn 
                                                  g k         m 
To understand whether tup is larger or tdn is larger, let us see the time change in the limit of
very small k. In that case (up to order k)



                                                     114
                             m  kv0  m  kv0 1 k 2 v0
                                                      2
                                                                v         2
                                                                       1 kv0
                    t up     ln 1                 ......  0 
                                                               
                             k  mg  k  mg 2 m 2 g 2
                                                              g 2 mg
                                                                            2



   And tdn is given by solving
                                                        v0 m        k      
                                      t up  t dn        1  exp  t dn 
                                                        g k         m 
                                                   v0 m      k       1 k2 2 
                                           t dn   1  1  t dn       t dn 
                                                    g k      m       2 m2 
                                            v     1 k 2
                                           0       t dn
                                            g 2m
   This gives
                                        2mv0 2m 2  kv0 
                             t   2
                                 dn                mg 
                                             2 ln 1  
                                         kg   k        
                                          2mv0 2m 2        kv0 1 k 2 v0 1 k 3 v0
                                                                       2        3
                                                                                        
                                              2             
                                                           mg 2 m g2 2
                                                                            3 3
                                                                                   ....
                                                                                        
                                           kg   k                         3m g         
                                           2      3
                                          v0 2 kv0
                                            
                                          g 2 3 mg 3
   Therefore
                                                                      12
                                                   v     2 kv0                      2
                                                                               v0 1 kv0
                                           t dn    0    3 mg 
                                                        1                    
                                                    g                        g 3 mg 2
   A comparison shows that tup is smaller than tdn. This makes sense because the while coming
   down, the average speed is smaller since the particle has lost energy due to viscosity and
   continues to do so.
   This also gives the total time of flight to be (up to order k)
                                                                               2
                                                                      2v0 5 kv0
                                                  T  t up  t dn       
                                                                       g   6 mg 2
                                                       2
                                                     kv0
   However, this approximation will be valid only if       1
                                                     mg 2


5.27 It is given that v0  100 ms 1 and   45 . Therefore v0 sin   v0 cos  50 2ms 1 . It is

     also given that g  10 ms 2 .




                                                              115
             v 0 sin 2 
               2
                                                    2v0 sin 
(i) Height =              250 m          Range =             v cos  1000 m
                 2g                                     g
(ii) When k  0 , from the expression derived in the problem above, we get with the initial
vertical speed v 0 sin 

                                          mv0 sin  m 2 g  kv0 sin  
                            Height =                2 ln 1 
                                                                     
                                             k       k        mg    
Substituting the values , we get
For k=0.1, Height = 202m
For k=0.2, Height = 172.3m
To find the range, we first fine the total time of flight and then use formula derived in example
7.13 to find the horizontal distance travelled. From the solution of the previous problem, we
know that the time of flight T is given by solving
                            mg    m            mg         k 
                               T   v0 sin      1  exp  T 
                             k    k             k         m 
For k=0.1, this gives
                  200 T  5414 1  exp( 0.05T ) OR T  27 .07 1  exp( 0.05T )
                                               2v 0 sin 
Since the time of flight without drag is                   14 .14 s , and as the result of the problem
                                                    g
shows the time of flight becomes smaller for k  0 , we tabulate T and the right hand side of the
equation above to find T for T  14


                                    T        27 .071  exp( 0.05T )
                                   14                 13.62
                                   13.5               13.29
                                   13                 12.93
                                   12.5               12.58


It is clear from the Table above that the expression on the right was smaller than T till T=13s and
becomes larger at 12.5s. Thus the time of flight will be between 13 and 12.5s. A little more
tabulation gives T=12.8s.



                                                    116
Substituting this in x(T ) 
                               mv o cos
                                   k
                                         
                                         1 e        
                                              k / m T
                                                         , we get Range = 669m

               m  kv0 sin  
Since t up     ln 1         6.05s , we get tdn= 6.75s. This confirms numerically that time
               k      mg  
taken to come down is greater.
For k=0.2, the equation to determine the time of flight is
                    100 T  107 1  exp( 0.1T ) OR T  17 .07 1  exp( 0.1T )

Making a table like above gives T=11.8s and Range x(T ) 
                                                                 mv o cos
                                                                     k
                                                                           
                                                                           1 e         
                                                                                k / m T
                                                                                            490m

(iii)




                                                                         k=0


                                                         k=0.1



                                        k=0.2




(iv) When drag is introduced, it is the range that is affected much more than the height.


(7.28) In the problem above, we have already found the range for   45 . Let us now take the
        case of k=0.1 and find the range for   40 and   50 .


                                                  117
For   40 , the equation
                             mg    m            mg         k 
                                T   v0 sin      1  exp  T 
                              k    k             k         m 
beomes
                 200 T  5286 1  exp  0.05T  or T  26 .431  exp  0.05T 
and gives
                                                T=11.7s.
This gives a range of

                           x(T ) 
                                     2  100 cos 40
                                           0.1
                                                                 
                                                     1  e 0.0511.7 = 678m.

Thus for   40 , the range increases.
For   50 , the equation
                             mg    m            mg         k 
                                T   v0 sin      1  exp  T 
                              k    k             k         m 
beomes
                 200 T  5532 1  exp  0.05T  or T  27 .661  exp  0.05T 
and gives
                                                T=13.8s.
This gives a range of

                           x(T ) 
                                     2  100 cos50
                                           0.1
                                                                 
                                                    1  e 0.0513.8 = 641m.

Thus for   50 the range decreases.
The two calculations above show that for maximum range, the projectile should be launched at
an angle less than 45 .
The reason for this is as follows. Since the horizontal speed reduces as the projectile moves, it
should cover a larger distance in the initial part of the flight. For this it is better to have a
relatively larger horizontal component of the velocity compared to the case when there is no
drag. Thus the angle should be smaller than 45 .




                                                   118
(7.29) In this case the drag force is proportional to the square of the speed. So the equation of
        motion will be given as follows for the motion up and motion down (taking vertically up
        direction to be the positive y direction)
      Motion up                  m  mg  ky 2
                                  y           

      Motion down                m  mg  ky 2
                                  y           

      Since we are only interested in height, we change  to
                                                        y                    
                                                                        1 d 2
                                                                        2 dy
                                                                             
                                                                             y to get the speed as a

function of the vertical distance of the ball from the ground. The first equation in that case is
                                         1 d 2
                                         2 dy
                                                
                                                 k
                                              y  y 2  g
                                              
                                                 m
                                                   

The solution of this equation is the sum of the solution y 2 ( y) of the homogeneous equation
                                                         
                                                                    h

1 d 2
2 dy
      
        k
     y  y 2  0 and a particular solution y 2 ( y ) . These solutions are
        m
                                          
                                              p



                                                2k                     mg
                             y 2 ( y )  A exp  
                                                  y  and y 2 ( y )  
                                                           
                                h
                                                m           p           k
Here A is a constant to be determined from the initial conditions. The full solution therefore is
                                                         2k      mg
                                      y 2 ( y )  A exp  
                                                               y 
                                                         m       k
The initial condition is that y 2 ( y  0)  vi2 . This gives
                              

                                                           mg
                                               A  vi2 
                                                            k
This the speed of the ball as it moves up is

                                              2k  mg         2k 
                           y 2 ( y)  vi2 exp 
                                                y   1  exp  m y 
                                              m  k                 
At the maximum height h, the speed becomes zero. Therefore

                                          2k  mg         2k 
                              0  vi2 exp  h   1  exp  m h 
                                          m  k                 
This gives

                                     m  mg k  m  kvi2 
                              h      ln             ln 1     
                                     2k  mg k  vi2  2k 
                                                              mg 
                                                                    



                                                     119
Now we consider the motion for downward motion. This can be rewritten as
                                          1 d 2
                                          2 dy
                                                   
                                                  k
                                               y  y 2  g
                                               
                                                  m
                                                    

Again the solution of this equation is the sum of the solution y 2 ( y) of the homogeneous
                                                               
                                                                                  h



equation
           1 d 2
           2 dy
                 
                y  y 2  0 and a particular solution y 2 ( y ) . These solutions are
                
                   k
                   m
                                                     
                                                         p



                   2k                   mg
y 2 ( y )  A exp 
                     y  and y 2 ( y ) 
                                            . Thus the complete solution is
   h
                  m            p         k

                                                          2k     mg
                                       y 2 ( y )  A exp 
                                                               y 
                                                         m       k

                                                                            m  kvi2 
Now the initial conditions are y( y  h)  0 . This gives, with h 
                                                                             ln 1     
                                                                            2k      mg 
                                                                                         
                                       kv2  mg                         mg k
                                 0  A1  i  
                                                               A
                                         mg  k
                                                                          kvi2
                                                                        1
                                                                           mg
This gives
                                                   mg k      2k  mg
                                   y 2 ( y)  
                                                        exp   y 
                                                            m  k
                                                       2
                                                     kvi
                                                  1
                                                     mg
If the final speed is vf, then
                                               mg k mg        vi2
                                     v2  
                                      f                  
                                                 kvi2   k      kvi2
                                              1            1
                                                 mg            mg

Note that of k=0 then v 2  vi2 The answer can also be written as
                        f


                                                   1    1    k
                                                    2
                                                       2 
                                                  v f vi    mg




                                                      120
                                               Chapter 8


8.1 Since there is no external force on the system in the horizontal direction, the total momentum
in the horizontal direction is conserved.
Initial momentum in the horizontal direction = momentum of the carriage + momentum of rain
                                                = Mv  0
                                                = Mv
Final momentum of the system after time t       = M  mt v f

Here vf is the final velocity. Equating the two moment gives
                                                     Mv
                                            vf 
                                                   M  mt 
8.2 Since the water leaking out of the carriage still has a horizontal velocity equal to the velocity
    of the carriage, total momentum of water after it came out for time t is = mtv
    If the initial amount of water in the carriage was m0, then the initial momentum of the
    system (carriage + water in it) = M  m0 v
    If the aped of carriage (with left over water in it) after time t is vf, then by momentum
    conservation
                           M  m0  mt v f    mtv  M  m0 v  v f  v

8.3 Exactly like in problem 8.2, there will be no change in the speeds of the two bicyclists. This
    is easily done by considering the momentum of the two friends before and after the books
    are given by one of them to the other person. Consider the person giving the books. Her
    momentum before transferring the books is Mv . After she gives the books, let her speed by
    vf. Then by momentum conservation
                                Mv  mv  M  m v f       vf  v

     Similarly, for the person receiving the books
                                Mv  mv  M  m v f       vf  v

8.4 Conserve momentum after the first bullet has been fired. Initial momentum is 0. Let the
    velocity (since the motion is one dimensional, we write only the symbol for it, the direction
    is taken care of by the sign) of the gun after the bullet is fired be v1. Since the relative
    velocity of the bullet when it leaves the gun is u, and the bullet leaves the gun when the gun


                                                   121
    is already moving with v, bullet’s speed ug with respect to the ground is calculated as
    follows:
                                    u  u g  v1           u g  u  v1

     Therefore momentum conservation gives

                    M 0  N  1mv1  mu  v1   0               v1  
                                                                                       mu
                                                                                    M 0  Nm

    Now the momentum of the gun and  N  1 bullets in it is

                                       M 0   N  1m
                                                                        mu
                                                                    M 0  Nm 
    Now let the speed of the gun after the second bullet is fired be v2. Then momentum
    conservation gives

                    M 0  N  2mv2  mu  v2   M 0  N  1m                   mu
                                                                                      M 0  Nm 
                                                          mu          mu
                                     v2                     
                                                       M 0  Nm M 0  N  1m
    Similarly one can now show that if the speed after the third bullet is fired is v3 then
                                     mu          mu              mu
                         v3                            
                                  M 0  Nm M 0   N  1m M 0  N  2m

    Generalizing this we get after N bullets have been fired
                                                  k  N 1
                                                                     mu
                                      v final      M
                                                   k 0             ( N  k )m 
                                                               0




8.5 (i) Momentum of the system = sum of the momentum of each particle
                                       = 0.2i  0.4 ˆ kg ms1
                                            ˆ       j

       (ii)    velocity of the centre of mass = total momentum/total mass

                                                     
                                                       1
                                                      0 .3
                                                              ˆ
                                                           0.2i  0.4 ˆ
                                                                      j       
                                                      2ˆ 4
                                                      i  ˆ ms 1
                                                               j
                                                      3     3


8.6 (i) acceleration of the CM



                                                         122
                                                    
                                                   Fnet
                                             
                                               total mass
                                               ˆ j
                                               iˆ
                                             
                                                0.3
                                             
                                                3
                                                    
                                               10 ˆ ˆ
                                                   i j  
(ii)      No, the acceleration is not in the same direction as the momentum of the CM.


8.7 If the base of the cylinder is in the xy plane, as shown in the figure, the x and y coordinates
       of the CM are (0, L/2). We thus have to calculate the z coordinate of the CM.




                                         z


                                                                 y

                                                                     L

                                                                         x
                                               R


To obtain the z coordinate of the CM, consider a rectangular sheet of thickness dz at height z, as
shown in the figure below.




                                                z
                                                             R




If the density of the material that the cylinder is made of is , the z coordinate of the CM, by
definition, is


                                                   123
                                       R
                                        2 L R 2  z 2 zdz                          R
                                                                               4
                          z CM                                                        R 2  z 2 zdz
                                        0

                                            L R 2      2
                                                                              R 2   0


To evaluate the integral, we substitute z  R sin  and dz  R cosd . This gives
                                                  2
                                          4
                               z CM    
                                         R 2         R cos  R sin   R cosd
                                                     0
                                                 1
                                                                 d cos 
                                            4R
                                                 cos
                                                             2

                                                0

                                            4R
                                       
                                            3


8.8 (i) CM of a cone shown in the figure below




                         h
                                                                  r
                                                     z



                                                                      R
                                                                      2

The CM is on the axis of the cone by symmetry. To calculate its height, we take a thin disc of
thickness dz at height z. By similarity of triangles, it radius r is given by

                                                                   r  h  z 
                                         r   R                                           R
                                           
                                        hz h                                            h
If the density of the material that the cone is made of is , then the position of the CM is given
by
                                 h
                                 z   r 2 dz
                                                                                               
                                                                      h
                                                           3 R2                        h
                      z CM      0
                                                          2  2 h 2  z 2  2hz zdz 
                                 1 2                      R h0 h                       4
                                   R h
                                 3




                                                                  124
It is reasonable that the location of the CM is more towards the base since larger mass of the
cone is concentrated there.
(ii)    Hemispherical bowl of radius R is shown in the figure below.



                                                  r

                                                  R       z
                                           




The CM will be on the line passing through the centre of the base. To calculate its height zCM, we
take a ring of height dz at height z. According to the figure

                                                                        z                    r     R2  z2
              z  R sin  and dz  R cos d ,            sin           ,       cos        
                                                                        R                    R       R
If the mass per unit area for the shell is , then the mass dm of the ring is
                                                                          dz
                        dm  2 r  R d  2 R 2  z 2                      2 R  dz
                                                                         cos
Thus
                                           zR                   zR

                                            zdm          2R     zdz           R
                                  z CM    z 0
                                                                z 0
                                                                              
                                            zR
                                                              2R 
                                                                  2
                                                                                  2
                                           z 0
                                                dm

                                                               N
                                                                                 
8.9 Given N particles of masses mi (i=1-N) with total mass M   mi at positions ri (i=1-N),
                                                                                      i 1
                            
       position of their CM RCM is given as




                                                      125
                                                           N
                                                                                  N
                                                                                               
                                                           mi ri                 m r       i i
                                               RCM        i 1
                                                              N
                                                                                  i 1


                                                           m
                                                                                       M
                                                                           i
                                                            i 1

Now let us make m subsystems of these masses with number of particles N1, N2, N3……Nm in
                                                                                                    Ni
them. Then we have the mass of each subsystem as M i   mi . The position of the CM can
                                                                                                    i 1

then be written as
                                                                  N1
                                               N
                                                                    N2 
                                               mi ri          mi ri   mi ri  .......
                              RCM             i 1
                                                  N
                                                                 i 1                 i 1


                                               m
                                                                                       M
                                                       i
                                                i 1

                                                                                 Ni
                                                                                        
By definition of the CM we have for the position RCMi of each subsystem M i RCMi   mi ri .
                                                                                                                                            i 1


Thus the expression above can be written as
                                 N
                                                                                                              m                 
                             m r       i i
                                                   M 1 RCM 1  M 2 RCM 2  .....                               M R          i       CM i
                      RCM    i 1
                                 N
                                                                      m
                                                                                                              i 1
                                                                                                                      m

                              m i 1
                                          i                        M
                                                                    i 1
                                                                               i                                      M
                                                                                                                      i 1
                                                                                                                                     i



This shows that the CM of the system can be calculated by treating each susbsystem as a point
particle of mass Mi located at the CM of each subsystem.


8.10   To find the CM, we will treat the cone and the hemisphere as two subsystems. It is also
       clear by symmetry that the CM will be on the extended axis of the cone. Taking the axis
       as the z direction with z = 0 at the base of the cone, we have
                     mass of the sphere  z CM ( sphere)  mass of the cone  z CM (cone)
            z CM 
                                                  total mass

Assuming the entire system is made of a material of uniform density, we get
                                        3R1 2R13 H R2 H
                                                       2
                                                  
                                                       3   3R1  R2 H
                                                                 4     2   2
                        z CM            8     3    4
                                           2R13 R2 H
                                                     2
                                                            8 R13  4 R2 H
                                                                       2
                                                 
                                             3      3




                                                                   126
8.11 Since there is no external force on the system in the horizontal direction, the position of the
     CM will remain unchanged as the small block moves from one side to the other. Taking
     horizontal direction to be the x-direction, let the position of the CM when the block in on
     the left be X1 and let it be X2 when the block is on the right. Then the poison of the CM of
     the block is (X1R) and (X2+R), respectively. Since the CM does not move, we have
                           X CM  m X 1  R   MX 1  m X 2  R   MX 2

     This immediately gives

                                      X 2  X 1        2mR
                                                         m  M 

8.12 When the ball is compressed, it looks like shown in the picture below




                                                  R


                                                      Rx




The radius r of the circular area of contact for x<<R is

                                    r  R 2  R  x   2 Rx
                                                         2



Thus, if the pressure in the ball remains unchanged, the force that the ball applies on the wall is
                                        F   r 2 p  2 pRx


8.12 As a photon hits the surface, it gives it an impulse proportional to its momentum. If it gets
                                                                  2
    absorbed, the impulse J        and if it is reflected then J      . And the force by the
                                  c                                  c




                                                 127
stream of photons hitting the surface will be nJ where n is the number of particles hitting the
surface per second. If the cross-sectional area of the parallel beam of light is a, then
   n  acN
 Thus the pressure P
                                                             acN 
 (i)     when the light is completely absorbed P                   N
                                                              a   c
                                                    acN 2
 (ii)    when light is perfectly reflected P               2 N
                                                     a   c


 8.14 By equation (8.42b), the force on the planar surface will be equal to momentum
 transfer per unit time. On hitting the surface, the component of momentum perpendicular
 to the plane becomes zero while that parallel to the plane remains unchanged.                      Thus all the
 momentum that water stream carries perpendicular to the surface is transferred to it. The
                                                                    d 2              d 2
 momentum carried by the stream of water per second is                     v  v           v 2
                                                                      4                4
                                                     d 2
 Its component perpendicular to the surface is               v 2
                                                     4 2
 When the water stream hits the surface, it makes an elliptical cross sectional area on the

                                                                                           d 2   d
 surface because the surface is slanted. The major axis of the ellipse is =                    
                                                                                            2     2
                                             d
 And the minor axis remains the same as
                                             2
                                                      d 2
 Thus the cross-sectional are of the ellipse is =
                                                     2 2
                                      d 2
                                             v 2
                                                                    v 2
 Thus the pressure on the surface = 4 2                      
                                                    d 2             2
                                                    2 2


 8.15 At steady state flow let the mass flow rate from the upper portion of the hour-glass be
        
        m . If the height through which it falls before hitting the lower surface is h then the
        amount of mass in the air is the rate at which the mass is falling and the time it takes



                                             128
                                                2h
                                            
     for it to reach the bottom. Thus it is m                        
                                                   and its weight is m 2hg . Thus the hour
                                                g
     glass should have weighed less by this amount. However as the sand hits the bottom,
     it transfers momentum to the hour glass that exactly compensates for the weight in the
     air. This is shown as follows. As the mass hits the bottom, its speed is      2hg . Thus

                                                           
     the momentum it transfers to the bottom per second is m 2hg .
8.16 Consider equation (8.43) for the rocket.
                                                    
                           (M  m)v  m u  v  Fextt
                                                          
  Now since the mass coming out leaves the rocket with u rel , we have
                                                   
                                    u rel  u  (v  v )

  This is because when the mass leaves the rocket, it has already achieved velocity
     
(v  v ) . This gives in the equation above
                                                  
                                    Mv  m urel  Fext t


8.17 Example (8.9) using equation derived above. The example is solved exactly as done
                                                                                    
     in the text except that in applying the equation derived above, u rel  u  (v  v ) each
     time the bullet is fired. This immediately leads to equation (8.50) and the rest is the
     same as done in the example.


