Conservation of Angular Momentum - Download as PowerPoint by os8lO1GE

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									Angular Momentum of a rigid object rotating about a fixed axis

                           L  I

                         dL d  I 
                            
                         dt   dt
            But for any rigid object the rotational
                     inertia is a constant

                         dL    d
                            I
                         dt    dt
                         dL
                             I
                         dt
                                                                
         Newton’s            dL                                 dp
         Second Law                    Analogous to   Fnet 
                             dt                                 dt
                 What if the system is isolated and closed?

Isolated – no external torques            Closed – no change in the mass


                                      dL
                                  
                                      dt
                                     dL
                                  0
                                     dt
                           L  constant

     Law of Conservation of Angular Momentum

           In any closed, isolated system, the
             angular momentum is constant
              Conservation of Angular Momentum Examples


1. The spinning volunteer.

                                  Li  L f
                              Iii  I f  f



                             Ii   i  I f
                                             f
               Conservation of Angular Momentum Examples


2. Stabilizing a Frisbee®.




         The classic Frisbee® has a heavy outer ridge which increases its
          rotational inertia and is then spun resulting in a large angular
                 momentum which resists changes to its motion.
                 Conservation of Angular Momentum Examples

3. Two disks are mounted on low friction bearings on the same axle and can be
   brought together so that they couple and rotate as one unit. The first disk, with
   mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev     . The second disk,
                                                             min
   with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev     in the same
                                                                   min
   direction as the first. They then couple together.

   a. What is their angular speed after coupling?

                                      Li  L f
                                l1i  l2i  L f
                        I11i  I 22i  I1  I 2  f
 1 mR 2    1 2m R 2    1 mR 2  1 2m R 2 
         1i              2i                       f
2           2               2         2          
                           1i  22i  3 f
                 Conservation of Angular Momentum Examples

3. Two disks are mounted on low friction bearings on the same axle and can be
   brought together so that they couple and rotate as one unit. The first disk, with
   mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev     . The second disk,
                                                             min
   with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev     in the same
                                                                   min
   direction as the first. They then couple together.

   a. What is their angular speed after coupling?

                           1i  22i  3 f
                                             1i  22i
                                     f 
                                                 3

                                     f 
                                             450 rev
                                                        min
                                                                    
                                                               2 900. rev
                                                                              min
                                                                                  
                                                                3
                                      f  750 rev min
                  Conservation of Angular Momentum Examples

3. Two disks are mounted on low friction bearings on the same axle and can be
   brought together so that they couple and rotate as one unit. The first disk, with
   mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev       . The second disk,
                                                               min
   with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev       in the same
                                                                     min
   direction as the first. They then couple together.
   b. If instead the second disk is set spinning at 900.rev        in the opposite direction
                                                              min
      of the first disk' s rotation, what is their angular speed after coupling?



                   f 
                           450 rev
                                       min
                                                  
                                              2  900. rev
                                                                 min
                                                                       
                                                 3

                    f  450 rev min
                     Conservation of Angular Momentum Examples

    2. Two children, each with mass M, sit on opposite ends of a narrow board with length
       L and mass M. The board is pivoted at its center and is free to rotate in a horizontal
       circle without friction. (Treat the board as a thin rod.)
      a. What is the rotational inertia of the board plus the children about a vertical axis
         through the center of the board?
                      l
                      2                      I total  I board  2 I child
M                              M                                                2
                                             I total
                                                          1       l
                                                         Ml  2M  
                                                            2
                l
                                                         12        2
                                                          1   1 2
                                             I total     Ml  Ml
                                                            2
                                                         12   2
                                                          7
                                              I total    Ml 2
                                                         12
                     Conservation of Angular Momentum Examples

    2. Two children, each with mass M, sit on opposite ends of a narrow board with length
       L and mass M. The board is pivoted at its center and is free to rotate in a horizontal
       circle without friction. (Treat the board as a thin rod.)
      b. What is the magnitude and direction of the angular momentum of the system if it
         is rotating with angular speed ωo in a clockwise direction as seen from above?


                                                   L  I
M                              M
                                                       7
                l                                  L  Ml 2o           Downward
                                                      12
                     Conservation of Angular Momentum Examples

While the system is rotating, the children pull themselves toward the center of the
board until they are half as far from the center as before.
  c. What is the ratio of the new rotational inertia to the initial rotational inertia?
                 l
                                                                                          2
                 4                      7
                                  I i  Ml 2
                                                              1       l
                                                        I f  Ml  2M  
                                                                2
                                       12                    12        4
  M                     M                                     5
                                                        I f  Ml 2
             l                                               24
                                                     5
                                            If         Ml 2
                                                   24
                                             Ii      7
                                                       Ml 2
                                                    12
                                            If       5
                                                  
                                             Ii     14
                    Conservation of Angular Momentum Examples

While the system is rotating, the children pull themselves toward the center of the
board until they are half as far from the center as before.
  d. What is the resulting angular speed in terms of ωo?
                l
                                                    Li  L f
                4
                                                 Iio  I f  f
  M                    M
            l                                          Ii
                                                   f  o
                                                       If

                                                       14
                                                   f  o
                                                        5
                    Conservation of Angular Momentum Examples

While the system is rotating, the children pull themselves toward the center of the
board until they are half as far from the center as before.
  e. What is the change in kinetic energy of the system as a result of the children
     changing their position? (From where does the added kinetic energy come?)
                l
                4           Ek  Ekf  Eki
                                 1         1
  M                    M    Ek  I f  f  I ii2
                                        2
                                 2         2
            l                                                  2
                                 1 5   2  14   1  7 Ml 2  2
                            Ek   Ml  o                  o
                                 2  24    5  2  12        
  *Note: L = constant                                   The added energy comes
                                 21 2 2
                            Ek  Ml o
                                                        from the work done by the
         Ek = increases
                                                           children when pulling
                                 40                        themselves forward.

								
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