8.18 (i) Force needed to hold the chain is equal to the force required to hold the part of the
     chain hanging vertically. This force is = gh
 (ii)If the chain is to be pulled at a constant speed v, its mass increases at the rate of v .
 This gives momentum change per unit time = v 2 . This is the additional force required
 to provide momentum. Thus the total force is gh  v 2 . This is also seen easily by the
 rocket equation
                     
    dv dM 
  M          u rel  Fext
    dt     dt




                                          129
                       
                      dv                  dM
 Now it is given that     0,                  v, u rel  v .          This substituted in the rocket
                      dt                   dt
 equation immediately gives the result derived above.
                                          dM
 (iii)      The rocket equation, with          v, u rel  v is
                                           dt
                                                     dv
                                            M (t )       v 2  F
                                                     dt
 If at a time t, the length of the chain on the table is x then M (t )   (h  x) and

  dv dv dx 1 dv 2
                . Thus the rocket equation can be rewritten as
  dt dx dt 2 dx
                                                     dv 2
                                          (h  x)           v 2  F
                                      2               dx
 This is integrated as
                                       V2                   x
                                            dv 2     2     dx'
                                         F  v 2    h  x'
                                        0              0

 Upon integration this gives

                                   2 h x                                h x
                                                                                        2
                      1     F                                       F
                       ln      2 
                                     ln                                    
                        F  V    h                         F  V 2
                                                                             h 
 Upon solving this gives

                                     V
                                                 1    F 2
                                              h  x 
                                                            
                                                        x  2hx       

8.19 Since the peg is frictionless and the length of the portion of chain passing over the
         peg is negligible, the other portion has length (Lx) and the tension in the chain is the
         same throughout. Taking the tension to be T, the equation of motion for the two
         portions is (see figure)




                                                 130
                                                              T
                                  T


                              ( L  x)



                                                                  x(t)
                               M
                                 ( L  x) g
                               L



                                                              M
                                                                xg
                                                              L


            M        M 
             x     x  g  T
                  x
             L        L 

            M           x      M                      M                  M
              ( L  x)(L  )    ( L  x) g  T        ( L  x)   T  ( L  x) g
                                                                   x
            L                    L                      L                  L


Adding the two equations to eliminate T gives
                                     2Mx               2g
                            M  
                              x          g  Mg   
                                                  x       x  g
                                      L                 L
This equation is also obtained by direct application. Since the total mass being moved is

                        g x  ( L  x) 
                      M                    2Mx
M, the net force is                            g  Mg and the acceleration is  .
                                                                              x
                      L                     L
The solution for the equation above is a sum of the solution of the homogeneous equation
       2g                                        L
 
x         x  0 and the particular solution x p  . The solution of the homogeneous part is
        L                                        2
                                              2g             2g 
                             x h (t )  A exp   t   B exp 
                                              l                t
                                                              L 
                                                                   
Where A and B are two constants to be determined by the initial conditions. The full
solution is


                                              131
                                          2g             2g  L
                            x(t )  A exp   t   B exp 
                                          l                t 
                                                          L  2
                                                               
                                         3L                                      L
The initial conditions are x(t  0)        and x(t  0)  0 . This gives A  B  . This give
                                                
                                          4                                      8
the solution for x(t) to be

                                  L     2g          L       2g  L
                            x(t )  exp    
                                        l t          exp 
                                                                t 
                                  8                 8        L  2
                                                                  
                                  L 1               2 g 
                                  1  cosh           t 
                                  2 2
                                                     l 


8.20 (i) Since the pressure inside the box is p, and the force is unbalanced over an area S,
        the force on the box will be pS.
                                                pS
(ii)     The acceleration of the box will be
                                                M
(iii)    In the simplest calculation, the rate at which the molecules are coming out in one
         second will be those contained in a cylinder of height v x , where v x is the average
         speed in the x direction in the rms sense. Thus the rate at which the gas will be
         leaking out is Snmv x where n is the number density of molecules and m the mass of
         each molecule.
(iv)     Equation (8.45)
                              
             dv dM 
           M          u rel  Fext
              dt    dt
                       dM                             
         In our case         Snmv x ; u rel  v x ; Fext  0 so
                        dt
              dv            1
          M       Snmv x2  Snmv 2  pS by equation (8.40)
              dt            3


                                       M            
8.21 By the rocket equation v f  u ln  i            . If the mass of the fuel is M, we have
                                       M            
                                        f           
        M i  1000  M , Mf = M. Thus we have




                                              132
                                              1000  M 
                                    6  5 ln           
                                              1000 
                  M         6
This gives 1          exp    M  2320 kg
                 1000       5
8.22 In this case the rocket equation becomes (assuming vertically up direction to be
     positive)

                                                               bv  u  g 
                          dv                               dv
                      m       mu  mg  bmv 
                          dt                               dt
Solution to the equation above is given by the sum of the solution for the homogeneous
part and the particular solution. This gives
                                                           u  g
                                 v(t )  A exp(bt ) 
                                                              b
                                                                                 u  g
A is determined by the initial condition v(t  0)  0 . This gives A                    . Thus
                                                                                   b
                                           u  g
                                 v(t )             1  exp(bt )
                                             b




                                             133
                                                                    Chapter 9


9.1         Consider a frame the origin O and another frame with origin O’. Frame O’ is moving
            with velocity V in the x-direction with respect to frame O. Let there be a force F act
            on a particle; the force is the same in the two frames.                                       Then by the work energy
            theorem in frame O, we have
                                                                                   xf
                                               1 2 1 2
                                                 mv f  mv i   F ( x)dx
                                               2       2       xi

      Now in the frame O’, the corresponding velocities are related to the velocities in frame O
      as follows
                                                 v f  V  v 'f                vi  V  vi'
      and
                          x  x'Vt  dx  dx'Vdt and t  t '  dt  dt '
      Substituting this in the work energy theorem gives

                                                                                              x'
                                                                                                f             tf
                        1
                        2
                             
                          m v 'f
                                   2    1
                                        m v i'
                                        2
                                                     2
                                                                       
                                                            mV v 'f  vi'          F ( x' )dx'  V  Fdt 
                                                                                                               ti
                                                                                              x'
                                                                                               i
              tf

      Now      Fdt  is noting but the momentum change mv                               f     vi   mv 'f  vi'  , which is the same
              ti

                                       tf

                                        Fdt   mv                       
                                                                     vi' . This, when substituted in the equation above
                                                               '
      in both the frames. Thus                                  f
                                       ti


      leads to

                                                                                        x'
                                                                                          f
                                            1
                                            2
                                                 
                                              m v 'f
                                                           2
                                                               
                                                                    1
                                                                    2
                                                                            
                                                                      m v i'
                                                                               2
                                                                                         F ( x' )dx'
                                                                                        x'
                                                                                         i

      Which is the work energy theorem in frame O’.


9.2   Kinetic energy of each particle is




                                                                    134
        1                                    1
           0.1  12  0.05J      and           0.2  2 2  0.4 J
        2                                    2
      Thus the total kinetic energy of the system is = 0.45J
      Momentum of the system is = 0.1  0.4  0.5kgms 1
                                             0.5
      Velocity of the CM therefore is =           1.667ms1
                                             0.3
                                                       2
                                                     PCM       0.25
      Thus the kinetic energy of the CM is =                          0.417J
                                                  2m1  m2  2  0.3
      Thus the kinetic energy of then particles in the CM frame is = 0.450.417 = 0.033J
      This can also be checked by calculating the kinetic energy of each particle in the CM
      frame, which is

                             0.1  1.0  VCM   0.05  1.0  1.667  0.022J
                          1                     2                      2

                          2
                             0.2  2.  1.667  0.1  2.0  1.667  0.011J
                          1                      2                    2

                          2


9.3   Since the momentum of the system is zero, the kinetic energy of the CM = 0
9.4   Masses, velocities, moment and kinetic energies o feach particle are shown in the table
          below


            mi(kg)                  vi(ms1)                pi(kgms1)             KEi(J)
               1                        1                       1                 0.5
               2                        2                            4              4.0
               3                        3                       9                13.5
               4                        4                        16                 32
               5                        5                      25                62.5


      Thus the total mass is = 15kg
      Total momentum is = 15kgms1
                                15
      Velocity of the CM =           1ms 1
                               15



                                                    135
      Total kinetic energy = 112.5J
                                            2
                                          PCM       15  15
      Kinetic energy of CM =                                7.5 J
                                    2  total mass 2  15
      Kinetic energy about the CM = 105J
      This is easily checked by calculating the kinetic energy of each particle about the CM and
      adding them all up. Thus

      KE about the CM =
                              1
                              2
                                
                                1   1  1  2  2  1  3   3  1  4  4  1  5   5  1
                                             2             2               2             2               2
                                                                                                             
                           = 105J


9.5   From equation (9.37), we have
                                                          m2
                                          v1  VCM             (v1  v 2 )
                                                        m1  m2
                                                          m1
                                          v 2 V CM            (v1  v 2 )
                                                        m1  m2
      Thus the kinetic energy of the system
                                                 2
                                        1 m1 m2
              1       1     2  1
                m1v1  m2 v 2  m1VCM 
                   2                2
                                                      v1  v2 2  m1m2 VCM v1  v2 
              2       2        2        2 m1  m2 2
                                                                   m1  m2 
                                              1 m2 m12
                                     1
                                     m2VCM 
                                          2
                                                            v1  v2 2  m1m2 VCM v1  v2 
                                     2        2 m1  m2 2
                                                                         m1  m2 
      Since v rel  v1  v 2 , we get

                                                             1 m1 m2
                            m1v12  m2 v 2  m1  m2 VCM               v1  v2 2
                          1        1     2  1            2

                          2        2        2                2 m1  m2 
                                                 1       1 2
                                                  MVCM  v rel
                                                      2

                                                 2       2


                                          dU ( x)
9.6   (a) We will be using F ( x)               to calculate the force. Thus U ( x)   F ( x)dx  C ,
                                           dx
           where C is a constant chosen according to the reference point x0, where U x0   0 .

                                                        1
        (i) F ( x)  kx        U ( x)    kxdx  C   kx2  C
                                                        2




                                                     136
                                1
             (ii) F ( x)                    To calculate the potential energy, substitute x  a sinh z so
                             x2  a2
                                                             x
        that dx  a cosh zdz to get U ( x)   sinh 1         C
                                                             a
                                                                           5
             (iii) F ( x)  5 sin 2 x      U ( x)   5 sin 2 x dx       cos2 x dx  C
                                                                          2
             (iv) F ( x)  5 sin 2 2 x

                                                                                5     sin 4 x 
                                                         1  cos 4 x dx   2  x  4   C
                                                      5
                   U ( x)    5 sin 2 2 x dx  
                                                      2                                        
(b)     The plots of force and potential in each of the cases above are as follows. The force is
        shown on the left and the corresponding potential on the right in each case.


      (i)        We have taken k = 1.5. We have chosen the constant such that U(0) = 0.




      (ii)       The plots are drawn for a = 5




                                                           137
(iii) We have chosen the constant such that U(0) = max.




(iv) We have chosen the constant such that U(0) = 0. Notice that the average force is always
positive, i.e. pointing in the positive x-direction. Therefore the potential energy curve keeps
going down as x increase.




 9.7 We use U ( x)   F ( x)dx  C and keep the potential continuous everywhere. In the

         plots the force is shown on the left and the corresponding potential on the right in each
         case.


   (i)
                                                 k if x  0
                                       F ( x)  
                                                 k if x  0




                                              138
               x  0 U ( x)    (k )dx  C  kx  C 
                                                       
                                                         U ( x)  k x  C
               x  0 U ( x)    kdx  C  kx  C 
                                                       
If we choose U (0)  0, U ( x)  k x

For the plots drawn below we have chosen k = 2.




 (ii) F ( x)  k x

                                                         1
                        x  0 U ( x)    kx dx  C   kx2  C
                                                         2
                                                          1
                        x  0 U ( x)    (kx) dx  C  kx2  C
                                                          2
 If we choose U(x) = 0, C=0.
For the plots below, we have chosen k=2.




                                        139
(iii)
                                             0 if x  a
                                            
                                   F ( x)  
                                             kx if x  a
                                            
The nature of force implies that the potential is a constant for  a  x  a
                                                           1 2
                            x  a U ( x)   kx dx  C      kx  C
                                                           2
                                             1 2
                            x  a U ( x)      ka  C
                                             2
                             1
Choosing U(0) = 0 gives C   ka2 and
                             2

                                 x  a U ( x) 
                                                1
                                                2
                                                     
                                                  k x2  a2   
                                 x  a U ( x)  0

For the plots below, we have chosen k=2 and a =15.




(iv)
                                             0 if x  a
                                             
                                    F ( x)   C
                                              2 if x  a
                                             x
The nature of force implies that the potential is a constant for  a  x  a
                                             C               C
                        x  a U ( x)        2
                                                 dx  const   const
                                             x               x
                                         C
                        x  a U ( x)       const
                                         a
                                   C                         C
Choosing U(0) = 0 gives const       for x  a and const   for x  a
                                   a                         a


                                          140
      Thus
                                                                    C C
                                                     x  a U ( x)  x  a
                                                    
                                                                    C C
                                                     x  a U ( x)  
                                                                    x a
                                                     x  a U ( x)  0
                                                    
                                                    
      For the plots below, we have chosen C=250 and a =20.




                              dU ( x)
9.8   We use F ( x)                 . This gives
                               dx
                                                    dkx
        (i) U ( x)  kx                F ( x)          k
                                                    dx
                                                 d kx
                                                 dx  k x  0
                                                
        (ii) U ( x)  k x              F ( x)  
                                                 d  kx  k x  0
                                                
                                                     dx
                           1                        d  1            2x
        (iii) U ( x)                  F ( x)         2    2 
                                                                  
                         x  a2
                          2
                                                                   
                                                    dx  x  a  x  a 2
                                                                    2
                                                                             
                                                                             2



                                1                           d      1        2x
        (iv) U ( x)                          F ( x)         2   2 
                                                                          
                              x  a2
                               2
                                                            dx  x  a  x  a 2
                                                                            2
                                                                                    
                                                                                     2




      (b) Plot for part (i) is straightforward; it is a constant force and linearly varying potential
        Plots for part (ii) are similar to that of part (i) in problem 9.7. This has been given above.




                                                          141
      (iii)   We have chosen a = 5. The force is shown on the left and the corresponding potential
              on the right.




      Notice that the force is negative for negative x and positive for positive x. Thus it is a force
      that pushes a particle away from x = 0 and the position at x = 0 is an unstable equilibrium
      point. Thus when a particle is displaced slightly from this point, it will runaway to infinity.


      (iv)    We have chosen a = 5. The force is shown on the left and the corresponding
              potential on the right.




      Notice that the force is positive for negative x and negative for positive x. Thus it is a
      restoring force and the position at x = 0 is a stable equilibrium point. Further the force is
      varying almost linearly with x near x = 0. Thus when a particle is displaced slightly from this
      point, it will perform simple harmonic oscillation.


9.9     Since all the potentials except (i) are time-dependent, only (i) is conservative.




                                                   142
9.10 Fo++923206487r the block not to fall off the track, its speed v at the top of the loop
          should be such that
                                       v2
                                           g  v 2  Rg
                                       R
     Now by the conservation of energy we have
                                               1 2 mgR      5
                           mgh  mg (2 R)       mv     h R
                                               2      2     2


9.11 For the potential energy U ( x)  C  x  a  we have the potential energy curve (with
                                                    3



          parameters C = 0.5 and a =2)




It is clear from the curve that a particle will always experience a force in the negative direction
except at a = 2. From this it is clear that the point x = a is a point of unstable equilibrium.


For the potential energy U ( x)  C  x  a  we have the potential energy curve (with parameters C
                                           3



= 0.5 and a =2)




                                                 143
This potential energy curve gives force opposite to the displacement, and the force is zero at x =
a. Thus this point is a point of stable equilibrium.           Notice that unlike the potential energy for a
       1    
spring  kx2  , this potential energy is quite flat near the equilibrium point (because
       2    

x  a 4  x  a 2 for x  a  1.   For larger displacements from x = a, the curve rises much faster

than the spring potential energy.


9.12       Let the velocities of the balls be v1 and v2 after the collision (since this is a one
            dimensional motion, we are not putting vector signs on top of the velocities). Then by
            momentum conservation
                                  mV  mv1  v 2   V  v1  v 2
                                  1
                                  2
                                          1
                                          2
                                                 
                                    mV 2  m v12  v 2
                                                     2
                                                                V 2  v12  v 2
                                                                                2




       The first equation gives V 2  v12  v 2  2v1v 2
                                              2



       This and the second equation implies that 2v1v2  0
       Thus either v1 is zero, i.e. the first ball is not moving. This is the situation after collision.
       Or v2 is zero, i.e. the second ball is not moving. This is the situation before collision.


9.13        According to the situation give, the two balls are undergoing an elastic collision (no
            loss in energy) when they are moving in the opposite directions with equal speed V.



                                                     144
    Let the mass of the lighter ball be m and that of the heavier one be M. If the velocity
    of smaller ball is v1 after the collision, then by equation 9.36 (taking vertically up
    direction to be positive)

                         v1 
                                M  mV   
                                                M
                                                    (2V ) 
                                                             3MV  mV
                                  M m         M m           M m
In the extreme limit, when M>>m, we have v1  3V

If the balls are dropped from a height h then V  2 gh

If after the collision the smaller ball goes to height H then 3V  2gH

These two equations give H  9h


9.14 By momentum conservation we have
                                                                m1v o
                                m1  m2 v  m1v0    v
                                                               m1  m 2

                         1    2
Totla initial energy =     m1v0
                         2
                      2
               1 m12 v0    m1
Final energy =                   initial energy
               2 m1  m2 m1  m2
As is clear, the final energy is less than the initial energy. Therefore some energy is lost in
the process.
Now consider a pile of beads, each of mass m, connected with strings of length l between
them. As the first bead falls over the edge, its initial speed is zero bu by the time the string
connecting it to the second bead is taut, it gains a velocity of v  2gl . However, as soon
as it pulls the second bead, by momentum conservation at that instant (neglecting the
impulse due to gravity) the speed of two beads is

                                               m 2 gl     gl
                                         v           
                                                2m        2
Now the two beads fall together and as the string connecting the second bead to the third
bead becomes taut, the energy of hse two beads is
                                     1       gl       5
                                        2m   2mgl  mgl
                                     2       2        2



                                               145
                                                                5gl
       This gives the speed of the two beads to be
                                                                 2
       Thus when the third bead is pulled over, momentum conservation at that instant gives the

                                               2m 5 gl 2          2 5 gl
       speed of the three beads to be =                       
                                                    3m            3 2
       One can go on like this and build up the solution up to pth bead falling by recognizing a
       pattern. We can also do the problem in the following way.
       Suppose when the (p-1)th bead has just fallen of the edge, the speed of the beads becomes
       v p 1 . Then when these beads fall and the string between the (p-1)th and pth string becomes

       taut, the energy of the system will be
                                            1
                                               ( p  1)m  v 2 1  ( p  1)mgl
                                                              p
                                            2
       Thus the speed of these beads just before the pth bead falls off the edge is given by
                     1               1
                       ( p  1)mv 2   ( p  1)m  v 2 1  ( p  1)mgl  v 2  v 2 1  2 gl
                                                      p                            p
                     2               2
       By momentum conservation, therefore, we get the speed when the pth bead falls off as
       follows

        pmv p  ( p  1)mv            pv p  ( p  1) v 2 1  2 gl
                                                         p                     p 2 v 2  ( p  1) 2 v 2 1  ( p  1) 2 2 gl
                                                                                      p                p


       We know that v1  0 i.e. when the first bead just falls off the edge of the table, its speed is
       zero. Now we can write
                                        ( p  1) 2 v 2  ( p  2) 2 v 2 1  ( p  2) 2 2gl
                                                     p                p


       And so on so that continuing in this manner we get ( v1  0 )

                                                                          
                                                                                        p 1
                                                                                                   ( p  1) p (2 p  1)
       p 2 v 2  v12  ( p  1) 2  ( p  2) 2  ( p  3) 2  ..........  1 2 gl  2 gl  i 2 
             p                                                                                                           2 gl
                                                                                        i 1                6
       This gives

                                                         ( p  1)(2 p  1) gl
                                                vp 
                                                                 3p

9.15      From problem 8.18, the total force required to move the chain is gh  v 2 . Thus the

            power delivered by the person pulling the chain is = ghv  v 3
        On the other hand, the energy of the chain is = kinetic energy + potential energy


                                                          146
                                                      dx
  If the length of the chain on the table is x then       v and
                                                      dt
                     1                                          1
  Kinetic energy =     ( x  h )v 2       potential energy =     gh 2  xgh
                     2                                          2
                            1               1
  Thus Total energy E =       ( x  h)v 2  gh 2  xgh
                            2               2
               dE 1 3
  This gives      v  vgh
               dt 2
Thus the rate of change of the energy of the chain is not equal to the power delivered. As
shown in the previous problem, the energy is lost in the inelastic collision.


One may ask a question: what if instead of the chain, it s a rope or a strin that is being
pulled? Where does the energy go in that case? The answer is that the energy difference is
then used in stretching the rope to generate the required force.


9.16 Let the length of the chain be L. If its length y is hanging from the table, the force
                            Mg
     pulling it down is =      y
                            L
Taking the vertically down direction to be positive, the coordinate of the end of the hanging
portion of the chain, when its length is y , is also y.
                                                          Mg               g
Thus the equation of motion of the chain is M  
                                              y              y or    
                                                                    y        y0
                                                           l               L
The equation can also be derived by considering the two protions, the one hanging from the
table and the other on the table, separately.
                
             1 dy 2
Writing  
        y           we get from the equation above
             2 dy
                                                          v         y
                   1 dy 2 Mg
                                                 1            Mg
                     M       y0                  M  d y2 
                                                                  y' dy'  0
                   2   dy   l                     2 0           l y1

This gives


               1
               2
                 Mv2 
                       1 Mg 2
                       2 l
                                      
                            y  y12  0 or
                                                      1
                                                      2
                                                        Mv2 
                                                              1 Mg 2
                                                              2 l
                                                                  y 
                                                                       1 Mg 2
                                                                       2 l
                                                                           y1



                                            147
The second term in the equation above is the change in the potential energy of the chain as
the length of its hanging portion changes from y1 to y. Thus the total energy of the chain
remains unchanged as it slips off or the energy is conserved. This can also be seen as
follows
When length y1 is hanging,
                                                  1 Mg 2                             1 Mg 2
kinetic energy = 0       potential energy =          y1          Total energy =        y1
                                                  2 l                                2 l
When length y is hanging,
                               1                                          1 Mg 2
               kinetic energy = Mv2               potential energy =         y
                               2                                          2 l
                                                 1       1 Mg 2
                              Total energy =       Mv2      y
                                                 2       2 l
It is clear from the above that the total energy remains unchanged.


9.17 As the chain unfolds, the mass of the hanging (and moving) portion keeps on
    changing. Thus the problem is like the variable mass problem. We take the mass per
    unit length of the chain to be λ. Taking the vertically down direction to be positive,
    the coordinate of the end of the hanging portion of the chain, when its length is y , is
                                                           dm
    also y. The external force on the chain is yg ,           y , and the relative velocity of
                                                                 
                                                           dt
                                                                                 dm
    the mass that is added to it is  y . Thus the equation of motion m 
                                                                      y            u rel  Fext of
                                                                                 dt
    the chain is
                          y  y 2  yg OR
                            y                         y   y 2  yg
                                                        y      

                 
              1 dy 2
 Using  
       y             we write this equation as
              2 dy

                                                
                                         1 dy 2 y 2
                                                   g
                                         2 dy     y
                                                                          A
We now solve this equation for y 2 as the sum of the solution y 2 
                                                                           of the homogeneous
                                                                          y2

                                   2
part and the particular solution     yg to get
                                   3


                                            148
                                                A 2
                                        y2 
                                                  yg
                                                y2 3

                                                                                   2 3
Here A is a constant. Using the condition that y 2 ( y  y1 )  0 , we get A  
                                                                                    y1 g and
                                                                                   3
therefore

                                       y 
                                        2      
                                           2 y 3  y13
                                                       g
                                                        
                                              3y 2
The same result can be obtained by letting p   and l  0 such that pl  y in the
solution for a coiled set of beads in problem 9.14.
Now let us compare the energy of the chain at the beginning o fthe motion and agter it has
slipped so that y of its length is hanging.
At the beginning, when length y1 is hanging,
                                                1                               1
  kinetic energy = 0        potential energy =  gy12          Total energy =  gy12
                                                2                               2
When length y is hanging,

              Kinetic energy =
                                  1
                                    yy 2  
                                      
                                                    
                                              y 3  y13    1
                                                        g  y 2 g 
                                                                     1 y13
                                                                            g
                                  2              3y        3         3 y
                    1
Potential energy =  gy 2
                    2
                                         Total energy =
       1 y13    1         2y                         1
             g  gy 2   1   Initial total energy  gy 2  Initial total energy
                           3y 
       3 y       6                                    6
Thus we see that the energy is lost during the motion. This happens due to the inelastic
collision between the chain links as each new link is pulled by the moving chain (see
problem 9.14).




                                              149
                                            Chapter 10


10.1 (a) The fields are shown on the axses. They are the same all over the place.

                                                       y
    
                   ˆ
(i) F ( x, y )  y i

                                                                          x




                                                       y

     
(ii) F ( x, y )  x ˆ
                    j
                                                                      x




(b) If we think of the force arrows as velocity of a fluid, we see that a matchstick thrown in
that fluid will tend to rotate clockwise in (i) and counterclockwise in (ii). Thus the curl of
both the fields is not zero; it is negatve for (i) and positive for (ii).


(c)        curlf of force field (i)
                                                ˆ
                                                i        ˆ
                                                         j      kˆ
                                                            
                                         F                         ˆ
                                                                    k
                                               x       y      z
                                                y       0       0

           Curl of force field (ii)
                                                 ˆ
                                                 i          ˆ
                                                            j   kˆ
                                                            
                                          F                      ˆ
                                                                   k
                                                x         y   z
                                                0           x   0



                                             150
10.2


                                     Y


                                                                  (2, 1)




                                     O                                      X


                
                               ˆ
(i) Force field F ( x, y )  y i .
Path 1: Work done is zero as the particle moves along the x-axis. Similarly as it moves from
(2,0) to (2,1), work done is again zero since the particle is moving perpendicular to the force
field. Thus alongh path 1, the net work is zero.
Path 2: Work done is zero as the particle moves along the y axis. However, as it moves from
(0,1) to (2,1), it is under a constant force of 1 unit in the x direction. So the work done by the
field is 2 units.
                               x                  xˆ
Path 3: Along path 3 y          . Thus F ( x, y)  i . The displacement along path 3 is given as
                               2                   2
 
       ˆ            ˆ 1 j
dl  dxi  dy ˆ  dxi  dx ˆ . Thus
              j
                       2
                                                12
                                         dW  F .dl   xdx  1 unit
                                                     20
                   
(ii)   Force field F ( x, y )  x ˆ
                                  j
Path 1: Work done is zero as the particle moves along the x-axis since the particle is moving
perpendicular to the force field. Similarly as it moves from (2,0) to (2,1), Field is constant and
equal to 2 units. Thus the work done by the field will be 2 units. Thus alongh path 2, the net
work is 2 units.
Path 2: Work done is zero as the particle moves along the y axis because the field is zero. As it
moves from (0,1) to (2,1), it is is moving perpendicular to the force field so the work done is zero
again. So the net work done by the field is zero.


                                                    151
                            x        
Path 3: Along path 3 y       . Thus F ( x, y )  x ˆ . The displacement along path 3 is given as
                                                    j
                            2
 
       ˆ           ˆ 1 j
dl  dxi  dyˆ  dxi  dx ˆ . Thus
             j
                      2
                                          12
                                  W   F .dl   xdx  1 unit
                                               20

In both the cases the work done depends on the path. It is therefore consistent with the curl of
both the fields not being zero.


                            ˆ
10.3 Force field is F  2 y i  3 x ˆ . The paths are shown below
                                    j


                              Y                                (1, 1)


                                         Path2

                                                                 path1



                              O                                            X


    We will calculate the work done along path 1 in two parts: first when the particle mones
   along the x-axis and secondly when it is moving from (1,0) to (1,1). Along the x-axis, the
            
   force is F  3 x ˆ and the displacement is dx iˆ . Thus the work done is zero when the particle
                    j
   moves along the x-axis. When the particle moves from (1,0) to (1,1), the force is
    
         ˆ
   F  y i  3 ˆ and displacement is dy ˆ . Thus the total work done is
               j                        j
                                              1
                                         W   3dy  3 units
                                              0

   This then is the total work done along path 1.
   Along path 2 y = x so that dy = dx. The work done is given by the integral
                              1             1        1         1
                                                                          5
                      W   F .dl   2 ydx   3xdy   2 xdx   3 ydy  units
                                    0         0        0         0
                                                                          2

   Since the work done along two paths is different, the force field is NOT conservative.


                                                 152
10.4 The figure for the path si shown below. The way we have set up the transformations

                                     Y
                                                   C



                                                                    (2,1)
                                 A
                                         (0,1)         
                                                               B

                                                                                              X


x  (1  cos ) and y  (1  sin  ) , the corresponding angle is as shown in the figure and

varies from π to 2π.       Thus the integral W ACB                  F dx   F dy
                                                                   ACB )
                                                                           x
                                                                                        ACB
                                                                                              y    from A to B over the

semicircular path is
                                                     x3
             W ACB  A     x ydx  A              3 dy
                             2

                         ACB )                   ACB
                         2                                                        2
                    A  1  cos  1  sin  sin  d                         1  cos 
                                                                               A
                                                                                                      cos d
                                             2                                                    2

                                                                              3   

The integral can be carried out easily and gives
                                                               8A
                                                     W ACB 
                                                                3
     
              ˆ
10.5 F  k1 yi  k 2 x ˆ . The force field will be conservative if its curl vanishes. The curls
                        j
of it is
                                          ˆ
                                          i      ˆj                 kˆ
                                                                
                                  F                                  k 2  k1
                                         x     y                  z
                                         k1 y  k 2 x              0

If the curl has to vanish, we should have k1  k 2 .
                        
                               ˆ
For the force field F   y i  x ˆ it is clear by inspection that the potential should be
                                    j
U ( x, y)  xy . However, to derive it formally we put




                                                       153
                                        U ˆ U ˆ
                                          i           ˆ
                                                 j   yi  x ˆ
                                                              j
                                        x    y
This gives
                                               U
                                                  y
                                               x
Which is integrated to
                                        U ( x, y)  xy  f ( y)
                                                         U
Differentiating this with respect to y and writing           x gives
                                                         y
                                     df
                                         0 OR           f  const.
                                     dy
By the condition U (0,0)  0 , the const. is zero. Thus U ( x, y)  xy


10.6 (i) Curl of the field
                                          ˆ
                                          i   ˆ
                                              j ˆ
                                                k
                                            
                                   F                ˆ
                                                    Ai
                                         x y z
                                         3 A Az 0

is not zero. Therefore the field is not conservative.
    (ii) The curl of the field
                      ˆ
                      i    ˆ
                           j   kˆ
              
              F 
                     
                     x
                          
                          y
                               
                               z
                                                   ˆ 
                                     A x y  z i  yz  x  ˆ  z x  y k
                                                                 j             ˆ       
                    Axyz Axyz Axyz

is not zero. Therefore the field is not conservative.


                                         ( x  2) 2  ( y  1) 2 
10.7 (i) The function f ( x, y )  2 exp                         will be a constant where its
                                                    4            
      argument ( x  2) 2  ( y  1) 2 is a constant. Thus the contours are given by

                                    ( x  2) 2  ( y  1) 2  const.

These are concentric circles with centre at 2,  1 .



                                               154
  (ii)      The gradient of the function is given by
                      x  22   y  12          x  22   y  12 
      ˆ 
      i
       x
               
            j 2 exp
              y 
                                               exp                         
                                                                             x  2i   y  1 ˆ
                                                                                     ˆ            j    
                             4                             4           


10.8 Kinetic energy of the system of N particles with masses mi i  1 N  and positions
     
     ri i  1 N  is

                                      1 N     
                                         mi ri 2
                                      2 i 1
                                          
Let the position of the CM be R so that V  R . If the positions and velocities of these
                                                                       
particles with respect to the CM are given respectively by ric i  1 N  and

ric i  1N  then


                                                   
                                                      
                                                                   
                                                                      
                                     r  R  ri and r  R  ric  V  ric
Substituting this in the equation for kinetic energy above, we get
                          1 N            1 N       1 N          N
                                                                       
                            
                          2 i 1
                                 mi ri 2   miV 2   mi ric2   mi ric  V
                                    
                                          2 i 1    2 i 1
                                                           
                                                                 i 1
                                                                      

                              M                                                      N
                                                                                            
Denoting the total mass        mi as M and using the property of the CM that
                              i 1
                                                                                    m r
                                                                                    i 1
                                                                                           i ic    0 , we

get
                                      1 N           1        1 N     
                                         mi ri 2  2 MV 2  2  mi ric2
                                      2 i 1                    i 1

This is the desired result.


10.9 When some energy is released during collision, the equations during the collision are
                                  
                                      
                      p1  p2  p1  p 2 (momentum conservation)
                         2                '
                  p12     p2          p1'2   p22
                             E               (energy conservation)
                 2m1 2m2             2m1 2m2
Here the unprimed quatities are before collision and the primed quantities are after collision.
Further E  0 . The easiest way to prove that in the bove situation, the paricles cannot




                                                   155
move stuck together is to write all quatities in the with respect to the CM. In that case
(referring to quantities in the CM frame with subscript c)
                                                                            
                                     
                 p1c  p 2c  p1c  p 2c  0  p1c   p 2c         and             
                                                                           p1c   p 2c

And using the splitting of the kinetic energy as in the problem above and the fact that
                       
                           
p1c   p 2 c and p1c   p 2c
               2                 '                                   
         p12c   p 2c         p1'2   p 22c         p12c    1   1         p1'2  1 1 
                     E      c
                                                       
                                                          m  m   E  2  m  m 
                                                                  
                                                                             c
                                                                                     
        2m1 2m2             2m1 2m2                2      1    2              1  2 


Now if the particles are moving together, their momenta after the collision with respect to the
CM will be zero. This is also seen mathematically as follows. By momentum conservation
                                               
                                                 
                                        p1c   p 2 c

And if they are moving together, then
                                                 
                                                   
                                            p1c  p 2 c
                                 
                                   
The above two equation give p1c  p 2 c  0
Putting this in the energy conservation equation gives
                                                      
                      p12c  1  1                     p2  1  1 
                           
                           m  m                        m  m 
                                     E  0  E   1c        
                       2  1     2                     2  1   2 


However that is not possible because two positive quantities cannot add up to zero. Thus the
particles cannot move stuck together.


10.10 The picture of the carom coin and the striker is shown below

                Y
                                                                                 Direction of
                                                                                 impulse on coin




                                                         1.55cm
                                      2.05cm                              2cm


                                                            2ms-1                         X




                                            156
The coin will move in the direction of the impulse that it receives from the striker. This
direction is perpendicular to their common surfaces and theefore along the line joining their
centres. Thus the coin moves at an angle  such that

                                2        5          56
                sin                      cos                 and   33.75
                          (2.05  1.55) 9           9
We will work in cgs units. Initial momentum of the system is that equal to the initial
                                                ˆ
momentum of the striker which is p  3000 i .
                                                       
                                                              ˆ
If after the coliision, the velocity of the striker is V  Vx i  V y ˆ and the speed of the coin is v
                                                                      j

                                                     v 56 ˆ 5v ˆ
                                      ˆ
then its velocity would be v  v cos i  v sin  ˆ 
                                                  j        i   j and the momentum
                                                        9     9
conservation will lead to
                                        v 56
                                     5       15V x  3000
                                          9
                                        5v
                                     5   15V y  0
                                        9
Since the collision is elastic, the total energy is conserved and we get
                           1
                           2
                                        1
                                        2
                                                          1
                              5  v 2   15  Vx2  V y2   15  2002
                                                            2
These equations simplify to
                                     v 56
                                            3V x  600
                                       9
                                     5v  27V y  0
                                     v 2  3Vx2  3V y2  120000

These are 3 equations for 3 unknowns so all the answers can be obtained from these
equations. Substituting

                                              56               5v
                               V x  200        v and V y  
                                             27                27
In the energy conservation equation gives
                                            4v 2993 
                                          v        0
                                            3    9 




                                              157
This gives v  249.4cms1  2.5ms1 after collision. The trivial answer v  0 of course refers
to the situation before collision.
This then leads to V x  1.30 ms 1 and Vy  0.46 ms 1 . This gives the angle of deflection of

                          0.46 
the striker to be tan 1         19 .5 below the x-axis.
                          1.30 
        (ii) KE o fthe CM remains unchanged during the collision. It is therefore
                                      3000 2
                                                 10 7  0.0225 J .
                                     25  15 
                                  15  2
(iii)   Velocity of the CM =               1.5 ms 1
                                 15  5
Therefore the striker is moving with velocity 0.5 ms1 and the coin with -1.5 ms1 along the x-
axis in the CM frame.        The magnitude of momentum of each body is 0.0075 kgms1 in
directions opposite to each other (see figure). In the CM frame the magnitude of velicities
does not change during an elastic collision. Thus the momentum and the velocity vectors of
the striker and the coin just rotate. Further, the impulse J is in the direction (see figure) such
that it makes and angle
                                                 2 
                                        sin 1       33 .75 
                                                 3.6 
from the x-axis. As is clear from the figure, the magnitude of J must be such that the final
momentum has the same magnitude as the initial momentum. This is seaily done by drawing
a circle of radius equal to the magnitude of the momentum with its centre at the tail of the
initial momentum vector. We then draw the impulse vector from the head of the initial
momentum vector at the angle  and take its length such that its head touches the circle. The
vector from the centre to this point gives the final momentum.                 The construction
immediately shows that
                                                   
                                             p f  pi  J

It is also clear from the figure that
                            
                            J  2  0.0075  cos33.75  0.0125kgms1

It is also clear from the figure that the angular momenta vector srotate by
                                  CM  180   2  33 .75   112 .5


                                                158
                                                          J


                        0.0075 kgms1                   33.75
                                                      0.0075 kgms1
                          CM




As a check we now calculate the velocities of the striker and the coin in the lab frame from
the information above and compare our answer. As is clear from the figure, the x component
of the striker’s velocity in the CM frame is
                                V xCM  0.5 cos112 .5  0.187 ms 1

This then gives Vxlab as follows (keep in mind that the CM is moving in x direction only)
             V xCM  V xlab  VCM       V xlab  VCM  V xCM  1.5  0.187  1.313 ms 1

And we get from the figure (the CM is moving in x direction only)
                           Vylab  VyCM  0.5 sin 112.5  0.462ms1
Thus

                                Vstriker ,lab  Vxlab  V ylab  1.39ms1
                                                  2        2



Similarly for the coin (see figure)
                            V xCM  1.5 cos180   112 .5  0.574 ms 1

This gives
                          V xlab  VCM  V xCM  1.5  0.574  2.074 ms 1

and we get from the figure
                         Vylab  VyCM  1.5 sin180  112.5  1.386ms1
This gives

                                    Vcoin,lab  Vxlab  V ylab  2.49ms1
                                                  2        2




                                                  159
These match with the previously obtained answers.


10.11 As in the problem above, the stationary balls will move in the direction of impulse
after the collision.   This is going to be along the line joining the centre of the striking ball
and the centres of the stationary balls. At the time of colliding, the three balls will look as
shown below.

                                                                 Direction of
                                                                 impulse on ball

                                                           30




                                                                 Direction of
                                                                 impulse on ball

Since the net impulse (due to the two stationary ball) will be along the line of its initial
motion by symmetry (see figure), it will continue to move along its original direction. This is
also seen by noticing that the statinary balls have a net momentum in the direction of v0.
Thus there will be no component of the momentum to be balanced in the direction
perpendicular to that.    Thus the striking ball will continue to move along its original
direction. Let the speed of the striking call be v1 after the collision and the speeds (equal by
symmetry) of the other balls be v. Then by momentum conservation

                                                   3
                         mv 0  mv1  2  m  v             v0  v1  v 3
                                                  2
Since the collision is elastic, we have by energy conservation
                       1 2 1 2          1
                         mv0  mv1  2  m  v 2             v0  v12  2v 2
                                                               2

                       2      2         2
There are two unknowns v1 and v and two equations so we can get their values. Squaring the
momentum conservation equation and subtracting it from the energy conservation equation
gives
                                         
                                        v v  2v1 3  0


                                             160
One of its solutions is the trivial solution v  0 referring to the situation before collision.
Theother solution is
                                           v  2v1 3
Substituting this in the the momentum conservation equation gives
                                          v0             2v 0 3
                                 v1          and v 
                                          5                 5




                                               161
                                          Chapter 11


11.1 Disc rotating about a point on its periphery. Consider first a rotation about a point on
       the periphery by an angle. This is shown on the left in the figure below. The dashed
       circle shows the initial position of the disc while the solid circle shows the position
       after rotation.




                               
                                                                              




On the right we show the same rotation carried out by translating the CM of the disc first to
its new position, as shown by the straight arrow and then carrying out the rotation about the
CM. We have made the disc in this position by dotted circle. As is evident from the figure,
the magnitude and the sense of rotation is the same as in the figure on the left.


11.2    Now we generalize the result of the problem above. On the right in the figure below,
we show the same rotation as in the figure on the left carried out by translating the opposite
end of the diameter of the disc first to its new position, as shown by the straight arrow and
then carrying out the rotation about the this point.         We have made the disc in this
intermediate position by dotted circle. As is evident from the figure, the magnitude and the
sense of rotation is the same as in the figure on the left. Similarly, it can be shown about any
other point.




                                            162
                              
                                                                 




                                                             
11.3 The angular momentum L about the origin is given by L  mr  v . Thus the answers in
      different cases are
                                                                            
                                                             
          ˆ ˆ
(i) L  2i  2i  0                  j ˆ        ˆ
                          (ii) L  2 ˆ  2i  4k                         ˆ j       ˆ
                                                         (iii) L  2 ˆ  2i  ˆ  4k
                                                                     j

                       
      
           ˆ    j     ˆ j     ˆ   ˆ
(iv) L  i  2 ˆ  2i  ˆ  k  4k  3k  ˆ



11.4 The position of the wheel and the stone stuck to its periphery is shown in the figure
     below.




                      Y




                                     Vt

                                  x(t)
                                                    t
                                           y(t)
                                                                                        X




                                           163
                                                                                  
Thus we have for the position r (t ) , velocity v (t ) (obtained by differentiating r (t ) with respect
                                                                          
to time once) and acceleration a (t ) (obtained by differentiating r (t ) with respect to time
twice) of the stone
                                       
                                       r (t )  Vt  R sin t i  R cos t ˆ
                                                                ˆ            j
                              
                              v (t )  V  R cos t i  R sin t ˆ
                                                       ˆ             j
                                                            ˆ
                                       a (t )   2 R sin t i   2 R cos t ˆ
                                                                              j
As is evident, the acceleration is towards the centre of the wheel. It is the centripetal
acceleration.    The force for this is provided by the force that keeps the stone stuck to the
wheel. This force is
                                                          ˆ
                                F  ma (t )  m 2 R sin t i  m 2 R cos t ˆ
                                                                              j
                                
The angular momentum L (t ) of the stone with respect to the origin is
                      
                                                               j    
     L (t )  mr (t )  v (t )  m Vt  R sin t i  R cost ˆ  V  R cost i  R sin t ˆ
                                                   ˆ                              ˆ             j   
                               
                              m VRt sin t  R 2  VR cost k    ˆ  
This gives
                                              
                                             dL                   ˆ
                                                 mV  2 Rt cos tk
                                             dt
The angular momentum changes due to the torque provided by the force that keeps the stone
stuck to the wheel and is equal to the centripetal force. Thus the torque
                 
                                                   j       
                   m Vt  R sin t iˆ  R cost ˆ   2 R sin t iˆ   2 R cos t ˆ
                                                                                       j   
                     mV 2 Rt cost kˆ

Thus it is equal to the rate of change of the angular momentum.


11.5 We assume that at time t = 0, the paricle is on the x axis. Thus after time t, its position
      vector is give as (see figure)


                                       
                                       r (t )  R  R cos t i  R sin t ˆ
                                                               ˆ            j




                                                      164
                                      Y


                                                        R
                                                            t
                                      O                                       X




                                                                    
Thus its velocity and acceleration are (obtained by differentiating r (t ) with respect to time
once for the velocity and twice for the acceleration)
                                                   
                             ˆ
         v (t )  R sin t i  R cos t ˆ
                                           j                              ˆ
                                                   a (t )   2 R cos t i   2 R sin t ˆ
                                                                                           j
As expected, the acceleration is the centriprtal acceleration. It is provided by the external
force towards the centre. The angular momentum of particle with respect to the origin is
                     
                                                            j   
         L  mr (t )  v (t )  m R  R cost i  R sin t ˆ   R sin t i  R cost ˆ
                                                ˆ                            ˆ            j  
            mR 2 1  cost k   ˆ

Its time derivative is
                                           
                                          dL                   ˆ
                                              m 2 R 2 sin tk
                                          dt
The torque due to the external force is
                   
                                                          j 
         mr (t )  a (t )  m R  R cost i  R sin t ˆ    2 R cost i   2 R sin t ˆ
                                              ˆ                              ˆ                j   
          m 2 R 2 sin t k   ˆ

This is the same as the rate of change of angular momentum.


11.6 The angular momentum for a collection of particles about an origin O is
                                               
                                   LO   mi ri  vi
                                                   i
                                                                                      
Now let is choose a different origin O’ such that the position vector of O’ from O is R (see
figure) so that
                                                      
                                      ri  R  ri ' and vi  vi'



                                                 165
                                                                      Z’
                                Z

                                                          
                                                          ri '
                                           
                                           ri
                                                     
                                                     R               O’                  Y’


                                O                     Y
                                                          X’


                      X
                                                         
Substituting the expressions above in the formula for LO , we get

                                                     
                                                                     
               LO   mi ri  v i  mi R  ri '  v i  R   mi vi   mi ri '  vi'
                       i               i                         i         i

                                  
where we have used the fact that vi  vi' .
Now the angular momentum about O’ is
                                             
                                 LO '   mi ri '  vi'

Thus if the total momentum of the system is zero, then
                                                          
                              mi vi   mi vi'  0 and LO  LO'
                                i               i




11.7 At the maximum distance Rmax and the minimum distance Rmin from the sun, the
      velocity vector is perpendicular to the radius vector. If the speed of the earth at these
      distances is Vmax and Vmin, respectively, then the angular momentum of the earth at
      these two points is MearthRmaxVmax and MearthRminVmin , respectively. By the conservation
      of angular momrntum we have
                                    MearthRmaxVmax = MearthRminVmin
      This gives
                                     Vm ax Rm in 1.47
                                                     0.967
                                     Vm in Rm ax 1.52




                                                    166
                                              
11.8 Initial momentum p i , the final mpmentum p f and the change in the momentum
               
      p  p f  pi are shown in the figure below.




                                                                          Y
                                         pf                     p


                                                1
                                                            
                                                    2
                                                                                            X
                                                    
                                                    pi

                                                                            
As is evident from the figure, the magnitude of p is 2 p sin   where p  pi  p f and it
                                                              2
is in the direction bisecting the angle     between the incident and the scattered
direction.   Mathematically it can be seen as follows. Choose the direction of incoming
particle to be the x direction. Then
                                             
                                   ˆ
                             pi  pi and                  ˆ
                                              p f  p cosi  p sin ˆ
                                                                     j
                              
                             p  pcos  1i  p sin ˆ
                                              ˆ          j
                                             ˆ               
                                  2 p sin 2 i  2 p sin cos ˆ  j
                                             2             2   2
                                                 ˆ         
                                  2 p sin   sin i  cos ˆ  j
                                           2      2         2 
                                                                   ˆ            ˆ .
Thus the magnitude of p is 2 p sin and it is in the direction   sin i  cos       j
                                   2                                 2        2      
(ii) Since the paticle is moving parallel to the x axis at a distance d, its angular momentum is
mvd going into the plane of the paper. This can be seen as follows.          The position and
velocity vectors of the particle are
                                                          ˆ
                                            ˆ j
                                       r  xi  dˆ and v  vi
This gives for the angular momentum
                                                 ˆ
                                  L  mr  v  mvd k
Further, since the force is central, the angular momentum about the origin is a constant.


                                              167
               
             dp                                                                       
(iv)   Since       F , we have p   Fdt . Therefore  Fdt is in the same direction as p .
             dt
                
To calculate  Fdt , we change the integration over time to integration over the angle by

using the angular momentum conservation. Using polar coordinates, we write the angular
                                                         d ˆ
momentum of a particle moving in the xy plane as L  mr 2    k . Since in the present
                                                          dt
                                         ˆ
problem, it is a constant equal to  mvd k , we have

                                  d                              r2
                               mr 2
                                      mvd                dt   d
                                  dt                              vd
          
The force F between the two charges is in the radial direction and is equal to

                                                                    
                                          Qq      ˆ
                                      F  k 2 cos i  sin  ˆ
                                                             j
                                           r
Thus
                                 
                                                              
                                   Qq                     r2
                          Fdt   k 2 cos i  sin  ˆ   d
                                   r
                                            ˆ         j
                                                           vd
                                           
                                k
                                   Qq
                                   vd 
                                               
                                       cos iˆ  sin  ˆ d
                                                        j     
                                k
                                   Qq
                                   vd
                                           
                                       sin i  1  cos   ˆ
                                             ˆ                j          
                                   Qq                ˆ                   ˆ
                                k     2 cos   sin i  cos                j
                                   vd        2        2        2             
                                         Qq    
Thus the magnitude of    Fdt     is 2k
                                          vd
                                             cos . Comparison of this expression with that for
                                                2
the momentum change shows that the two are in the same direction and
                                                         Qq     
                                      2 p sin         2k    cos
                                                   2      vd     2
Or
                                           Qq        Qq       
                                  d k        cot  k    2
                                                           cot
                                           pv    2    mv       2
This is the Rutherford formula.




                                                    168
11.9 (i) At angle  1 the potential energy of the girl is MgL1  cos1  
                                                                                              1
                                                                                                MgL12 with
                                                                                              2
         respect to the equilibrium point, i.e. when the swing is in the vertical position. The
         kinetic energy of the system at the lowest point , if the angular speed is 1 , is
         1
           ML212 . By energy conservation
         2
                              1         1                                    g
                                ML212  MgL12          1  1
                              2         2                                    L
 (ii)       Whe the child stands up at the lowest point, all the external forces on the her are
            passing through the pivot so her angular momentum remains unchanged. However
            as she stands up her moment of inertia about the pivot is M L  d   I CM where
                                                                                              2



            ICM is her moment ofinertia about her CM. Neglecting it gives the moment of
            inertia to be M L  d  . Then by conservation of angular momentum
                                   2



                                                                L2                  2d 
                    M L  d  2  ML21  2                          1  1       1
                              2

                                                             L  d 2
                                                                                     L 
                                            d
            keeping only linear terms in      since d<<L.
                                            L
 (iii)      The final kinetic energy of the girl is
                                                                         2
                                                   2    2d  2
                                    I  M L  d  1 
                                  1 2 1
                                                             1
                                  2      2               L 
                                       1
                                                        
                                                     4d  2
                                       M L2  2dL 1      1
                                       2                 L 
                                            1         2d  4d 
                                             ML212 1    1    
                                            2            L     L 
                                            1         2d 
                                             ML212 1    
                                            2            L 
                                    d
 keeping only linear terms in         since d<<L. Thus we see that the kinetic energy of the
                                    L
 system has increased in comparison with the original energy. Threfore the swing would go
 higher on the other side.




                                                169
(iv)   Suppose the swing goes up to angle  2 on the othwr side, we have by energy
                                          1         1
       conservation and the fact that       ML212  MgL12
                                          2         2
                      2d  1                                     L  2d  2
                              Mg L  d  2
            1
              ML212 1                     2
                                                          22         1    1
            2            L  2                                   L  d     L 
Now we have the approximation d<<L, which gives
                         L  2d   d  2d                  3d
                               1      1  1      1
                         L  d     L       L    L        L
                               d
keeping only linear terms in     . Thus
                               L
                                           1
                                 3d  2           3d 
                           2  1   1   2  1  1
                                   L             2L 
This gives the increase in the angle of swing in half a period. While going back, the new
amplitude  1 by the time the swing reached its starting point will be.
                                                              2
                                       3d        3d        3d 
                        1   2 1         1 1    1 1   
                                       2L        2L           L
In the full period, the net increase in the amplitude will therefore be
                                                      3d
                                          1  1       1
                                                       L




                                               170
                                        Chapter 12


12.1 Moment of inertia of a ring. Let the mass of the ring be m and its radius R. Since all
the points on the radius are at the same distance from the axis, each small mass m on the
periphery gives the same contribution mR 2 to the moment of inertia. Thus the total
moment of inertia is
                                    I   mR2  mR2

A disc can be thought of made of many rings of width dr at different radii (see figure).
                                                                                M
                            The moment of inertia of each ring is r 2              2 rdr . Thus
                                                                                R2
                            the total moment of inertia is

                                                            M 3        1
                                                     I      2
                                                                2r dr  MR 2
                                                            R          2


For the moment of inertia about the diameter of a ring, consider a small portion of it of

extent d at and angle  It moment of inertia about the diameter is
                                                                          M
                                                                             R cos 2 Rd .
                                                                         2R




                                                            




                                           2
                                    MR 2                      1
                                            cos      d 
                                                 2
Thus the total moment of inertia is                             MR 2
                                    2     0
                                                              2


12.2 Moment of inertia of a rectangular sheet. The sheet is shown in the figure below


                                           171
                                         Z

                                                              Y

                           y

                                                                    b       X


                                     a



We first calculate its moment of inertia about the x axis. For this, let us take a thin strip of
                                                                                M         M
width dy parallel to the x axis and at a distance y from it. Its mass is           a dy    dy .
                                                                                ab        b
Since the perpendicular distance of all its points is y from the x-axis, its moment of inertia
                          M
about the x axis is y 2     dy . Thus the total moment of inertia about the x-axis is
                          b
                                                 b
                                           M         2
                                                              Mb2
                                      Ix         y dy 
                                                    2

                                           b     b
                                                              12
                                                      2

Similarly by taking a thin strip parallel to the y axis, we will get the moment of inertia Iy
about the y axis to be
                                                 a
                                           M         2
                                                          Ma 2
                                      Iy          y dy  12
                                                          2

                                           a     a
                                                      2

Let us now calculate the moment of inertia about the z axis. Consider again the strip
parallel to the x axis.    By parallel axis theorm the moment of inertia of this strip about the
z axis is = (moment of inertia of its CM with respect to the z-axis + its moment of inrtia
about the axis passing through its CM and parallel to the z-axis)
                                                M      1   M
                                   dI z  y 2     dy  a 2   dy
                                                b     12   b
which upon integration leads to

                                          Iz 
                                                 M 2
                                                 12
                                                    a  b2   


                                                172
12.3 The disc and a strip on it perpendicular to the axis of rotation (y-axis) is shown in the
      figure below. The strip is of width dy and at a distance y fron the centre.

                                         Y




                                                   R              y
                                                                                 X




                                                           2M
Length of this strip is 2 R 2  y 2 and its mass is                   R 2  y 2 dy . Thus its moment of
                                                            R2
inertia about the y-axis is

              dI y 
                        1
                       12
                                    
                           4 R2  y2 
                                        2M
                                         R2
                                                  R 2  y 2 dy 
                                                                    2M
                                                                   3 R 2
                                                                            
                                                                          R2  y2       3
                                                                                             2
                                                                                                 dy

The integration is easily carried out by substituting y  R cos and gives the resuly
                                                   1
                                           Iy       MR 2
                                                   4


12.4 This problem can be done exactly in the same manner as above. We now divide the
      shell inti thinn rings at an angle  (see figure).




                                                       




                                             173
                                                                                        M                       M
The radius of the ring is r cos and therefore its mass is                                    2 r cos  rd    cos d
                                                                                       4 r 2
                                                                                                                2

and its moment of inertia about the rotation axis is
                                                                           M
                                               dI  r 2 cos2               cos d
                                                                           2
Thus the total moment of inertia is
                                                    
                                       Mr 2             2
                                                                           Mr 2 4 2
                                    I                cos  d                 Mr 2
                                                          3

                                        2           
                                                                            2   3 3
                                                         2




12.5 In this problem, we divide the sphere into thin discs and add the moments of inertia of all

     discs. Thus for a disc of thickness dy at height y (see figure), its radius is                          R 2  y 2 , its
     mass is
                                        M
                                      4R 3
                                         3
                                             R 2  y 2 dy  
                                                             3M 2
                                                             4R 3
                                                                       
                                                                  R  y 2 dy .                 

                                                                 Y




                                                                           R            y
                                                                                                    X




     Therefore, the moment of inertia of the disc is

                         dI 
                                    1 3M 2
                                    2 4R
                                                
                                      3 R  y 2  R 2  y 2 dy  
                                                                  3M 2
                                                                  8R 3
                                                                               
                                                                               2
                                                                       R  y 2 dy .                
     Integrating it from R to R gives the total moment of inertia

                                                    3M  5 2 R 5 4 R 5  3M 16R 5 2
                          R             
                         R
                  3M
               I               2
                                    y   2 2
                                               dy       2R 
                                                      3 
                                                                      
                                                                        8R 3  15  5 MR
                                                                                          2

                  8R 3   R                         8R        5   3 



                                                                     174
     12.6 For the moment of inertia about the y-axis, the problem is done exactly in the same
           way as in 12.5 by taking a disc at height y. The only difference is that the mass M is
           now distributed over half the volume and the y integration runs from 0 to R. Thus the
           mass of the disc is
                                        M
                                      2R 3
                                         3
                                                  
                                             R 2  y 2 dy 3M 2
                                                             2R 3
                                                                    
                                                                  R  y 2 dy 
     And its moment of inertia is

                           dI 
                                   1 3M 2
                                   2 2R
                                                      
                                     3 R  y 2  R 2  y 2 dy 3M 2
                                                                 4R 3
                                                                             2
                                                                      R  y 2 dy 
     Thus the total moment of inertia is given as

                                                 3M  5 R 5 2 R 5  3M 8R 5 2
                        R        y 2  dy 
                       R
                3M
                                                     R          
                                       2
             I               2
                                                   3 
                                                                  4 R 3  15  5 MR
                                                                                      2

                4R 3   0                         4R      5   3 


Similarly, to calculate the moment of inertia about the x axis, we make a cylindrical shell as we
did in example 12.3 (see figure)

                                                  Y




                                                            R                y

                                                                                     X
                                                            



Thus the moment of inertia will be calculated exactly like in example 12.3. The mass of this

shell is given by

                                            1   M
                                       dm             2 y dy  2 
                                            2 2 R 3 3
                                            3M
                                            3 R 2  y 2 y dy
                                            R




                                                       175
               1
Th factor of     comes because there is only half the shell in a hemisphere. Therefore the moment
               2

of inertia of the hemisphere is

                                                       R
                                                  3M                                2
                                I   y 2 dm          y
                                                               3
                                                                   R 2  y 2 dy      MR2
                                                  R3   0
                                                                                    5

12.7 A conical shell is shown below. To calculate the moment of inertia, we take a ring at a

      distance x from the base extending from x to x+dx. Then by similarity fo triangles, its

      radius is y  r
                        h  x  .   Similarly the length of its side is
                                                                                     dx
                                                                                          
                                                                                                h
                                                                                                         dx
                           h                                                        cos      h2  r 2


                                                Y
                                                           x
                                           r
                                                                                                        X


                                                                        h



                                                     h2  r 2
                                                              dx  2 h  x dx . Its moment
                                                 m                2m
Thus the mass of the ring is dm             2 y 
                                  r h r
                                      2   2            h          h

of inertia therefore is
                                                                    2
                                   dI  y 2 
                                                2m
                                                   h  x dx  2mr h  x 3 dx
                                                h2               h4
The moment of inertia of the shell is then given by integrating the expression above over x from
0 to h and gives


                                h  x dx  h 4  h  3h x  3hx  x dx  2
                               h                       2 h
                     2mr 2                   2mr                             mr                      2
                   I 4
                                        3             3    2       2   3

                      h        0                           0

Now we calculate the moment of inertia about the y-axis. To do this we use the parallel axis
theorem and write the moment of inertia of the ring considered above as that of its CM plus its
moment of inertia about the axis parallel to the y axis and passing through the CM. Thus



                                                           176
                                                                             
                                                                                   2
        dI  x 2 
                     2m
                        h  x dx  1 y 2  2m h  x dx  2m hx 2  x 3 dx  mr2 h  x 3 dx
                     h2              2       h2              h2                 h
Integrating this we get
                                                      mr 2 mh 2
                                               I         
                                                       4    6
We now do the calculations for a solid cone. The calculations proceed in exactly the same
manner as for the shell bu now instead of a ring, we take a disc at distance x. The mass of the

                      y 2 dx  3 h  x 2 dx . Thus its moment of inertia is
                 m             3m
disc is dm 
               r h 3
                 2
                               h

                                                            3mr 2
                                        y  3 h  x  dx        h  x 4 dx
                                      1 2 3m
                              dI 
                                                      2
                                                                5
                                      2    h                 2h
Thus the total moment of inertia is
                                                  h
                                          3mr 2
                                                   h  x  dx  10 mr
                                                                     3
                                       I
                                                             4            2

                                           2h 5   0

For the moment of inertia about the y axis, we again consider the same disc and use the parallel
axis theorem to write the moment of inertia of the ring considered above as that of its CM plus
its moment of inertia about theaxis parallel to the y axis and passing through the CM. So
                                                                                         2
      dI  x 2 
                   3m
                      h  x 2 dx  1 y 2  3m h  x 2 dx  3m x 2 h  x 2 dx  3mr5 h  x 4 dx
                   h3                4       h3                h3                     4h
Now the integration over x from 0 to h gives
                                               1        3
                                                 mh 2     mr 2
                                              10        20


12.8 (i) The total angular momentum = angular momrntum of large disc + angular momentum
     of small disc
                                       L  I1 (large disc)  I 2 (small disc)
                                                             1
                                                      I1      MR2
                                                             2
                                                                             1
     I2 is calculated using the parallel axis theorem and its value is md 2  mr 2
                                                                             2

     Thus L 
                 1
                 2
                                 
                   MR2  mr 2   md 2 



                                                       177
(ii)     Let the large disc start rotating with angular speed  in the opoosite direction.
         Then by conservation of angular momentum
                                                              mr 2
           
               1
                                      1
                 MR2  mr 2   md 2   mr 2  0   
                                                                                     
               2                        2                MR2  m r 2  2d 2
(iii)    For M  0 and d  0 , we have
                                             
This keeps the total angular momentum zero since the large disc rotates in the opposite
direction with exactly the same angular speed as given to the small disc.
(iv)     In a helicopter also, as the large rotor starts, the body of the helicopter will tend to
         rotate in the opposite direction. To prevent this, a tail rotor is fitted that provides
         the counterbalancing torque.


12.9 The position of the person (represented by the filled circle) after time t is shown on
        the platform in the figure below.




                                                                           B
                                                                     u


                                                      O    ut


                                                                 A




As the person moves on the platform, the platform starts rotating clockwise with angular
speed  so that the total angula momentum remains zero. As the platform rotates, the
                                              
person also moves with it. The net velocity v of the person with respect to the ground is
                                                                                     
equal to (velocity u of the person with respect to the platform + rotational velocity v rot of
the person due to the rotation of the platform). This is expressed as
   
v  u  v rot

Thus the angular momentum of the person is


                                            178
          
l p  M r  u  vrot 
       
Here r is the position vector of the person from the origin O on the axis.              The position,
various distances and the velocities, as seen from the top, are shown in the figure below.


                                                                               B




                                                                           
                                                                           u

                                                O              a 2


                                                               
                                                               r

                                                                                  ut
                                                                   v rot


                                                                               A



It is clear from the figure that

    a a
            2
                  
                         2
                                       a                                   a  2  a   
                                                                                             2

 r       ut                   r u  u            r  vrot   r        ut  
                                                                        2

     2 2                               2                                   2   2
                                                                                            
Here the sign takes care of the ddirection of angular momentum along the aixs of rotation;
it is positive for counterclockwise rotation and negative for clockwise rotation. Thus the
angular momentum of the perso is

                                     Mau       a  2  a   
                                                             2

                              lp         M      ut  
                                      2        2   2
                                                            
The angular momentum of the platform is
                                                      ma 2
                                         l plat          
                                                       6
Now equation the total angular momentum to zero (by conservation of angular momentum)
gives



                                            179
                          Mau       a  2  a    ma
                                                  2      2
                               M      ut        0
                           2        2   2
                                                   6

This gives
                                                            Ma u 2
                                 (t ) 
                                           ma    2 a  2  a   
                                                                 2

                                               M      ut  
                                            6      2   2
                                                                
                                                                     a
Total time that the person takes to move from A to B is
                                                                     u
Thus the angle  through which the platform rotates by the time the person reaches B is
                           au               au
                                                               Ma u 2
                        (t )dt                                               dt
                                                     ma 2      a  2  a   
                                                                             2

                                                           M      ut  
                           0                 0

                                                      6        2   2
                                                                            
This integration can be done by substituting
                                          a
                                     z      ut         and     dz  udt
                                          2
So we get
                     a2                                         a2
                           Ma 2          a           1
                                  dz                      dz
                  a 2 
                         m M 2          2 a 2  m 1  2
                         a  Mz                  a  z
                                   2                         2

                       6 4                     6M 4 

                      m 1                            m 1
Now substitute z  a      tan  which gives dz  a      sec2  d to get
                      6M 4                           6M 4 



                                                  m   1
                                            0a        sec2 
                                       a          6M 4 
                                                              d
                                       2  0  m    1 2
                                                    a sec 
                                                             2

                                               6M 4 


                                     1
                                 
               1 m   1 2
where tan  0        . This gives
               2  6M 4 




                                                      180
                                                                0
                                          
                                                            m 1
                                                               
                                                            6M 4 
                                             
If m = 0, we have  0              and             . Thus although on the platform, the person has
                                4                 2
moved from A to B but overall he remains at his original position since the platform has
                        
rotated back by           .
                        2
For m  M , we have
                                                           1                     1
                                                                            
                                 1 m   1                 2    2    4m        2           m
                        tan  0                             1                 1
                                 2  6M 4                      2  6M                     3M
             
Thus  0          , where   1. We can therefore write
             4
                                                     
                        tan  0  tan     tan  sec2    1  2
                                     4          4      4
              m
Thus  
             6M
This gives
                                              1
                                          
                  m  1  m                2       m     m    m                            
                                               1                                  1
                4 6M  4 6M                      2 3M  3M  2 3M                           2   


12.10 This is similar to the variable mass problem except that in this problem we will get
     the answer by applying the principle of conservation of angular momentum because
     this is an extended body and if we try to apply Newton’s second law to each point
     particle, it is impossible to solve the problem because of a large number of internal
     forces involved.
         Let a small mass m come out at a given time t during the time interval from t to
     t+t from one of the containers. Let the angular speed of the display be  at t. Then
     the speed of gases coming out with respect to the ground is u  l  , as shown in the
     figure




                                                      181
                                                               u-l

                                 u-l
                                                     




If in time t the angular speed of the display becomes    , then by conservation of
angular momentum
                      4  m l 2  4m  m l 2      4ml u  l 
Here factor of 4 comes because of 4 containers. Here m is the mass of each container at
time t. Obviously m  M  m   t . Neglecting the second order term m , we get

                          m l 2   mlu                     m l  mu
Thus
                                              d dm u
                                                 
                                              dt   dt m l
                                                             dm
This equation can not yet be integrated because                 should be written in terms of m for
                                                             dt
                  dm    dm
that. Obviously          . This gives
                  dt    dt
                                            d    dm u
                                               
                                            dt    dt m l
Integrating this gives, starting from  = 0 at t = 0
                                         M  m t
                                     u               dm u  M  m 
                           (t )  
                                     l      
                                          M m
                                                      m
                                                         ln           
                                                         l  M  m  t 
                                                                       




Thus after the entire gubpowder is exhausted, we have
                                                 u M m
                                                ln   
                                                 l  M 


                                                 182
     12.11 This problem is exactly the same as the problem above except that the moment of
           inertia of the display is different because of different positioning of the containers of
           gunpowder. Conservation of angular momentum now gives
                                     2                                                 2
                           l                                         l
               4m l   4m     4m  m l 2      4m  m      
                    2

                           2                                         2
                                                           l       l
                                    4ml u  l   4m  u   
                                                           2       2
     Neglecting second-order terms and simplifying leads to
                         2      l2           lu  5 2       3
                         m l  m   m lu   
                                                     m l   mlu
                                4             2  4         2

     Following the steps in the solution of problem 12.10, this leads to
                                     6u  M  m                 6u  M  m 
                           (t )          M  m  t  and   5l ln  M 
                                       ln             
                                     5l                                  


12.12 As the water comes out of the sprinkler pipes, it carries angular momentum with it. If the
     opposing torque were not there, the system will rotate in the opposite direction with
     increasing angular speed to conserve the angular momentum. However it does not happen
     because of the opposing torque which is the external torque on the system. Taking the
     direction of rotation of the rods to be positive, we have
                                     Lt  t   L(t )  external torque  t
     Here external torque is   with the minus sign indicating that it is in the direction
     opposite to the rotation. Since the mass of the rod remains unchanged, if the mass of water
     coming out from each end in time interval t is m , we have

                            Lt  t   2 
                                                ml 2
                                                          4  m l  u  l 
                                                                                  
                                                12                      2      2

                                         
                                             ml 2
                                                       2ml u  l 
                                                                          
                                              6                        2
     and
                                                         ml 2    ml 2
                                             L(t )  2             
                                                         12       6


                                                     183
Thus we have
                               ml 2               l 
                                      2ml  u     t
                                6                 2 
This gives
                                 ml 2 d    dm      l 
                                         2    l u    
                                  6 dt      dt      2 
                                                dm
Since the area of each hole is S, we have           Su , where  is the density of water. In
                                                dt
                                                                                      d
steady state, the sprinker system rotates with constant angular speed so that             0.
                                                                                      dt
               dm           d
Substituting       Su and     0 in the equation above gives
               dt           dt
                                               2u     
                                                
                                                l   Sul 2
Assuming that at the beginning of turning the spinker on, all pipes are filled, we also solve
how the angular speed changes with time until it approaches the above vaue.               The
differential equation for  can be written as
                                 d 6 Su    12 Su 2 6 
                                                   2
                                 dt   m         ml    ml
This equation has the solution

                                          6Su   2u    
                            (t )  A exp    t  
                                                    l        
                                           m        Sul 2 
                                                              
Since  (t  0)  0 , we get

                                             2u    
                                       A   
                                             l         
                                                Sul 2 
                                                        
This gives
                                2u                 6Su 
                           (t )           1  exp 
                                           2 
                                                              t
                                l
                                     Sul               m  
                                                          
12.13 Let the angle between the unit vector n and vector r be  (see figure).
                                             ˆ




                                               184
                                                                        ˆ
                                                                        n

                                                                        


                                                                         
                                                                        r
                                                  
                                                           
                                                           r
                                    O




                                                     
It is clear from the figure that the magnitude of r is r sin   and it is in the direction of
                                                                       
n  r . Since the magnitude of n  r is also r sin   , we get r  n  r .
 ˆ                                 ˆ                                  ˆ


12.14As the disc hits the rough surface, let the surface apply an impulse J on it (see figure).


                                                                    

                                              V                                        V1
           J


As a consequence the disc moves with a smaller velocity and also starts rotating because
the impule applies a torque on it about its CM. By rolling condition we have V1  R .
By the equation for the CM, we get
                                        J  mV  mV1
And by the torque equation
                                 1            1     1
                          JR      mR 2  J  mR  mV1
                                 2            2     2
               2
This gives V1  V
               3



                                           185
12.15 Free body diagram of the disc is given below.




                                                                  F
                          f


Here f is the frictional force on the disc. By the equation for the motion of its CM we get
                                             F  f  ma
Taking torque about the CM of the disc we get
                                         1                     1
                                  fR      mR 2        f      mR
                                         2                     2
For pure rolling we have a  R . Substituting this in the above two equations we get
                                                    2F
                                              a
                                                    3m


12.16 (i) Free body diagram of the cylinder is shown below.



                              d                                    J


                          Jf


Here Jf is the frictional impuse on the disc. By the equation for the motion of its CM we
get
                                            J  J f  MV

Similarly the angular momentum equation about the CM gives
                                                    MR 2
                                          Jd  fR       
                                                     2
                                                    MRV
                                                  
                                                     2




                                              186
Where the second equality follows from the rolling condition V  R . Solving the above
two equations leads to
                                           3MV    
                                       d       1 R
                                           2J     
(ii)    For a sphere the equations are the same as for a cylinder except that its I about the
        CM is different. Thus the equations become
                                           J  J f  MV

                                                      2MR 2
                                       Jd  fR             
                                                         5
                                                      2 MRV
                                                    
                                                         5
Their solution gives
                                           7 MV    
                                       d        1 R .
                                           5J      


12.17 The rod in example 12.11 bounces back with angular speed  and speed V. Before it
       impacts the ground, its angular momentum about the point of impact is equal to the
       angular momentum of its CM because it is not rotating, and it is pointing into the
       page (see figure).


                                 l/2
                                       

                                                V

                                                    
                            h




Thus

                                                ml 2gh
                                   Linitial           cos 
                                                   2




                                            187
After the impact, the rotation about CM gives angular momentum coming out of the page
and the translational motion gives angular momentum going into the page. Thus
                                                ml 2    mlV
                                    L final              cos 
                                                12       2
By conservation of angular momentum, we have

           ml 2    mlV         ml 2 gh
                     cos           cos                l  6V cos   6 2 gh cos 
           12       2             2
By energy conservation we have
                1 ml 2 2 1
                        mV 2  mgh                               l 2 2  12V 2  24 gh
                2 12     2
These are two equations for two unknowns. Substituing l  6 cos V  2gh from the          
first equation into the second equation, we get the equation

                                                          
                   V 2 1  3 cos2    6 2gh cos V  2gh1  3 cos2    0
This is a quadratic equation in V, and its solution is
                                                1  3 cos2  
                                               1  3 cos2  
                                    V   2gh                 
                                                              
The plus sighn in front of 1 above gives the trivial solution that the velocity of the rod
before impact is       2 gh downward. And after the impact its velocity is gives by the second
root which is
                                              1  3 cos2  
                                              1  3 cos2  
                                     V  2gh               
                                                           

                               
From l  6 cos V  2gh , this gives

                                                12 2 gh cos
                                           
                                                l 1  3 cos2  
                                                      
It is evident from the equations above, for              , the rod rebounds without any rotation.
                                                       2




                                                188
12.18 This problem is similar to the problem 12.17 . However, the CM and the moment of
      inertia about the CM are different in this problem. In this case the CM is at distance
      3ml
           l from mass m. Thus the moment of inertia about the CM
      3m
                                  I CM  2ml 2  m  4l 2  6ml 2

Like the previous problem, before the system impacts the ground, its angular momentum
about the point of impact is equal to the angular momentum of its CM because it is not
rotating, and it is pointing into the page. Its value is

Linitial  3ml 2gh cos
Let the system bounce back with angular speed  and speed V after the impact. Then
                                  L final  6ml 2  3mlV cos
Thus angular momentum conservation about the point of impact gives

          6ml 2  3mlV cos  3ml 2gh cos  2l  V cos  2gh cos
Similarly energy conservation gives
              1           1
                6ml 2 2   3mV 2  3mgh                         2l 2 2  V 2  2 gh
              2           2

Again we substitute l 
                            1
                            2
                                            
                              V  2 gh cos in the second equation to get

                       1          
                  V 2 1  cos 2                           1          
                                             2 gh cos V  2 gh1  cos 2    0
                       2                                      2          

Its solution gives one trivial solution      2 gh and for speed after impact

                                                1        
                                            1  cos  
                                                      2

                                   V  2gh      2        
                                            1  1 cos2  
                                                         
                                                2        
and

                                                   2 2 gh cos
                                         
                                                    1         
                                                 l 1  cos2  
                                                    2         




                                                 189
12.19 The figure below shows the rod and the stone just before the impact.




                                     3l
                                     4
                                                     l




(i)    Right after the impact the angular momentum about the pivot is conserved. Thus,
       if the rod and stone stuch with it move with angular speed  after the impct, we

                                                                      3l  2 ml 2 
       have (moment of inertia of the rod and the stone together is m0          )
                                                                     4
                                                                               3  

                                       3l   3l    ml 2 
                                                   2

                                 m0 v 0   m0         
                                       4  4
                                                     3  
            m
With m0      and v0  gl , we get
            3
                                                12   g
                                          
                                                25   l
(ii)   The kinetic energy of the rod and the stone system after the impact is

                   1   3l    ml 2  2 1 25 2  12  g
                              2                        2
                                                           3
               KE  m0              ml           mgl
                   2  4 
                                3     2 48     25  l 50

If the rod rises by an angle , the potential energy of the system at this point is

                 PE  mg
                            l
                              1  cos   m0 g 3l 1  cos   3mgl 1  cos 
                            2                    4                 4
Equating the KE and PE, we get
                                  cos  0.92    23




                                            190
(iii)      The impulse by the pivot. As soon as the stone hits the rod and gets stuck with it,
           the horizontal momentum of the system changes. This change is brought about by
           the impulse that the pivot applies on the rod. Thus by calculating the change in the
           momentum of the system, we obtain the impulse that the pivot applies on the rod.

                                                m gl
Initial momentum of the system = m0 gl 
                                                 3
Final momentum of the system =
                    momentum of the CM of the rod  momentum of the stone
                        l      3l
                     m   m0 
                        2      4
                      3ml 12 g
                         
                       4    25 l
                        9
                         m gl
                       25
Thus after the stone gets stuck, the momentum of the system changes by
                                    9 1       2
                                     m gl     m gl
                                    25 3     75
in the same direction that the stone was moving initially. Thus the impulse given by the
pivot is
                                    2               2
                                       m gl    or      m0 gl
                                    75              25
in the same direction that the stone was moving initially.
It is also instructive to solve this problem by angular momentum consideration by
calculating the change in the angular momentum of the system and attributing it to the
torque provided by the impulse. For this we apply the equation
                                       dLCM
                                              applied,CM
                                        dt
Here CM is the centre of mass of the entire system (rod+sone). Note that this equation can
be applied only about the CM of the system and NOT about the CM of the rod alone.


                                                           1  l m 3l  9l
              Distance of the CM from the pivot point =        m    
                                                          4m 3  2 3 4  16



                                            191
                           9l l  l
Thus the CM of the rod is         above the system CM. Similarly the stone is
                           16 2  16
 3l 9l  3l                                                 m0 v 0  v
         below the system CM. The velocity of the CM is          0 in the
 4 16  16                                                 m  m0    4
same directon as the stone. Thus befor the stone strikes the rod, it is moving with respect
                                    3v 0
to the system CM with speed              in the same direction and the CM of the rod is moving
                                     4
                v0
with speeed       (in the opposite direction). This is shown in the figure below in the
                4
system CM frame.




                                                        9l
                                                        16
                                       v0
                                       4

                         3v 0
                          4



Thus the angular momentum of the system in its CM before the impact is
                Linitial  Lrod  Lstone
                          ( L of CM of rod )  Lrod (about CM of rod )  Lstone
                                l v0         3l 3v
                          m      0  m0   0
                               16 4          16 4
                           mlv0
                         
                            16
                                            m
where we have used the fact that m0          . The sense of rotation is counter clockwise.
                                            3
Immediately after the stone hits the rod, rod starts moving with speed
                                     9l     9l 12v0   27
                                                    v0
                                     16     16 25l 100




                                             192
With respect to the CM of the system, the CM of the rod and the stone are moving with
speed (keep in mind that  is the same about any point and we are thinking of the motion
of the system as the translation of the system CM plus rotation about the system CM. Thus
with respect to the system CM, the motion is roational)
                      l      l 12v0 3v0                               3l 12v0 9v0
             vCM 
              rod
                                              and    v stone          
                     16     16 25l 100                                16 25l 100
Thus the angulat momentum of the system about the system CM after the impact
               L final  Lrod  Lstone
                         ( L of CM of rod )  Lrod (about CM of rod )  Lstone
                              l 3v0 ml 2 12v0       3l 9v
                         m                m0   0
                             16 100 12    25l       16 100
                          19mlv0
                        
                           400
                                             m
where we have used the fact that m0           . The sense of rotation is counter clockwise.
                                             3
Thus the impuse gives a torque equal to
                                                       19   1  3mlv 0
                           L final  Linitial  mlv 0      
                                                       400 16   200
The negative sign here shows that the change is in clockwise direction. Its magnitude is the
torque impulse which is equal to the impulse J provided by the pivot times the distance fo
the pivot from the system CM. For clockwise sense, J is in the same direction as the
firction of stone’s velocity. Thus
                             9l 3mlv0               2mlv0             2m0 lv0
                        J                J               or
                             16   200                75                 25


12.20 Let the distance of the sweet spot be ls. The ball hitting the bat and its swing is
     shown in the figure below.




                                            193
                                                                        F



                                             ls




                                             0

The point where the bat is held is the pivot point of the bat. Further, let the ball come
horizontally with speed vi and retun alonh the same line with speed vf. Considering
clockwise rotation to be represented by the positive direction, we have:
Initial angular momentum Li  I 0  mv i l s

Final angular momentum L f  mv f l s

Since there is no external torque about the pivot, we have
                                                                                  I 0
                  Li  L f        I 0  mv i l s  mv f l s or mv f  vi  
                                                                                   ls

Finally the momentum change of the system is caused by the force applied on the bat.
Initial momentum is the sum of the momentum of the bat and that of the ball. The final
momentum is that of the ball only. This gives
                              F  Pf  Pi  mv f  (mv i  MLc  0 )

Since for the sweet spot force is zero, we have
                                       mv f  vi   MLc  0

                                                    I 0
This combined with the equation mv f  vi             gives
                                                     ls

                                                     I
                                             ls 
                                                    MLc




                                             194
(i)      For M=2 kg, LC=40 cm and I=0.3kgm2, we get
                                         0.3
                                 ls           m  37.5cm
                                        2  .4
(ii)     From the momentum equation above, we get
                                            mv f  vi        0.154  60
                                 0                                      11.55 rad s 1
                                               MLc               2  0.4
(iii)    With the wrapping of the tape, the mass the moment of inertia of the bat and its CM
         all change. We have
                                 I new  0.3  0.050  0.65 2  0.321 kg m 2

and
              M new Lc new  MLc  0.05  0.65  2  0.4  0.05  0.65  0.8325 kg m

This gives
                                                     0.321
                                        l s new           m  38.6cm
                                                    0.8325
Thus the sweet spot is lowered by 1.1 cm.


12.21 The problem is solved as example 15.2 in chapter 15.


12.22 For generality we first take the masses on the dumbel to be of mass M each and the
        mass striking to have mass m. Since there are no external forces or torques acting on
        the system, its momentum and angular momentum remains the same before and after
        the collision. For convenience we take angular momentum about the origin.            We
        show various distances in the figure below.

                                                    Y
                                             M

                                                    l/2
                         v                     O
                                        a                                         X
                             m
                                                    l/2

                                             M



                                                     195
The momentum components of the mass m are as follows
                                                             a                  2mva
                                  p x  mv                             
                                                     a l 4
                                                         2         2
                                                                               4a 2  l 2
                                                             l 2                 mvl
                                  p y  mv                              
                                                     a2  l 2 4                4a 2  l 2
Initial angular momentum about the origin arises only from the motion of mass m. When
particle is on the x axis, only the y component of its momentum gives the angular momentum
about the origin and it is
                                               l 2                            mvla
                        Linitial  mv                        a                          clockwise
                                          a l 4
                                           2         2
                                                                             4a 2  l 2
After the mass m gets stuck with M after hitting it, the CM of the system is at

                       xCM  0        y CM 
                                                M  m l 2  M l                  2
                                                                                       
                                                                                              ml
                                                    2M  m                               22M  m 
immediately after the impact.
By momentum conservation the momentum of the system after the imact is the same as before it.
Thus the angular momentum of the CM of the system after the impact is (see figure below)

                                                     Y

                                       (M+m)                 py

                                                                        px
                                          yCM
                                          M
                                                                                       X
                                                             O



                                               M


                                                                            m 2 vla
                                LCM  yCM  p x 
                                                                 2M  m        4a 2  l 2
Since angular momentum is conserved, the rest of the angular momentum comes from the
angular momentum about the CM of the system. Thus the angular momentum about the CM is
                                                                              2Mmvla
                             LaboutCM  Linitial  LCM 
                                                                       2M  m        4a 2  l 2


                                                              196
This then gives the rate of rotation  through the equation
                                                  I aboutCM   LaboutCM

For this we first calculate I aboutCM . It is

                                                                                                   M M  ml 2
                                                        2                                2
                            l                 l              
         I aboutCM  M  m 
                                   ml                    ml
                             2 22M  m   M  2  22M  m 
                                                                                          
                                                                                                    2M  m
                                                             
Thus
               M M  ml 2        2Mmvla                                 m         2va
                                                                           
                2M  m       2M  m 4a 2  l 2                        M  m  l 4a 2  l 2
With these we easily calculate the velocity of mass (M+m) and mass M right after the impact.
These will be given by adding the velocity of the CM to their velocity due to rotation about the
CM. Thus
                          l     ml            px         Ml                                                              py
v x ( M  m )  v xCM    
                           2 22M  m    2M  m   2M  m 
                                                                                            v y ( M  m )  v yCM 
                                                                                                                     2M  m

                 l
v xM  v xCM    
                        ml            px       M  ml                                                              py
                  2 22M  m    2M  m   2M  m 
                                                                                            v yM  v yCM 
                                                                                                                2M  m
                               

For m  M these answers become
                                                                                                        Mvla
Angular momentum of the system CM about the origin after the impact = 
                                                                                                   3 4a 2  l 2
                                                                    2 Mvla
Angular momentum of the system about its CM 
                                                                  3 4a 2  l 2
and
                                       va                                                    vl
                      v x( M m)                            v y ( M  m )  v yCM 
                                     4a 2  l 2                                        3 4a 2  l 2

                                                                        vl
                      v xM  0                    v yM  v yCM 
                                                                   3 4a 2  l 2


                                                     Chapter 13




                                                            197
13.1     (i) I xx   mi  y i2  z i2   2mw 2                                          
                                                                 I yy   mi xi2  z i2  2ml 2
                         i                                                i


                                              
              I zz   mi xi2  y i2  2m l 2  w 2                  I xy  I yx   mi xi y i   mlw
                     i                                                                 i


              I xz  I zx   mi xi z i  0                    I yz  I zy   mi y i z i  0
                                  i                                                i

       Thus
                                                   2mw 2  mlw      0                         
                                                                                              
                                               I   mlw 2ml 2
                                                                     0                         
                                                    0
                                                           0   2m l  w 2
                                                                    2
                                                                                              
                                                                                               
                                                                                               
       (ii)      To find the principal axes, one rotates the frame with respect to the z-axis by an
                 angle  so that the coordinates for each particle transform by the formulae
                                                      x'  x cos   y sin 
                                                      y '   x sin   y cos 

       and the resulting coordinates make I x ' y '  0 . Numbering the masses as shown in the

       figure, we get

                                                      Y
                                       Y’
                                                      4                            3
                                                                                                            X’
                                                                                       w

                                                      1                           2
                                                                                                      X
                                                                  l


                             x1'  0       x2  l cos
                                            '
                                                                x3  l cos  w sin 
                                                                 '
                                                                                                   x4  w sin 
                                                                                                    '


                             y1'  0       y 2  l sin 
                                             '
                                                                y3  l sin   w cos
                                                                 '
                                                                                                   y 4  w cos
                                                                                                     '



       Substituting these in the formula for I x ' y ' and equating the resulting expression to zero

       gives
                                                                           lw
                                                            tan 2 
                                                                         l  w2
                                                                          2


       Substitution of new (x’, y’) also gives




                                                                198
                               
                 I x ' x '  m l 2  w 2  
                                                  l 2  w 2 2       l 2 w2       
                                                                                    
                               
                                              l 4  w4  l 2 w2  l 4  w4  l 2 w2 
                                                                                    
and

                 I y'y'
                               2
                           m l  w  
                                     2       l 2  w 2 2       l 2 w2                        
                                                                                                
                              
                                         l 4  w4  l 2 w2  l 4  w4  l 2 w2                  
                                                                                                


13.2 The principal axes (1,2) of the rectangle passing through its CM are shown in the
figure when it is in the plane of the paper. Axis (3) is coming out of the paper.

                                      2
                                                                         1

                                                   


                                           a
                                          a
                                                                b


(i)     Components of the angular velocity are
                                      b                                             a
           1   cos                            2   sin                               3  0
                                    a2  b2                                       a2  b2
(ii)    Components of angular momentum along the principal axes are
                                ma 2 b                                 mab 2
              L1  I11                          L2  I 2 2                                L3  0
                             12 a 2  b 2                             12 a 2  b 2
                                                                 ˆ ˆ      ˆ
Thus if the unit vectors along the principal axes are denoted as 1, 2 and 3 then
                  
                  L
                            ma 2 b       ˆ
                                          1
                                                mab2           ˆ
                                                                2
                                                                       mab
                                                                                        a1  b2
                                                                                          ˆ    ˆ
                          12 a  b
                                2     2
                                               12 a  b
                                                   2        2
                                                                     12 a  b
                                                                          2         2


(iii)   Kinetic energy of the rectangle
                                  1   1         1         ma 2 b 2 2
                           K .E.    L  I112  I 22 
                                                       2

                                  2       2       2        12 a 2  b 2                  


                                                  199
13.3 The system and its principal axes (1,2) are shown in the figure below. Principal axis
(3) is coming out of the paper.

                                                           2
                                                                    




                                                                
                                           a                                         1

                                                                        L


                                                         b



We have

           I 1  2  2m 
                            a2
                            4
                                ma 2          I 2  2  2m 
                                                                b2
                                                                4
                                                                    mb 2                   
                                                                                     I3  m a2  b2      
                                       b                                         a
            1   cos                           2   sin                               3  0
                                  a b
                                   2       2
                                                                            a  b2
                                                                             2



                                 ma 2 b                            mab 2
                 L1  I 11                       L2  I 2 2                          L3  0
                                 a2  b2                            a2  b2
Rate of change of angular momrntum can be found easily by either the Euler’s equations or
by taking components of angular momentum in directions parallel to and perpendicular to
                 
the direction of  .
                                               
By Euler’s equations: Since the components of  in the direction of the principal axes
remain unchanged, we have for the change in angular momentum
                  
                dL  
                    L
                 dt
                                        ˆ                   ˆ                    ˆ
                     ( 2 L3  3 L2 ) 1  (3 L1  1 L3 )2  (1 L2   2 L1 )3
With the components calculated above, we get
                        
                      dL                     ˆ            a 2  b 2 ˆ
                           (1 L2   2 L1 )3  mab 2  2
                                                          a  b 2 3
                      dt                                            


                                                  200
          
         dL                                                                      ˆ
Thus the    vector is always in the direction opposite to that of principal axis 3 . Thus at
         dt
the instant shown, it is pointing into the paper.
By taking components of angular momentum in directions parallel to and
                                          
perpendicular to the direction of  : If we decompose the angular momentum
                                                                     
components L|| in the direction parallel to  and L perpendicular to  , then as the body
rotates, L|| remains unchanged and L rotates with angulae speed  and changes at the rate
L and that gives the rate of change of angular momentum. From the figure above, it is
clea that
                                                          a2  b2 
                          L  L1 sin   L2 cos  mab  2
                                                          a  b2 
                                                                  
This is also shown in the figure above. It is then clear that at the instant shown this vector
is going to rotate into the plane of the paper. Thus the direction of change of angular
momentum is into the paper and its magnitude is
                                               a2  b2 
                                   L  mab  2
                                               a  b2 
                                                    2
                                                        
                                                       


13.4 The picture of the system of eight particles is shown below. We also show the
       diagonal about which we consider it rotating.

                                                        Y


                                                                                 




                                                                                 X




                                    Z



                                           201
   (i)       To show that the coordinate system chosen is is also the principal axes system,
             it is sufficient to show that all the off diagonal elements of its moment of inertia
             tensor vanish. This is easily shown. We do it for Ixy here.
                                          l  h h h h  l  h h h h 
           I xy  I yx   mi xi yi  m             0
                          i               2  2 2 2 2  2  2 2 2 2 
Thus the set of axes chosen is the principal set of axes.


                                                                          ˆ j       ˆ
                                                                         li  hˆ  wk
   (ii)      For the diagonal shown, the unit vector is =
                                                                         l 2  h 2  w2

                                            ˆ j
                                            li  hˆ  wkˆ
Thus the angular velocity is given as   
                                            l 2  h 2  w2
To calculate the angular momentum, we also need the moment of inertia about the principal
axes. These are
                                                           h 2 w2 
                     I 1   mi ( y i2  z i2 )  8  m   
                                                           4   4 
                                                                                
                                                                     2m h 2  w 2               
                           i                                      
                                                           w2 l 2 
                     I 2   mi ( z i2  xi2 )  8  m   4  4   2m w  l
                                                                   
                                                                            2
                                                                                2
                                                                                              
                            i                                     
                                                     l 2 h2 
                     I 3   mi ( x  y )  8  m      2 m l 2  h 2
                                      2
                                      i
                                          2
                                          i         4    4 
                                                                                             
                           i                                
Thus we have

           L1 
                             
                  2m h 2  w 2 l
                                          L2 
                                                            
                                                 2m w 2  l 2 h
                                                                           L3  
                                                                                          
                                                                                     2m l 2  h 2 w          
                    l h w
                     2    2       2
                                                   l h w
                                                    2    2       2
                                                                                       l  h  w2
                                                                                          2           2


The kinetic energy
                         1    2  1     2  1
                  K .E.  I 11  I 2 2  I 3 3 
                                                2                    
                                                    2m 2 h 2 l 2  w 2 l 2  w 2 h 2                     
                         2       2        2               h 2  l 2  w2


13.5 As the rod rotates, its angular momentum also changes. The required torque is
         provided by the gravitational force pulling it down.                       Thus the rod can be in
         equilibrium at two positions. Absolutely vertical or at an angle  from the vertical.
         The first solution is trivial. For the second solution, we find  by equating the
         angular momentum change about the pivot (a staionary point) to the torque. Later


                                                  202
              we will check the answer by applying the torque equation about the centre of mass
              also.
       The rotating rod in the plane of the paper is shown in the figure along with its principal
       axes at the pivot; axis (3) is coming out of the plane fop the paper. Also shown is the free
       body diagram of the rod. The pivot applies a vertical force N to balance the weight mg of
       the rod and a horizontal force F to provide the required centripetal accelerartion to the CM
       of the rod.

                     2                                                     N
                                             1
                                  

                                                              F

                                                                                




                                                                                     mg




 The moment of inertia of the rod about the principal axes at the pivot are
                                             ml 2                         ml 2
                                      I1            I2  0        I3 
                                              3                            3
The angular velocity components are
                               1   sin            2   cos         3  0
Thus the angular momentum components are
                                       ml 2 sin 
                                  L1                     L2  0          L3  0
                                            3
Since the angular velocity and its components along the principal axes are constant, we have for
the torque
                     
                   dL                           ˆ                    ˆ                    ˆ
                        L  ( 2 L3   3 L2 )1  ( 3 L1  1 L3 )2  (1 L2   2 L1 )3
                    dt
Thus


                                                      203
                                                                   ml 2 2 sin  cos
                            1  0       2  0           3  
                                                                            3
Here the negative sign shows that the torque is pointing into the paper. This is proided by the
weight of the rod. The torque of the weight about the pivot is pointing into the paper and its
magnitude is
                                                     l
                                                   mg sin 
                                                     2
Thus we should have
                        l         ml 2 2 sin  cos                                  3g 
                   mg     sin                                             cos 1    2 
                        2                  3                                          2l 
                                                                                  
     The same answer can also be obtained by taling horizontal (perpendicular to  ) LH and
                           
     vertical (parallel to  ) LV components of the angular momentum and then calculating the
     rate of change of angular momentum as LH.
Now we calculate F. This provideds the centripetal force so we have
                                                       l      
                                                  F  m sin   2
                                                       2      
Check: To check our answers, we apply the torque equation about the CM and see if the forces
calculated by us satisfy this. About the CM we have
                                         
                                        dLCM   
                                               applied ,CM
                                         dt
Further, for the principal axes at the CM (parallel to those at the pivot)
                                          ml 2                              ml 2
                                     I1             I2  0            I3 
                                          12                                12
So that (angular velocity components are the same)
                                    ml 2 sin 
                               L1                          L2  0          L3  0
                                        12
                           3g 
Thus we have, with cos      2 
                           2l 

                                                          ml 2 2 sin  cos    mgl sin 
                   1  0      2  0            3                        
                                                                  12               8




                                                      204
Let us see if , N and F calculated by us give the same torque. This is in the direction
perpendicular to the paper and its magnitude is


                      l        l         l2 2              l
                     F cos  N sin   m  sin  cos  mg sin 
                      2        2         4                 2
                     3g 
Substituting cos      2 
                             we get the torque as
                     2l 
                             l2 2         3g       l          mgl sin 
                         m      sin          mg sin   
                             4           2l 2
                                                   2             8
Thus our answers also satisfy the torque equation about the CM and are therefore correct.


     13.6 (i) Since the wheel is rolling without slipping, it rotates about the horizontal axis
                                          V
            with an angular speed  H 
                                          R
                                                                              V
     Further, it rotates about the vertical axis with an angular speed V      . This can also be
                                                                              L
                                                                                  2L
     seen easily as follows. Since the wheel rotates a full 2 radians in time        , its angular
                                                                                   V
                                          2L  V
     speed about the vertical is V  2        in counterclockwise direction looking at it
                                          V  L
     from the top.
(ii) Angular momentum about O = angular momentum of the CM + angular momentum about
                                    the CM
     Angular momentum of the CM about O = MLV in the vertically up direction
     Angular momentum about the CM has both the horizontal and the vertical components.
                                              1      V                                 1
     Angular momentum about the CM =            MR 2   in the vertically up direction + MRV in
                                              4      L                                 2
     the horizontal direction pointing towards O.
     Thus total angular momentum about O is
                                   1
     Horizontal component LH =       MRV in the horizontal direction pointing towards O.
                                   2




                                                  205
                                  MR 2V
Vertical components LV= MLV            in the vertically up direction.
                                   L
   (iii)   As the wheel rotates, the vertical component of the angular momentum remains
           unchanged while the horizontal component rotates.             Its rate of change is
           therefoe equal to
                                                  1 MRV 2
                                       V L H 
                                                  2 L
   If seen from the top, its direction is as shown in the figure below




                                                            O
                                                  LH


                                                  LH




(iv) The free body diagram of the wheel, when it is going into the page is shown below.

                                     N1

                                           O
                           F
                                                       L              mg


                                                                  N



As the wheel rotates, the change in the angular momentum is such that at the position
shown above, it will point out of the paper. On the other hand, torque due to the weight




                                          206
about O is pointing into the paper. Thus to generate torque coming out of the paper, the
normal reaction of the ground has to be more than the weight of the wheel so that
                    Torque  change in angular momentum
                                    1 MRV 2                           1 MRV 2
                    L N  mg                           N  mg 
                                    2 L                               2 L2
                                                    1 MRV 2
By balancing the vertical forces, we get N 1 
                                                    2 L2
Thus when the wheel is moving, it is pulled down by the vertical shaft at the centre. In
turn, the shaft is pulled up. Thus with time it will tend to come out of the ground.
The horizontal force applied by the shaft provided the centripetal acceleration. Thus
                                                    mV 2
                                              F
                                                     L
The explanation for why the wheel presses theground harder has been given above.
Another way to think about it is as follows. If there were no gravity, the horizontal shaft of
the wheel will tend to turn clockwise about a horizontal axis as the wheel moves so that the
change in the torque is zero. In turn the wheel presses the ground and generated enough
torque so that the rate of change of its angular momentum is equal to the torque generated.


13.7 This problem is like the previous problem except for position of the centre of mass
and the moment of inertia of a cone.          We again obtain the horizontal component of the
angular momentum of the cone and multiply it with the vertical component of the angular
velocity of the cone to get the rate of change of the angular momentum. The horizontal and
the vertical components of the angular velocity and the free body diagram of the cone are
shown below.

                           V                                         N1


               H
                                                                 F

                                                                                mg
                                                                                         N




                                              207
                                                                                 V
Since the speed of the centre of cone’s base is V and its radius is R,  H        . Similarly
                                                                                 R
       V
V      .
       h
The principal axes of the cone at its vertex are its axis and two axes perpendicular to it.
                                                         3
The moment of inertia about the axis of the cone is        mR 2 . Thus the horizontal moment
                                                        10
of inertia is
                                        3       V  3
                                LH       mR 2   mVR
                                       10       R 10
And its rate of change is
                                   3             3 mV 2 R
                                     mVR  V 
                                  10            10 h
Coming out of the plane of the paper at the position of the cone shown. This shoud be
equal to the torque about the pivot. Thus
                     3       3 mV 2 R                         3       3 mV 2 R
                 hN  hmg                             N      mg 
                     4      10 h                              4      10 h 2
And
                                                           1      3 mV 2 R
                    N1  mg  N                    N 1   mg 
                                                           4     10 h 2
Similarly, the force F provides the centripetal force. Thus
                                         3         3 mV 2
                                  F  m  h  V 
                                               2

                                         4         4 h
13.8    This is a problem where we take the components of the angular velocity according
to the convenience of applying the rolling condition. So we split the angular velocity so
that it has component  about the vertical and  about its axis.             We then find the
relationship between them by demanding that the speed of all the points on the ground
vanishes so that the cone rolls without slipping.
(i) Now a point at distance x from the vertex (pivot point) on the line touching the ground
moves with speed x due to the rotation about the vertical axis through the vertex and
speed x sin  (see figure below) in the opposite direction due to rotation about the axis of
the cone.


                                            208
                                                             



                                                                        
                                         
                                                                    x
                                   


Thus for rolling we get
                                                    sin 
Now if the centre of the base moves with speed V, we have
                                                                       V
                                   V  h cos           
                                                                    h cos
This also gives
                                                          V
                                             
                                                     h cos sin 
As is clear from the components, the net angular velocity  of the cone is  cos parallel
to the line touching the ground an in direction shown in the figure. Thus

                                                       V       V h2  R2
                                cos                     
                                                     h sin       hR
(ii)    To calculate the angular momentum of the cone, we calculate the angular momenta
        along the principal axes shown in the figure below.

                                                 2                           1


                                         
                                             



From the figure it is clear that

                                        3        V h2  R2     h         3
             L1   I 1 cos           mR 2                       mRV
                                       10           hR       h2  R2    10




                                                  209
                    3            V h R                 3           V
                                      2   2
                            3                    R                3
 L2  I 2 sin    mR 2  mh 2                      mR 2  mh 2 
                    20    80        hR      h 2  R 2  20     80    h
In this case, the angular momentum changes because its vertical component rotates with
angular speed . Thus the rate of change of angular momentum is given by
        
            LV   L2 cos  L1 sin  
      dL
       dt
                      3
               V h2  R2       3 2 V    h       3       R     
                     mR  mh  
                           2
                                                mRV          
                  hR  20    80   h h 2  R 2 10     h2  R2 
              3 mRV 2 3 mhV 2
                   
              20 h     80 R
and it is in the direction of axis 3. Its sign depends on the relation between h and R.
Check: The answer can be checked easily by applying the Euler equations that give
                   
                   L1   2 3 ( I 3  I 2 )  0
                   
                   L2   31 ( I 1  I 2 )  0
                                                              3       3    
                   L3  1 2 ( I 2  I 1 )   2 sin  cos   mR 2  mh 2 
                                                               20     80    
                                             V  3
                                               2
                                                                 3 2
                                                  mR  mh 
                                                            2

                                             hR  20            80  
which is the same answer as obtained above.


13.9 In the figure below, we show the horizontal angular momentum LH of the rotating
      disc, its change LH as the platform rotates and its free body diagram.

                                                                                          N1
                                                                  N2
                        LH
             LH
                                                                                 mg

We have
                                                      
                              1                      dL        1
                          LH  mR 20                    LH  mR 20 
                              2                      dt        2




                                                   210
If the torque applied on the disc is zero, i.e., N1 and N2 are equal, there cannot be any
change in the angular momentum. Hence the disc will tend to rotate clockwise in the
position shown so that N1 will become larger than N2. Finally, N1 and N2 will be such that
the torque is equal to change in the angular momentum. This gives

                           N1  N 2  mg
                                            l
                                              N1  N 2   1 mR 20 
                                            2               2
Solving these two equations gives
                                 mg mR 2 0            mg mR 2 0 
                          N1                   N2       
                                  2    2l                2    2l
13.10: It has been worked out in section 13.5.3 (see figure 13.18). Carry it out further for
each axis step by step.


13.11 We show in the figure below the principal axes for the system at the pivot point and
     the forces that act on the vertical rod at the bearings. Principal axis 1 is along the rod
     and 2 and 3 are perpendicular to the rod with axis 3 coming out of the plane of the
     paper at the instant the rod is shown. In the free body diagram of the rod, we have
     not shown the vertical forces. The cetre of mass of the system is also indicated in the
     figure.

                                                 N1



           2       
                                1
                                                                     
                                                 LH


                           d                                         d


                                                                             N2



The moment of inertia about the principal axes are as follows




                                           211
                                                         l2    9l 2 5 2
                            I1  0      I2  I3  m         m      ml
                                                         4      4   2
The angular velocity components are
                            1   cos              2   sin       3  0
Thus the angular momentum components are
                                                   5 2
                              L1  0     L2         ml  sin         L3  0
                                                   2
The torque required to keep the rod rotating can be calculated either by Euler’s equations or
by the horizontal component LH of the angular momentum and multiplying it by .
Euler equations give
                      1  ( 2 L3  3 L2 )  0            2  (3 L1  1 L3 )  0

                                                5
                      3  (1 L2   2 L1 )  ml 2 2 sin  cos
                                                2
This is the same answer as obtained through LH as is easily checkd.
The torque is provided by the reactions of the bearings. The difference in these reactions
also provides the centripetal force as the CM of then system is moving in a circle of radius
l
  sin  . Thus
2

          N1  N 2 d  5 ml 2 2 sin  cos           N1  N 2 
                                                                          5
                                                                            ml 2 2 sin  cos
                          2                                              2d
                         l
          N1  N 2  2m  sin    2  ml 2 sin 
                         2
Solving these two equations gives
                  ml  2 sin     5l cos                         ml  2 sin     5l cos  
           N1                   1                       N2                    1        
                       2             2d                                2             2d 


13.12 In the figure below we show the principal axes for all the three rigid bodies. Axes
     (1, 2) are in the plane of the paper and axis 3 is coming out of the the paper in all
     three cases.




                                                212
                                                                                                     2             
                                              1           2
                                                                             
                     2                                                                                             
                                                                                                                      1
                                                                                       1




                         (a)                                     (b)                                     (c)


     In all the three cases we have
                                          1   cos            2   sin                3  0
     and
                                          L1  I1 cos          L2  I 2 sin              L3  0

     Thus we have
                                dL                                          dL 
                                  ( 2 L3   3 L2 )  0                     ( 3 L1  1 L3 )  0
                                dt 1                                        dt  2


                                dL 
                                  (1 L2   2 L1 )  I 2  I 1  sin  cos 
                                                                      2

                                dt  3
     Now in case (a)
                                 mb 2              ma 2                          b                             a
                          I1              I2                 sin                          cos 
                                 12                12                        a2  b2                      a2  b2
This gives

                                                    dL    m 2 ab  a 2  b 2 
                                                                            
                                                    dt  3  12  a 2  b 2 
                                                                               
In case (b)
                     mb 2                 ma 2 mb 2                                   dL    ma 2 2
              I1                  I2             .          This gives                          sin  cos
                     12                   12    12                                    dt  3  12
In case (c)
                           mR 2                mR 2                           dL    mR 2 2
                 I1                    I2         .     This gives                       sin  cos
                            4                   2                             dt  3   4



                                                               213
13.13    As the ring rotates, the tension in the string does two three things: it balances the
     weight of the ring, it provides the centripetal acceleration and it provides the torque
     for the angular momentum change of the ring. The figure below shows the principal
     axes of the ring at its CM and the free body diagram of the ring. Axes (1, 2) are in
     the plane of the paper while axis 3 is coming out of the paper.


                  
                                                                      O
                   2
                                                                 
                                     A
                                            1


        B
                                                                          A
                                                                                   T


                                           B                     mg




The components of the angular velocity are
                             1   cos          2   sin         3  0
Therefore components of the angular momentum along the principal axes at the CM are
                             1
                      L1      mR 2 cos         L2  mR 2 sin         L3  0
                             2
Thus the change in the angular momentum is  times the horizontal component of the
                                        1                  1
angular momentum =   mR 2 sin  cos  mR 2 cos sin    mR 2 2 sin  cos
                                        2                  2
And it points in the direction of axis (3) i.e. coming out of the page at the instant shown.
Euler equations also give the same answer through




                                                214
                     dL                                          dL 
                       ( 2 L3   3 L2 )  0                     ( 3 L1  1 L3 )  0
                     dt 1                                        dt  2


                     dL 
                       (1 L2   2 L1 )  I 2  I 1  sin  cos 
                                                           2

                     dt  3
     However, at the position shown, the tension and the weight of the ring give no torque about
     the CM of the ring. Thus the internal forces will move the ring so that there is an opposing
     change in the angular momentum. This ring moves so that its B end moves up. This
     implies that the ring moves towards position 1. However, as it moves up, the tension
     gets misaligned with the CM and starts giving a torque about it and finally the ring stops at
     an angle such that the torque equals the angular momentum change.
     Now balancing the forces gives
                         T cos  mg
                         T sin   mR  l  2 sin                       T  mR  l  2
     These equations give
                          g                                        
              cos                1                             
                        R  l 
                         2
                                                                                     g
                                                                     sin  cos  2
                                   g2                    g 2
                                                                                   R  l 
              sin   1                        1
                               4 R  l 2         2 4 R  l  
                                                                 2 



     If the ring moves up by an angle , The perpendicular distance of the tension and the CM
     is about R . This gives a torque

                                               3   RT   mR R  l  2
     Equating this to the rate of change of angular momentum gives

                      mR R  l  2 
                                               1                     1          g
                                                 mR 2 2 sin  cos  mR 2 2 2
                                               2                     2        R  l 
     This gives
                                                             Rg
                                                   
                                                        2 2 R  l 
                                                                        2




13.14      Following the notation of example 13.7, we have (using parallel axis theorem for
          calculating I  )


                                                       215
                I
                    mR 2 50  10 3  2  10 2
                          
                                                             2

                                                      10 5 kgm 2
                     2                 2
                         2
                     mR
                I         ml 2  5  10 6  50  10 3  25  10 4  1.3  10  4 kgm 2
                       4
Given  S  100  rads 1
(i) The presseion angular frequency will then be
                               mgl 50  10 3  9.8  5  10 2
                    p                                         7.8 rads 1  450 s 1
                               I        10 5  100 
Seeing the change in the angular momentum for the given sense of rotation, the top will be
coming out of the paper.
(ii) We see that  p   S . Therefore for the nutation motion we can write

                                            p sin  0
                            (   0 )                      1  cos  t  with      
                                                                                            I
                                                                                              S
                                                                                          I

                                                                                     2 p sin  0
The maximum difference between  and                             0   is therefore                   .   Substituting all the
                                                                                           
numbers, we get
                                                 2 p sin  0
                                 m ax   0                          30   18 .6  48 .6
                                                         
     (iii)   The frictional force is needed to provide the centripetal force. Thus
                   Ffriction  ml sin 302  50 103  5 102  0.5  7.82  0.076 N
                                          p




     13.15      The soulution of the Euler equations governing the motion is the same as in
     example 13.7 but the initial conditions are different. Thus we have
                                                                                sin 
                                    2  A sin  0 t  B cos  0 t                    cos S t 
                                                                                 I S

                                                                                 sin 
                                    3   A cos  0 t  B sin  0 t                   sin  S t 
                                                                                  I S

                  I  I  
     Here  0                  S . Let the initial angular speed given about the vertical be  0 . Then
                     I

     the initial conditions for the subsequent motion are




                                                              216
                                2 (t  0)   0 sin  0         3 (t  0)  0
These give
                                                                   
                                    A0          B   0 
                                                                     sin  0
                                                                     
                                                           I S     
Thus we get
                                                                 
                       2   0 
                                           sin  0 cos 0 t  
                                                                       sin  cos S t 
                                  I S                           I S

                                                                   
                      3   0 
                                            sin  0 sin  0 t  
                                                                         sin  sin  S t 
                                 I S                              I S

If the precession velocity of the top is  p then

         p sin    2 cos s t   3 sin  s t
                                                    
                 0 sin  0 cos 0 t   S t        sin  cos S t  sin  0 cos 0 t cos s t
                                                   I S
                                                  
                                                     sin  sin  S t  sin  0 sin  0 t sin  s t
                                                 I S
                                 I                                      
                 0 sin  0 cos  S t      sin   sin  0 cos I  S t  
                                I       I                      I        
                                         S                             
This gives

                              sin  0     I        sin  0        I     
                   p  0            cos  S t     1        cos  S t  
                              sin       I       I      sin     I      
                                                  S                    
And
          
            2 sin  s t   3 cos s t
                                                   
               0 sin  0 sin  0 t   S t        sin  cos S t  sin  0 cos 0 t sin  s t
                                                 I S
                                                
                                                    sin  sin  S t  sin  0 sin  0 t  cos s t
                                               I S
                                            I     
                0 
                                sin  0 sin   S t 
                                             I      
                       I S                       
                                    
Upon integration, the equation for  gives




                                                     217
                                      I                                   I     
                   (t )    0             0 
                                                            sin  0 1  cos  S t 
                                                                             I      
                                     I S           I S                         
The variation of  (t ) with time is similar to that shown in figure 13.23.
To get the angle through which the top has precessed, we substitute the value of  (t ) in the
             
equation for  p (t ) and integrate it numerically. Finally to see the motion of the tip of the

axis of the top,  is plotted against   p (t )dt .
                                        




                                                218
                                              Chapter 14

                              k         25
14.1 For this spring                     3.54 rad s1
                              m          2
             A sin   0.1                                               
     (i)                      , A  0.1 giving x(t )  0.1sin  3.54 t  
             A cos  0         2                                         2

                             A sin   0.1       
     (ii)                                          A  0.11, tan   2.38 or   1.17rad
                  A cos  0.15  A cos  0.042
     leading to x(t )  0.11 sin 3.54t  1.17 


                        A sin   0.1         
     (iii)                                      A  0.104, tan   3.57
              A cos  0.1  A cos  0.028
                          sin   0 
                                     
                          cos  0                 which gives   1.84rad
                                      2
                          tan   0 
                                    
                                        x(t )  0.104 sin 3.54t  1.84 


                       A sin   0.1       
     (iv)                                     A  0.115, tan   1.76
             A cos  0.2  A cos  0.0565
                        sin   0 
                                    
                        cos  0      0 which gives   1.06rad
                                     2
                        tan   0 
                                  
                                        x(t )  0.115 sin 3.54t  1.06 


                       A sin   0.2      
     (v)                                     A  0.202, tan   7.14
             A cos  0.1  A cos  0.028
                        sin   0 
                                    
                        cos  0      0 which gives   1.43rad
                                     2
                        tan   0 
                                  
                                        x(t )  0.202 sin 3.54t  1.43 



                                               219
                      A sin   0.2         
      (vi)                                     A  0.208, tan   3.57
             A cos  0.2  A cos  0.056
                      sin   0 
                                            
                      cos  0                   which gives   1.84rad
                                             2
                      tan   0 
                                
                                       x(t )  0.208 sin 3.54t  1.84 


14.2 It is given that x(t )  5 sin(2t   ) . Thus the velocity is v(t )  10 cos(2t   ) and the

      acceleration is x(t )  20 sin(2t   ) . This is shown below for different values of .
      Displacement curve is the one with smallest amplitude of 5, the velocity curve has
      intermediate amplitude of 10 and the acceleration curve has the largest amplitude of
      20. Notice that phase difference of π and π give identical curves.

                                            0




                                                                   
                                                            
                            4                                        4




                                              220
                          
                   
     3                     3




                          
                   
          2                2




          2               2
                   
           3                3




               221
                          3                                         3
                                                             
                           4                                          4




                                                                     




                                                                                  dV ( x)
14.3 Equilibrium        points       of   the    potential     are   given   by            0.   For
                                                                                   dx
    V ( x)  2 x3  9 x 2  12 x , this gives

                               6 x 2  18 x  12  0  x 2  3x  2  0
The roots of this equation are at x = 2 and x = 1. The second derivative of the potential at
these points is
                                 d 2V ( x)               6    x  1
                                            12x  18  
                                   dx 2
                                                         6     x  2
Thus the potential is minimum at x = 2.                  The corresponding spring constant is

k  V  x2  6 . This gives   k m  6  2.45rad s 1 .




                                                222
14.4                       
                V ( x)  C 1  e               
                                   a ( x  x0 ) 2
                                                       
                                                                  dV
                                                                  dx
                                                                                               
                                                                      2C 1  e a ( x  x0 )  ae a ( x  x0 )

       Equating the derivative to zero gives x = x0. At this point the second derivative
                                        d 2V
                                        dx 2
                                                                                          
                                              2C 2a 2 e  2 a ( x  x0 )  a 2 e  a ( x  x0 )  2Ca 2

       is positive and therefore at this point the potential is minimum.
       (i)       The corresponding spring constant k  2Ca 2 .
       (ii)      To plot the potential and harmonic approximation to it, we have taken C=2, x0=1.0
                 and two values of a: a=0.4 and a=1.2


                               a = 0.4                                                           a = 1.2




                         k    2C
       (iii)             a
                         m     m
       (iv)      As is clear from the figure, depending on the value of a the potential becomes softer
                 or harder for larger displacements from x = x0. Thus the frequency will become
                 smaller and time-period larger for a=1.2 and the frequency will become larger and
                 time-period smaller for a=0.4.


                            12    6                        dV          12   6
14.5           V ( x)                                          12 13  6 7 
                           x 
                                  x                           dx         x      x 

       Equating the derivative to zero gives x  21 6  . At this point the second derivative

                 d 2V         12    6              12           6         18
                        156 14  42 8    156 14          42 8      13 
                                                                                 13 2
                 dx 2
                             x       x            4 2 13
                                                                     2 2  2 


                                                                 223
                                              18
Thus the corresponding spring constant is             and the frequency of small oscillations is
                                             21 3  2
                                   k   1      18      9  21 3   
                                                  
                                   m         21 3 m             m


                                                                             x
14.6 The surface charge density    x gives a electric field E              that pulls the
                                                                              0
      displaced electrons back, i.e. it applies a restoring force. We consider volume V of
                                                      x V 2 x
      the displaced block, the force on it is V                .
                                                       0   0
The total mass of the block is = number of electrons in volume V electron mass
                                  V
                              =      me
                                   e

So the equation of motion for the displaced block of electrons is (writing x  y )

                          V           V 2           e
                             me   
                                y           y   
                                                y           y0
                           e            0            0 me
This gives the plasma frequency (here n is the number density of electrons)

                                             e       ne 2
                                   p             
                                             0 me    0 me


14.7 In this problem k  2000 Nm 1       m  75 kg

          k      2000
(i)                 5.16 rad s 1
          m       75
(ii) Velocity of platform and person on it is calculated by principle of linear momentum
conservation and gives

                                        55 2 gh
                                   v              4.59ms 1
                                        55  20




                                           224
(iv)      If we take the equilibrium point of (person+platform) as y=0, then the displacement
                                                             55  9.8
          without the person on the platform is =                      0.27m . Thus this is a problem
                                                              2000
          with the initial conditions
       y (t  0)  0.27       y (t  0)  4.6ms 1
                              

       Representin the motion as y(t )  A sin(t   ) , we have

                          A sin   0.27   
                                             A  0.93 and tan   0.30
            A cos  4.6  A cos  0.89
                          sin   0 
                                     
                          cos  0                 which gives   2.85rad
                                      2
                          tan   0 
                                    
Thus the displacement is given by x(t )  0.93 sin 5.16t  2.85 


14.8 In this case, it is the motion of a rigid body about a pivot point. Thus the equation of
         motion is written in terms of the moment of inertia, angular acceleration and the
         restoring torque. If the square is displaced by an angle  about the vertical (see
                                           a             mga
         figure) the torque is   mg          sin           
                                           2              2




                                                         


                                                                      a



                                                              mg


The moment of inertia about the pivot is calculated by the parallel axis theorem and is
                                                    2
                                            a     a2 2
                                      I  m    m    ma 2
                                            2     6 3



                                                  225
The quation of motion is
                                     mga                2      mga   0
                            
                           I                 or      ma 2 
                                      2                 3          2
This can be rewritten as
                                                       3g
                                           
                                                          0
                                                  2a 2

                                                                    3g
This gives the frequency of oscillation to be  
                                                                2a 2


14.9 These two pendulums with their displacements are shown in the figure below.




                                       1                                         2

                                 L
                                                            K


                                           M



For visualizing the force applied by the spring, let us take  2  1 . Then the bob on the left
feels a force to the right due to the spring and to the left due to the weight of the bob. The
spring force is in the opposite direction on the pendulum on the right.
(i) Since the rods are rigid, and therefre generate internal forces, we use the angular-
momentum torque equation for describing the motion of the pendulums.                                    Thus the
equations of motion for the two pendulums are

                                  k  2  1                                            2  1 
                               L    L                                         g         k
             
          ML2  Mgl1 
               1                                                        
                                                                           1 
                                                                           1
                               2    2                                         L        4M

                                 k  2  1                                               2  1 
                              L    L                                          g           k
           
        ML22  Mgl 2                                               
                                                                         2   2  
                              2    2                                          L          4M



                                               226
(ii)       The equations above are solved by adding them to get the equation for
            (t )  1 (t )   2 (t ) and for  (t )  1 (t )   2 (t ) . These are
       g                          g   k 

    0                  
                                      0
       L                           L 2M 
These give
                                                  g                                                         g   k   
 (t )  A cos1t  B sin 1t  1 
                              
                                                               (t )  C cos 2 t  D sin  2 t   2               
                                                  L
                                                    
                                                                                                 
                                                                                                              L 2M    
                                                                                                                       


Combination of these two gives 1 (t ) and  2 (t ) . There are four unknowns that will be given
in terms of four initial conditions.


14.10 Angular displacement of the rod by and angle  and the corresponding angle  that
       the strings make with the vertical are shown in the figure below. Also shown are the
       tensions in the strings; these are equal by symmetry.




                                                                             T           l
                                                     T
                                                           



                                                                 L

Since we are dealing with a rigid rod, the correct equation of motion to be used is the
angular momentum-torque equation. There are three unknowns in the problem: ,  and
T. We thus need three equations. These are
                                              mL2                                                  L
              2T cos   mg                          L  T sin         and           l sin       sin 
                                              12                                                     2
In the small angle approximation we have
                                   cos  1          sin       and sin   
Then


                                                         227
                                            mg                 L
                                      T                        
                                             2                 2l
And the equation of motion becomes
                     mL2        mg L                                       3g
                           L                                   
                                                                              0
                     12            2 2l                                       l

                                             3g
Thus the frequency of oscillation is          .
                                              l


14.11We show in the figure below a displacement of the rod when the left spring is
     stretched by y1 and the right one by y2.




                                                           L

                                                                                               y2
                            y1                            yCM


The equation of motion for the CM is
                                                                                  y1  y 2
                  M CM  Fext  k ( y1  y 2 )
                    y                                  with           yCM                 .
                                                                                     2
And the equation for rotation about the CM is
                                              
                                             I   ext

where we mesure counterclockwise  as positive. Then
                                 y1  y 2                       L
                                                 ext   k ( y1  y 2 )
                                    L                           2
Thus we have for the CM
                                     2k                                      2k
                           CM 
                           y            y CM  0   CM 
                                     M                                       M
And for rotation about the CM



                                            228
                     ML2    kL2               6k                                  6k
                                        or     0   rot 
                     12        2                   M                                 M
With the initial conditions y1  A1 and y 2  A2 and the rod held stationary in the
beginning, we have
                           A1  A2                                       A1  A2    
           yCM (t  0)            ,   yCM (t  0)  0,  (t  0) 
                                                                                ,  (t  0)  0
                              2                                             L
This then gives the answer
                           A  A2                                       A1  A2 
               yCM (t )   1       cosCM t           and      (t )            cosrot t
                           2                                            L 


14.12 Since we are writing the displacement as y(t )  A sin t    , the projection of the
phasor on the y-axis gives the displacement. The phase angle is measured from the x-axis.
In the following we show the position of he phasor at t = 0.                         After that it rotates
counterclockwise.




     (i)                                 (ii)                                (iii)




       (iv)                                 (v)                              (vi)




14.13 Assuming that the swing is performing simple harmonic motion, the period of the
     swing is
                                                g       9.8
                                                          1.28s 1
                                                l        6



                                                  229
Therefore the speed of the person on the swing as the swing passes through its equilibrium
point is = A  1.28  2  2.56ms1
(i)    When the swing is at the extremes and the child is handed over, the amplitude will
       not change. It is like strating a swing, ireespective of its mass, from a distance A
       from the equilibrium with zero initial speed.             Energetically it can be seen as
       follows:
                                                                1               M gA2
 When only the man is on the swing the total energy is =          M man 2 A 2  man
                                                                2                 2l
When the child is handed over to the person at the extreme, the child brings in additional
potential energy (with respect to the equilibrium point)
= Mchildgthe height of the extreme point with respect to the equilibrium point

                                                    
                              M child g l  l 2  A 2 
                                                           M child gA 2
                                                               2l
Thus the total energy that the swing posseses after the child is handed over is
                                      M man  M child gA2
                                               2l
If the new amplitude is Anew, this should equal (keep in mind that  remains unchanged)
                                                   M man  M child gAnew
                                                                       2
                      1
                        M man  M child  Anew 
                                           2 2

                      2                                     2l
A comparison in the total energies calculated in two ways immediately gives Anew  A .
(ii) When the child is handed over at the equilibrium point, the speed os the swing decreses
by the conservation of linear momentum. It is
                                M manVinitial    50  2.56
                         V                                2.13 ms 1
                               M man  M child      60

And the total energy of the system is now
                                1
                                  M man  M child V 2  136.53J
                                2
If the new amplitude is Anew, then (keep in mind that  remains unchanged)
                    1
                      M man  M child  2 Anew  30  1.64  Anew  136.53
                                             2                  2

                    2
This gives Anew  1.67 m .



                                           230
      (iii)   As is clear from the calculations above, energy is conserved in case (i) only.


14.14 The equation of motion now will be
                                                m  kx  f
                                                 x
Its general solution is
                                                                          f
                                       x(t )  C coso t  D sin o t 
                                                                          k
With the initial conditions x(0)   A and x(0)  0 , the solution is
                                           

                                                     f             f
                                        x(t )   A   cos  o t 
                                                     k             k
                   
Since the velocity x of the block is

                                                    f 
                                            x   A   o sin  o t
                                            
                                                    k

It becomes zero at  0 t   , At this time

                                                    f  f      2f
                                        x(t )   A     A 
                                                    k k       k


14.15 (i) Since this is the motion of a rigid body pivoted at a point under an external force, its
      motion is described by the angular momentum-torque equation. The torque on the body is
      applied by the weight acting at its CG, and it is a restoring torque (see figure). The value of
      the torque about the pivot point is
                                                  l
                                                mg sin 
                                                  2
                                                                   ml 2
The moment of inertia of the rod about the pivot point is I 
                                                                    3




                                                   231
                                                   mg


Thus the equation of motion for the rod is
                                      ml 2    l
                                              mg sin 
                                       3        2
For small angles this equation bocomes

                     ml 2    l                3g   0              3g
                             mg         or  
                      3        2                    2l                    2l
(ii)    As the pendulum swings about, the frictional torque causes it to lose energy and
        therefore the amplitude of the pendulum decreases slowly. The equation of motion
        can be written and solved exactly as in the problem above. When the pendulum in
        the figure above is moving clockwise, the equation of motion is
                          ml 2    l                     3g    3
                                  mg            or     2
                           3        2                        2l  ml
                                               
With the initial conditions  (t  0)   0 ,  (t  0)  0 , the solution of the equation above is
(just like in the problem of spring-mass system with friction)
                                                 2           2
                                 (t )   0 
                                                      cost 
                                                      
                                                 mgl          mgl
                                                                                      4
Like in the problem above, this then leads to a reduction in the amplitude by             by the
                                                                                      mgl
time the pendulum reaches the left extreme. Thus in one cycle the amplitude reduces by
8
    . We will now derive this result from energy considerations also.
mgl




                                             232
     Suppose at a certain instant the pendulum starts with the maximum displacement of  1 and
     goes to the maximum angle  2 on the other side. Then the total angle covered by it is
      1   2     and the work done against the frictional torque is therefore  1   2  . The
                                                 1 2 2 1 2 2
     energy loss during the motion is              I 1  I  2 . Equating this to the work done
                                                 2        2
     against friction gives
                                                                                2         2      4
                      I 1  I  2   1   2           1   2  
                    1 2 2 1 2 2
                                                                                                
                    2        2                                                 I  2
                                                                                       ml 2
                                                                                              3g mgl
                                                                                            
                                                                                        3     2l
                                                            8
     Thus in one cycle the amplitude reduces by                 .
                                                            mgl
     Now if the pendulum starts with an angle  0 and completes N cycles, it will stop swinging

                                           mgl                       2
     when it is at angle  such that                               . Thus we have
                                            2                        mgl

                                       8   2                      mgl  0 1 mgl  0
                              0  N                        N            
                                       mgl mgl                       8     4  8


                                                                                   1 dm 2 1 2
14.16 As the mass leaks out, it carries with it the energy at the rate                  v  Cv . If at time
                                                                                   2 dt    2
     t, the amplitude is A(t) and the frequency (t), then v(t )   (t ) A(t ) sin t    and the
                               dE (t )
     average rate of loss              of energy E(t) of the oscillator is
                                dt

                                    C 2 (t ) A 2 (t ) sin 2 t      C 2 (t ) A 2 (t )
                           dE(t )    1                                      1
                            dt       2                                      4
     In time averaging A(t) and (t) come out of the integral because they vary slowly over a
     few oscillations (that is precisely why they can be defined). The negative sign shows that
     the energy is being lost with time.
                                                            k       k
                                               2 (t )         
                                                           m(t ) m0  Ct




                                                       233
                                                               1 2
     And if amplitude is A(t), the energy E (t )                kA (t ) and the rate of change of energy
                                                               2
      dE (t )          dA(t )
               kA(t )        . Combining all this gives
       dt               dt
                            dA(t )   1    k                          dA(t )        C
                   kA(t )           C        A 2 (t )                                 A(t )
                             dt      4 m0  Ct                        dt       4m0  Ct 
     If the amplitude at t = 0 is A0, this equation is integrated as

                                              dA        dt 
                                            A              t

                                             A   4m0  Ct 
                                            A       0
                                            0


     This gives
                                                                        14
                                                            Ct     
                                             A(t )  A0 1 
                                                         m         
                                                                    
                                                             0     


14.17 The energy loss is given by the formula
                                           dE     4 e 2 A2
                                                          cos2 t
                                           dt    6 0 c 3
     Thus the average energy loss is
                                    dE    4 e 2 A2             1  4 e 2 A2
                                                   cos2 t  
                                    dt   6 0 c 3              2 6 0 c 3

                                                                        dE
     If the damping factor of the system is , then                         E where the energy E is
                                                                        dt

           1
      E     me 2 A 2 . Thus for the current problem we get
           2
                                    1 dE      2      1  4 e 2 A2    2e2
                                                             
                                    E dt   me 2 A2 2 6 0 c 3 6 0 me c 3


14.18 We consider an oscillator with initial conditions so that its motion is given by
                                                             t
                                                x(t )    Ae 2 sin 0 t




                                                      234
                                                                                   
     For undamped oscillatot the siplacement is maximum when 0 t                         . For the damped
                                                                                   2
     oscillator, it will be given at time when
                                                      
                   dx                              2t             2
                       A  sin 0 t  0 cos0 t e  0  tan 0 t  0  2Q
                   dt     2                                          
                                                                   
     Since Q is very high, the maximum occurs when  0 t                where  is very small. Since
                                                                   2
                                   
     the maximum occurs before       , it is advanced by phase angle . Thus
                                   2
                                          
                                   tan     2Q  cot   2Q
                                      2    
     For small , we have
                                                        cos  1
                                              cot          
                                                        sin  
                       1
     This gives        .
                      2Q


14.19 If the person gives an impulse J, the energy gained by him is Jv, where v is the speed at
     the equilibrium point. Since v  A , we have
                                    Energy gained per cycle = JA
     This will equal the energy loss per cycle due to damping.
                                                                                           E
                      Energy loss per cycle = 2πenergy loss per radian = 2
                                                                                           Q
     Here E is the energy of the swing. This follows from the definition of the quality factor.
                                               1            mgA 2
                                         E      m 2 A 2 
                                               2             2l
     Thus we get

                                     1 m 2 A2                    mA         mA g
                        JA  2                       J              
                                     Q   2                         Q           Q       l
     Putting in all the numbers gives
                                               45  2 9.8
                                        J                      7.9 Ns
                                                 50        5


                                                  235
14.20 It is given that k  5000Nm1 and the mass of the platform is 50kg. Since the system is
     critically damped for 500kg put on the platform, we have
                                                k      5000
                              2  4 2  4       4                      6.03
                                                m      550
     So the coefficient of drag force b  m  550  6.03  3316 .5N/ms 1
                                                                               b 3316.5
     Thus when a 200kg weight is on the platform, the value of                        13.27s 1
                                                                               m   250
                                            k       5000
     And the angular frequency  0                      4.47 rad s 1
                                            m        250

     Thus  2  176 .1  4 0  80 , which makes the system hevily damped.
                            2



     The solution then is
                                                       t    t
                                            y(t )  Ae 1  Be 2
     with

                                2                                2
                  1                 0  1.735 and
                                         2
                                                             2       0  11.535
                                                                          2

                         2       4                                2        4
     So

                                         y(t )  Ae1.735t  Be11.535t
     Here A and B are to be determined by the initial conditions.
     Initial conditions:
     Since we are taking the equilibrium position of the (load+platform) as y = 0, the initial
                                                         200g
     displacement from that position is y(t  0)              0.4m .
                                                           k
     (i)    The speed of the load when it lands on the platform is 2gh  20  4.47ms 1 .
            Then by momentum conservation, the load+platform will move with the speed
             200  4.47
                         3.58ms 1 .
                250
     (ii)   From the above, the initial conditions are y (t  0)  0.4          y (t  0)  3.58 ms 1 .
                                                                                

     Thus we have from the solution above
                                  A  B  0.4      1.735A  11.535B  3.58
     Thses give


                                                   236
                                         A  0.106 and         B  0.294
     The solution therefore is

                                     y(t )  0.106e 1.735t  0.294e 11.535t




14.21 For a forces oscillation the angle  by which the displacement lags the applied force is
     given by equations

                                                                         ( o   2 )
                                                                                2

                 sin                                        cos 
                           ( o   2 ) 2   2  2                    ( o   2 ) 2   2  2
                                 2                                         2


                                                   
                 and                 tan  
                                               ( o   2 )
                                                  2



     For a very heavily damped oscillator    . Thus we have
                           sin   1             cos  0       and      tan   
                      
     This gives  
                      2




                                                      237
                                             Chapter 15


15.1 We perform our calculations in a noninertial frame attached with the box and take the
        origin (x = 0) at the point where the spring is connected to the box. In this frame,
        there is a pseudo force ma acting in the negative x direction. Thus the system in the
        noninertial frame looks as shown below.



                           ma                                               x=0



                                                            l


The equilibrium point x0 is given by

                                k x0  (l )  ma  0  x0  l 
                                                                       ma
                                                                       k
Thus the equation of motion for the mass in this frame is
                                         ma        k         ma 
                    m  k  x    l 
                     x                             x  l 
                                                   x                 0
                                         k         m         k 
At t  0 , the initial conditions are x(t  0)  l , x(t  0)  0.
                                                      

                                        ma 
We change the variables to y   x  l      and write the above equation as
                                         k 
       k                                               ma,
 
y        y  0 with the initial conditions y (t  0)            y (t  0)  0.
                                                                                 This gives the
       m                                                k
soulution

                               ma     k                  ma         k 
                    y (t )       cos   t  x(t )  l     1  cos  t
                               k      m                  k         m 
                                                                       


15.2 This problem is very easy to solve in the accelerating frame attached with the car.
        The free body diagram of the door in the car frame with pseudo force included is
        shown below.




                                              238
                                                           car


                                                          Ma                       w


In the car frame therefore the door is being pulled at its CM by a force Ma and rotates
about the hinges. Thus the problem becomes like examples 12.7 and 15.3 where a
ruler/rod, pivoted at one of its ends’ rotates about an axis at that point when the pivot point
starts accelerating with an acceleration. Then by energy conservation this gives the answer
for the angular speed of the door when it is about to close

                                                                              3a
                                                                     
                                                                              w


15.3 This problem is similar to problem 12.7 and the problem above and is solved in
         exactly the same manner.

                    
15.4 We substitute A  vrotating in
                                               
                                 dA            dA           
                                                          A
                                  dt inertial dt rotating

to get
                                                                     
                                     dv rotational                   dv rotating                 
                                                                                                v rotating
                                         dt           inertial           dt        rotating

                                                                        
                                                                  dv rotating
Now from equation 15.8 v rotating  vinertial    r and a rotating                                                       so we have
                                                                          dt                                     rotating

                                                          
                            dvinertial                  d   r                     
                                                                        arotating    vrotatingl
                               dt          inertial        dt inertilal
                      
                    dr
Since vinertilal                    , the above equation is
                     dt   inertial

                                  
                                 dvinertial                                           
                                                             vinertial  arotating    vrotating
                                    dt         inertial




                                                                     239
                                                                                      
                                                                                dv inertial
Again substituting vinertial  v rotating    r from equation 15.8 and ainertial                           ,
                                                                                        dt         inertial

we get
                                                                          
                    ainertial    v rotational      r   a rotating    v rotational

which gives
                                                                        
                            ainertial  a rotating  2  v rotating    (  r )


15.5 In the rotating frame the rod is stationary and there is a pseudo force – the centrifugal
force – acting on it. Thus the free-body diagram of the rod in the rotating frame is as
shown below. Notice that we are not taking the centrifugal force to be acting at the centre
of mass because it is different for different portions of the rod; it increases with the distance
of fom the axis.




                                                                    Centrifugal
                                                                    force




                                    mg



(i)      It is clear from the figure above that the rod is pulled out horizontally by the
         centrifugal force (in the rotating frame) and would therefore move out when
         displaced from the vertical position. Since we are dealing with a rigid body, we
         should be working in terms of totques. Thus at the given position, the centrifugal
         force applies a counterclockwise torque on the rod but at the vertical position,
         neither its weight nor the force by the pivot apply any torque. Thus the rod will
         tend to swing away from the axis. It will be clearer mathematically in part (ii).
(ii)     When the rod is at an angle  from the vertical, the component of force on a portion
                                                                m
         of length ds at distance s from the pivot is             ds( s sin  ) 2 (see figure).
                                                                l


                                                   240
                                             s


                                                                    m
                                                                      ds( s sin  ) 2
                                                                    l



                                        mg



Thus the torque due to the centrifugal force is

                                                                ml 2 2
                                    l
                                              m
                   centrifugal     s cos  ds( s sin  ) 
                                                             2
                                                                        sin  cos
                                    0
                                              l                    3

At the same time there is an opposing torque due to gravity and its value is
                                                               mgl
                                                  gravity        sin 
                                                                2
Balancing these two gives
                                                                 3g
                                                    cos 
                                                                2l 2
                                                       3g
Thus as long as cos  1, i.e. l 2                      , the rod will be at equilibrium at the angle
                                                        2
whose cosine is given above. Thus when disturbed from the vertical position it will move
out to this equilibrium position (with very small friction damping the motion but not
affecting   the     equilibrium              position).             For       angular    speeds   such   that
                            3g
cos  1, i.e. l 2           , the rod will not move out because the torque due to the
                             2
centrifugal force will not be able to overcome the torque due to the rod’s weight.


15.6 This problem is similar to the problem above except that there is no gravitational force
    involved here. If looked at from the top along CD, the free body diagram of the rod in
    the rotating frame, when it is at an angle  from the frame, looks as shown below.




                                                       241
                                                    m
                                                      ds( s sin  ) 2
                                                    l



                                                          s
                 A                                                                                         B
                                                                            




If we take a strip of width ds at a distance s from the axis CD, the mass of the strip will be
m       m                                     m
   wds  ds and the centrifugal force on it =   ds( s sin  ) 2 . The net torque on it is
lw      l                                     l
therefore

                                                                   ml 2 2
                                        l 2
                                                 m
                      centrifugal     s cos  ds( s sin  ) 
                                                                2
                                                                           sin  cos
                                      l 2
                                                 l                  12

                                                      
This shows that both   0 and                          are equilibrium positions of the sheet.
                                                      2
(i)    Thus       when          disturbed            slightly         ~ 0 ,   the   torque   on   the   sheet     is

                         ml 2 2
        centrifugal             and tends to take it away from that position. The equation of
                          12
       motion for slight disturbance from the position parallel to the frame is therefore
                                              ml 2  ml 2 2
                                      
                                     I                                
                                                                         or    2  0
                                              12       12
This gives
                                               (t )  A exp(t )  B exp(t )
Therefore the equilibrium position at   0 is an unstable equilibrium position.
                                                                                                               
(ii)   If we call the angle from the perpendicular position of the sheet  then                                   .
                                                                                                               2
       The expression for torque in terms of  remains
                                                                 ml 2 2
                                                centrifugal            sin  cos 
                                                                  12



                                                               242
     However, as the sheet is disturbed from this position, the torque has restoring nature. For
     small  therefore the equation of motion is
                                             ml 2    ml 2 2
                                      
                                     I                                   
                                                                             or    2  0
                                             12         12
     Thus the motion about this position is oscillatory and is given by
                                                 (t )  A cos(t )  B sin(t )
     or equivalently by
                                                 (t )  A cos(t )  B sin(t )
                                                                  
     Therefore the equilibrium position at                          is a stable equilibrium position.
                                                                  2


     15.7 We follow the convention for the directions as in the main text. There is no force in
          the horizontal plane. The equations of motion in the northern hemisphere are therefore
                                                  dv x
                                                        2v y  sin   2v z  cos 
                                                   dt
                                                  dv y
                                                        2v x  sin 
                                                   dt
                                                    dv z
                                                           g  2v x  cos
                                                     dt
     If   0 , vx and vy would remain zero for a particle thrown up. Thus correct to the linear
     order in  , the equations above are
                                         dv x                                     dv z
                                               2v z  cos                and         g
                                          dt                                       dt
     This gives
                                                           gt 2              dv x d 2 x
              v z (t )  v z 0  gt ; z (t )  v z 0 t               and         2  2v z 0  gt  cos 
                                                            2                 dt  dt
     Upon integration, the second equation gives
                                                                                   gt 3 
                    v x (t )   2v zo t  gt 2  cos and x(t )    v zo t 2 
                                                                                         cos
                                                                                    3  
Straightforward approach to solve for the deflection by the time the particle reaches the earth
again would be to find the time for it to come back to the ground and substitute it in the equation
for x(t). We have


                                                                  243
                              v z 0  2gh  2  9.8 100  44.27ms1 ,
Time taken to reach the highest point
                                                    vz0
                                               T        4.52 s
                                                     g
And the total time of flight = 2T = 9.04s.
Thus the total deflection by the time the stone comes back to the ground is
                                      9.8  9.043 
                    44.27  9.04 2 
         x(2T )                                   cos  1206.65 cos  8.8 cos cm
                                                   
                                           3      
The negative sing in front indicates that the deflection is towards the west. This is an interesting
result. One’s immediate answer would normally have been that the stone deflects one way while
going up and the other way while coming down, since the sign of vz changes, the net deflection
would have been zero. This does not happen because by the time the stone reaches its highest
point, it already has a westward velocity. To see its effect, let us calculate the distances in two
steps: first when the stone goes up and the second when it comes down.
While going up, the deflection of the particle by the time it reaches the highest point is
                                        9.8  4.523 
            x(T )    44.27  4.52 2 
                                                     cos  603.33 cos  4.4 cos
                                                     
                                             3      
At this point it has the horizontal velocity
                                                   
       v x (T )   2  44 .27  4.52  9.8  4.52 2  cos   200 .22  cos   1.46 cos  cm s1

The negative sing in front indicates that the velocity is towards the west. Now, as shown in
example 15.9, if a stone is dropped from a height of 100m, it deflects to the east by 2.2 cos cm.
However, now the stone has an initial westward speed of 1.46 cos cm s1. Thus the nest
deflection fo th stone while it is coming down will be
                               1.46 cos  4.52  2.2 cos  4.4 cos
This gives a total deflection of  8.8 cos during the entire flight.


15.8       The initial components of the velocity in the east, north and vertical direction are
                       v x 0  v0 cos cos         v y 0  v0 cos sin    v z 0  v0 sin 

       Equations of motion in are




                                                        244
                                           dv x
                                                 2v y  sin   2v z  cos 
                                            dt
                                           dv y
                                                 2v x  sin 
                                            dt
                                           dv z
                                                  g  2v x  cos 
                                            dt

Since  is very small, we will do our calculations correct to the first order in . This means that

for vx and vy, we substitute their initial values and for vz we substitute v z (t )  v0 sin   gt in the

equations above. This gives

                     dv x d 2 x
                          2  2v0 cos sin   sin   2v0 sin   gt  cos 
                      dt  dt
                     dv y d 2 y
                          2  2v0 cos cos  sin 
                      dt  dt
                     dv z d 2 z
                          2   g  2v0 cos cos  cos 
                      dt  dt

Integration of the last equation, with proper initial conditions v z 0 (t  0)  0, z (t  0)  0 , gives

                             v z  v0 sin   gt  2v0 t cos cos  cos 
                                                1 2
                             z  v0 sin  t      gt  v0 t 2 cos cos  cos 
                                                2

For time of flight T calculation, we substitute z = 0 and get

                             2v0 sin         2v sin   2v0  cos cos cos 
               T                             0       1 
                                                                             
                                                                              
                    g  2v0  cos cos cos     g                g          

Thus the time of flight of the projectile increases. Now we integrate the equation for motion in

the x direction, with the initial conditions v x 0 (t  0)  v0 cos cos  , x(t  0)  0 to get

                                                                                1     
             v x (t )  v0 cos cos  2v0  t cos sin  sin   2 v0 t sin   gt 2  cos
                                                                                2     
                                                                                  gt 3 
             x(t )  v0 t cos cos  v0  t 2 cos sin  sin    v0 t 2 sin  
                                                                                        cos
                                                                                   3  




                                                      245
Similarly, integration of the equation for the motion in the y direction gives

                            v y (t )  v0 cos sin   2v0  t cos cos sin 
                            y(t )  v0 t cos sin   v0  t 2 cos cos sin 

Thus the deflection of the projectile in the east and north direction are known. As a check, we

take a projectile thrown staright up   90  and find that the answer matches with that obtained

in problem 15.7. To know the deflection x(t ) in the direction of launch and y (t ) perpendicular

to it, we transform these as (see figure)


                                    Y’
                                                Y

                                                                            X’


                                                       α
                                                                           X
                                            O


                                         x'  x cos   y sin 
                                         y '   x sin   y cos 

This transformation gives

                                                              gt 3 
                         x(t )  v0 cos t   v0 sin  t 2 
                                                                    cos cos
                                                               3  

                                                                     gt 3 
                    y (t )  v0  t 2 cos sin    v0 sin  t 2 
                                                                           sin  cos
                                                                      3  

To calculate the range and deflection from the path during the flight, we substitute the time of

flight T for t. To the first order in , we then have

                                   2v0 sin   2v0  cos cos cos 
                              T             1 
                                                                   
                                                                    
                                       g                g          



                                                    246
                                     4v0 sin 2   4v0  cos cos cos 
                                       2
                                 T 
                                   2
                                                 1 
                                                                       
                                                                        
                                         g2                 g          

                                        8v0 sin 3   6v0  cos cos cos 
                                          3
                                 T3                1 
                                                                          
                                                                           
                                            g3                 g          

Therefor the range, up to the first order in  is

                                         2v0 sin     2v0  cos cos cos  
                  x (T )  v0 cos                 1 
                                                                            
                                                                             
                                             g                  g           
                                       4v 2 sin 2  g 8v0 sin 3  
                                                         3
                            v0 sin   0 2
                                                                 cos cos 
                                                                   
                                            g       3     g3      

                              2v0 sin  cos 4v0  sin  
                                2              3
                                                                  1 2 
                                                        cos   sin   cos cos 
                                                              2

                                                                        
                                                   2
                                     g           g                3

                                                               4v0  sin   2
                                                                 3
                                                                                    1 2 
So the change in the range due to the coriolis force is                     cos   sin   cos cos .
                                                                                          
                                                                     2
                                                                   g                3

The sideways deflection is given by

                        4v0 sin 2 
                          2
                                                            4v 2 sin 2  g 8v0 sin 3  
                                                                              3
    y (t )  v0                 cos sin    v0 sin   0 2
                                                                                      sin  cos
                                                                                        
                            g2                                   g       3     g3      
           4v0  sin 2 
             3
                              1                              
                              sin  sin  cos  cos sin  
                g2            3                              

                                        4v0  sin 2   1
                                          3
                                                                                      
Thus the sideways deflection is                        sin  sin  cos  cos sin   .
                                                      3                              
                                               2
                                             g

This shows that a projectile fired eastward   0  will deflect towards the south and that fired

westward   180  will deflect to the north. If we look at the equations of motion in the

northern and the southern hemisphere, they differ by the sign of the terms containing sin  ,

which is equivalent to changing the sign of λ to λ. From the expression derived above, it is then




                                                        247
clear that the change in the range will not change but the sideways deflection will change to

4v0  sin 2   1
  3
                                              
               sin  sin  cos  cos sin   in the southern hemisphere.
              3                              
       2
     g

15.9 Exact solution of the equations of motion on the surface of the earth. Thse are

                                    dv x
                                          2v y  sin   2v z  cos 
                                     dt
                                    dv y
                                          2v x  sin 
                                     dt
                                    dv z
                                           g  2v x  cos 
                                     dt

To get the exact solutions, we differentiate the first equation once and get

                                 d 2vx    dv y            dv
                                     2
                                       2       sin   2 z  cos 
                                  dt       dt              dt

Substituting for the derivatives of the velocity components appearing on the right, we get

                  d 2vx                                    d 2vx
                      2
                         4 2 v x  2 g cos               2
                                                                  4 2 v x  2 g cos 
                   dt                                       dt

The solution of this differential equation is given by the sum of the solution for its homogeneous

part and the particular solution. Thus it is

                                                                        g cos
                              v x (t )  A cos2t   B sin 2t  
                                                                          2

Here A and B are determined by the initial conditions. Subztituting this solution in the equation

for vy gives

                       v y (t )   A sin 2t   B cos2t   gt cos  sin   C

Here C is another constant. Similarly from the last equation we get

                    v z (t )   gt  A sin 2t   B cos2t   gt cos  cos   D

Here D is another constant.



                                                    248
No wlet us determine the constants appearing in the equations above for a stone dropped from

height h. Since we have solved a second-order differential equation for vx , we need two initial

                                                        dv x
condtions for it. These are v x (t  0)  0,                 (t  0)  0 . Then we have
                                                         dt

                                                        g cos
                                               A             , B0
                                                          2

This gives

                          g cos                                           sin 2t  
             v x (t )           1  cos2t              v y (t )  g             t  cos  sin   C
                            2                                             2            

                                                         sin 2t  
                                    v z (t )   gt  g            t  cos 2   D
                                                            2         

For the y and the z components of the velocity we have v y (t  0)  0, v z (t  0)  0 . These give

C = 0 and D = 0. Thus the complete solution for the velocity is

                               g cos                                      sin 2t  
                  v x (t )           1  cos2t         v y (t )  g             t  cos  sin 
                                 2                                        2            

                                                           sin 2t  
                                      v z (t )   gt  g            t  cos 2 
                                                              2         

In the limit of   0 this gives, to the first order in ,

                                 vx (t )  gt 2 cos       v y (t )  0     vz (t )  gt

This leads to the same answers as in example 15.9.



15.10 Cyclone building up in the Bay of Bengal. If only the motion in the horizontal plane is

considered, the equations of motion are




                                                           249
                                       dv x                   dy
                                             2v y  sin   2  sin 
                                        dt                    dt
                                       dv y                      dx
                                             2v x  sin   2 sin
                                        dt                       dt

We choose the coordinate system such that its origin is where the storm starts from. Thus the

initial conditions are x(t  0)  0, v x (t  0)  50 kmph , y (t  0)  0, v y (t  0)  0 . The second

equation above can be immediately integrated to obtain

                                            v y (t )  2x(t ) sin   C

where C is a constant. Since x(t  0)  0, v y (t  0)  0 , we get C = 0. Thus

                                              v y (t )  2x(t ) sin 

We substitute this in the first equation above to get

                          dv x d 2 x                             d 2x
                               2  2 sin   x or                  2 sin   x  0
                                                2                                  2
                                                                    2
                           dt  dt                                dt

This gives

                                   x(t )  A sin 2t sin    B cos2t sin  

With the initial conditions x(t  0)  0, v x (t  0)  50 kmph  13 .9ms 1 , this gives


                                                                                   sin 2t sin  
                             13.9                                          13.9
                   A               , B0                   x(t )  
                            2 sin                                       2 sin 

Thus

                           13.9 sin 2t sin                                      cos2t sin    C
                       dy                                                    13.9
          v y (t )                                           y(t )  
                       dt                                                   2 sin 

                                                                                 13.9
With the initial conditions y (t  0)  0, v y (t  0)  0 we get C                     . Thus finally we have
                                                                                2 sin 

for the coordinates of the eye of the cyclone




                                                        250
             x(t )  
                          13.9
                                  sin 2t sin   and y (t ) 
                                                                 13.9
                                                                         1  cos2t sin  
                         2 sin                                2 sin 

(i) This gives the equation for the trajectory as

                                                          2               2
                                               13.9       13.9 
                                    x2   y                      
                                              2 sin     2 sin  

       13 .9            13.9  10 -3
Since          m                             278 km , the above equation can also be written in
      2 sin     2  7.3  10 5  sin 20 

kilometers as

                                          x 2   y  278   278 2
                                                          2




Thus the trajectory is a circlular one with the centre at 0, 278  km.

(ii)   Thus the sense of the trajectory is clockwise as seen from above. On the other hand, since

we are in the northern hemisphere, the winds around the eye move in counterclockwise direction.

(iii) To find out where the cyclone hits the coast, we substitute in the equation above

x  10 km and get y  18.6 km and y  537.4 km . The second coordinate is for the return path.

Thus the cyclone hits the coast 18km north of where it started from.


15.11 The solution here is similar to the solution above except for the initial conditions. We

take the origin to be where the striker is played from. The x and the y axes are shown in the

figure below.




                                                    251
                                               y
                                          B
                                                                  
                                         1         2
                                                                      x
                                              A




                
For the problem    z . Thus the equation of motion in the xy plane are
                       ˆ

                                             dv x                 dy
                                                   2 v y   2 
                                              dt                  dt
                                             dv y              dx
                                                   2v x   2 
                                              dt               dt
Note that the signs on th right hand side are opposite to what they are in the equations of motion
in the orther hemisphere of the earth because the direction of the angular velocity is opposite.
Assuming that the striker is played at an angle  from the x axis, the initial conditions are
                x(t  0)  0, v x (t  0)  v cos ,     y (t  0)  0, v y (t  0)  v sin 

The second equation above is integrated to give
                                          v y (t )  2 x(t )  C

With the initial condition x(t  0)  0, v y (t  0)  v sin  , we get C  v sin  . Thus

                                        v y (t )  2 x(t )  v sin 

Substituting this in the first equation of motion, we get
                dv x d 2 x                                        d 2x
                     2  2  x  2v sin                           2  x  2v sin 
                                2                                             2
                                                             or      2
                 dt  dt                                           dt




                                                       252
The solution for the equation is then obtained as the sum of the solution for the homogeneous
part and the particular solution. Thus
                                                                        v sin 
                                 x(t )  A sin 2t   B cos2t  
                                                                          2
                                                                                      v cos     v sin 
With the initial condition x(t  0)  0, v x (t  0)  v cos , we get A                    ,B         . Thus
                                                                                        2         2
                                     v cos              v sin              v sin 
                           x(t )           sin 2t           cos2t  
                                       2                  2                  2
When substituted in v y (t )  2 x(t )  v sin  , this gives

                                     v cos sin 2t   v sin  cos2t 
                                 dy
                                 dt
Solved with the initial condition y(0) = 0, this gives
                                      v cos             v sin               v cos
                          y(t )            cos2t           sin 2t  
                                        2                 2                   2
Thus the trajectory is described by the coordinates
                                     v cos              v sin              v sin 
                           x(t )           sin 2t           cos2t  
                                       2                  2                  2
                                      v cos             v sin               v cos
                          y(t )            cos2t           sin 2t  
                                        2                 2                   2
These can be combined together to give the equation of trajectory as

                                     v sin       v cos 
                                                 2                 2              2
                                                              v 
                                x           y             
                                      2           2      2 
                                                v sin  v cos 
Thus the trajectory is a circle with centre at        ,        and the sence of direction is
                                                  2      2 
therefore counterclockwise. This implies that the striker should be played in direction 2. Now
we want the trajectory to be such that it pass through point B, i.e. through coordinates 0, L  .
When substituted in the equation for the trajectory, this gives
                                                              L
                                                     cos 
                                                              v
If the angle with AB is α, then
                                                L                  L 
                                      sin              sin 1     
                                                v                   v 


                                                        253
                                                                 L 
The striker is played in direction 2 making an angle   sin 1      from AB.
                                                                 v 
                                                                                   L
(ii) We first solve this problem for v  L correct to first order in                in an easy way. We
                                                                                    v
then solve the problem exactly.
                                                                     L
Easy approximate solution correct to first order in                     :
                                                                      v
                                                     dv x
                                                           2 v y 
                                                      dt
                                                     dv y
                                                           2v x 
                                                      dt
It is given that initially v y  v and v x  0 . Thus as we integrate the equations above v x   and

            dv y
therefore            2 . So up to order , vy can be taken to be a constant equal to v. Thus
            dt
                                                       y (t )  vt
and the integration of the first equation above gives
                                                 v x (t )  2y  C

Since v x ( y  0)  0 , we get C = 0. This gives

                        v x (t )  2vt       and        x(t )  vt 2 because x(t=0)=0.

                                                                                                 L
Therefore by the time the striker reaches the other end, time taken by it is T                    and its has
                                                                                                 v
                                  L2
moved a distance x(T )                   and has x-component of the velocity x(T )  2L . Now the
                                                                               
                                    v
                                                                                             L
striker starts its journey back after hitting then board. It again takes time T               to reach the side
                                                                                             v
it started from. Now the initial x-componet of the velocity is x(T )  2L and the displacement
                                                               

            L2
x(T )            . Now if we integrate the equation with these conditions, then
             v




                                                          254
                                   vx                T t

                                    dvx  2  v y dt   2y(T  t )  y(T )
                                   2L               T

                                                                      2 y (T  t )  L
                                                                      2 L  vt  L   2vt
This gives
                         v x (T  t )  2L  2vt                   or     v x (T  t )  2L  2vt

Keep in mind that t is being measured from T onwatds. Thus t = 0 inplies when the striker starts
                                                                                                 L2
its journey back. Integrating this with the initial condition x(T )                                  , we get
                                                                                                   v
      x (T  t )     t
                                                                                                          L2
           dx    2L  2vt dt   2Lt  vt                                   x(T  t )  
                                                                                                           v
                                                                                                                 2Lt  vt
          L2        0
      
           v

                   L
For t  T           this gives
                   v
                                                                       2L2
                                                            x(2T )  
                                                                         v
This is the final displacement when the striker comes back. One would have thought that the
striker will come back to its original position; that does not happen because when it starts its
journey back, it has an x-component of velocity that makes it drift. Had the x-component been
zero when it started back, the striker would have reached its original position.
Exact solution:
If the striker is played along line AB then   90 . The trajectory of the striker is then
                                                                 2                   2
                                                     v        v 
                                                x       y     
                                                             2

                                                    2        2 
and with time x(t) and y(t) vary as

                                             cos2t                                       sin 2t 
                                           v              v                                v
                              x(t )                                            y(t ) 
                                          2             2                               2
Thus for y = L, we have from the trajectory equation

                                   v                                             v   v    4 2 L2
                          x2           x  L2  0                       x         1
                                                                               2 2       v2




                                                                     255
Thus the striker gets displaced in x direction as it reached point B. One of the two answers

                             v   v    4 2 L2
(smaller magnitude              1         ) is when the particle is moving upward. The other
                            2 2       v2

                               v   v    4 2 L2
one (larger magnitude             1         ) is when the striker would have returned had it
                              2 2       v2
not hit the other side of the board. Further, when the striker reaches the other end, it would have
taken time T such that y(T) = L. This gives

                                                  2L                   4 2 L2
                        sin 2T   sin 2T       , cos2T   1 
                      v
               L
                     2                            v                      v2
The x and y components of the velocity at the other end are therefore

             x(t ) t T  v sin 2T   2L
                                                   y(t ) t T  v cos2T   v 2  4 2 L2
                                                    

From the above, one sees that the speed       x 2  y 2 remains unchanged during the motion. This is
                                                   

expected since the coriolis force is perpendicular to the velocity and therefore does no work on
the striker. As the striker hits the opposite side of the carom and returns back, its x velocity
remains unchanged and the y component of the velocity changes direction and it traverses a new
trajectory. Taking the time when it starts back to be t = 0, the new trajectory is thus determined
by the initial conditions

                    v   v    4 2 L2
        x(0)           1         , x(0)  2L,
                                                             y(0)  L,   y(0)   v 2  4 2 L2
                                                                          
                   2 2       v2
One could solve this problem with these initial conditions. However, we can make use of the
equations already derived for the trajectory if we describe the journey back with new coordinate
system with the origin at the point where the striker starts back. This is shown in the figure
below. In this coordinate system the initial conditions are

                      x(0)  0, x(0)  2L,
                                                 y(0)  0,   y(0)  v 2  4 2 L2
                                                              
So that we have initial speed v at an angle  such that

                               v 2  4 2 L2             v 2  4 2 L2          2L
                    tan                    , sin                   , cos 
                                   2L                        v                  v




                                                   256
               x                                         B
                                          

                                   v

                                                                                          v    4 2 L2    v
                                                                                   x      1         
                                                                                         2      v 2
                                                                                                         2



                                                y
                                                         A



Thus the x and y displacement of the striker in the new coordinate system is
                             v cos              v sin              v sin 
                   x(t )           sin 2t           cos2t  
                               2                  2                  2
                                                     4 2 L2                   4 2 L2
                         L sin 2t                       cos2t  
                                                v                         v
                                                  1                        1
                                               2      v2                2      v2


                                          v cos             v sin 
                                                 cos2t           cos2t  
                                                                                 v cos
                             y (t )  
                                            2                 2                 2
                                                             4 2 L2
                                     L cos2t                   sin 2t   L
                                                        v
                                                          1
                                                       2      v2
These can be combined together to give the equation of trajectory as
                                                              2
                                                            
                                 x  v 1  4 L
                                                                                     2
                                                                y  L 2   v 
                                               2 2
                                                                                  
                                    2       v2                              2 
                                                            
For y = L this then gives

                                                     v    4 2 L2    v
                                              x      1         
                                                    2      v 2
                                                                    2
While cutting the y = L line for the first time, the value of x will be

                                                     v    4 2 L2    v
                                              x      1         
                                                    2      v 2
                                                                    2
This is in addition to the previous displacement of the origin in A  B journey. Thus the total
displacement is


                                                         257
                                             v        4 2 L2 v
                                    x         1          
                                                       v2     
It is easily seen that for v  L this goes over to the approximate answer obtained earlier.




                                                 258
                                              Appendix A


A.1 In this problem there are four variables. So we make one dimensionless variable and equate
it to a constant. The variables of the problem and the corresponding dimensions are
                        p  ML1T 2        mM            n  L3    v  LT 1

Let the dimensionless variable be   p a m b n c v d . Since this is a dimensionless variable we
have, on equating the dimensions of the right hand side in the equation above to zero
                            ab  0      a  3c  d  0          2a  d  0
Since we want an expression for p, we take a = 1.                    Then the equations above give
b  1, d  2, and c  1 . Therefore the dimensionless variable is
                                                    p
                                              
                                                   mnv 2
Equating it to a constant k, we get
                                               p  kmnv 2


A.2 This problem has three variables. Therefore to apply Buckingham’s π theorem we form one
dimensionless variable and equate it to a constant. The variables of the problem and the
corresponding dimensions are
                               p  ML1T 2       T  MT 2          RL

Let the dimensionless variable be   p a T b R c . On equating the dimensions of the right hand
side in the equation above to zero we get
                             ab  0         2a  2b  0        ac 0
Taking a = 1, since we want an expression for p, we get b  1 and c  1 . Therefore the
dimensionless variable is
                                                    pR
                                               
                                                    T
Equating it to a constant k, we get
                                                       T
                                                pk
                                                       R




                                                 259
A.3 This problem has four variables. Therefore to apply Buckingham’s π theorem we form one
dimensionless variable and equate it to a constant. The variables of the problem and the
corresponding dimensions are
                       E  ML2T 2         h  ML2T 1        mM      RL

If the dimensionless variable is   E a h b m c R d , on equating the dimensions of the right hand
side in the equation above to zero we get
                        abc  0           2a  2b  d  0      2a  b  0
Since we want an expression for E, we take a = 1. Then the equations above give b  2, d  2,
and c  1 . Therefore the dimensionless variable is
                                                  EmR 2
                                             
                                                   h2
Equating it to a constant k, we get
                                                     h2
                                              pk
                                                    mR 2


A.4 This problem has four variables. Therefore we form one dimensionless variable and equate
it to a constant. The variables of the problem and the corresponding dimensions are
                      E
                         ML1T 2         h  ML2T 1        mM      n  L3
                      V
If the dimensionless variable is   E a h b m c n d , on equating the dimensions of the right hand
side in the equation above to zero we get
                        abc  0           a  2b  3d  0     2a  b  0
                                      E
Since we want an expression for         , we take a = 1. Then the equations above give c  1 and
                                      V
             5
b  2, d   , . Therefore the dimensionless variable is
             3

                                             
                                                  E V m
                                                  h 2n5 3
Equating it to a constant k, we get
                                            E    h2 5 3
                                              k   n
                                            V    m



                                                  260
A.5 This problem has five variables. Therefore to apply Buckingham’s π theorem we form
twodimensionless variables and write one of them as a function of the other. The variables of the
problem and the corresponding dimensions are
              dV
                  L3T 1      P  ML1T 2           ML1T 1       LL       RL
              dt
                                                                    a
                                                     dV 
If the dimensionless variables are expressed as        P  L R , on equating the
                                                             b c d e

                                                     dt 
dimensions of the right hand side in the equation above to zero we get
                     bc 0        3a  b  c  d  e  0       a  2b  c  0
                                                                                  dV
These give b  a, c  a and d  e  3a . Since we want an expression for           , we form one of
                                                                                  dt
the dimensioless variables with a = 1.        Then the equations above give b  1, c  1, and
d  e  3 . Taking d = 0 gives e  3 giving one dimensionless variable as
                                                dV  
                                        1        
                                                dt  PR
                                                          3



For the other dimensionless variable, we take a = 0. This gives b = c = 0 and d  e . Taking e
= 1, the second dimensionaless variable is
                                                       R
                                               2 
                                                       L
Thus the functional relationship between the variables can be expressed as
                                              1  f  2 
Now we have several choices for the function f. If we take it to be a constant, we get
                                          dV    PR 3
                                             k
                                          dt     
But this is not consistent with observations. We next take f  2   k 2 where k is a constant.
Then we get
                                          dV    PR 4
                                             k
                                          dt     L
This gives answer consistent with observations. For corrections to higher orders, we can take
higher powers of  2 and construct higher order functional relationships.




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