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KINEMATICS

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					                                     KINEMATICS

1. INTRODUCTION TO KINEMATICS

Dynamics is the study of moving objects. The subject is divided into kinematics and
kinetics. Kinematics is the study of a body’s motion independent of the forces on the
body. It is the study of the geometry of motion without consideration of the causes of
motion. Kinematics deals only with relationships among the position, velocity,
acceleration, and time. Kinetics deals with both forces and motion.

2. PARTICLES AND RIGID BODIES

Bodies in motion can be considered particles if rotation is absent or insignificant.
Particles do not possess rotational kinetic energy. All parts of a particle have the same
instantaneous displacement, velocity, and acceleration.

A rigid body does not deform when loaded and can be considered as a combination of
two or more particles that remain at a fixed, finite distance from each other.

3. COORDINATE SYSTEMS

The position of a particle is specified with reference to a coordinate system.
A coordinate can represent a position along an axis, as in the rectangular coordinate
system or it can represent an angle, as in the polar, cylindrical, and spherical
coordinate systems.

In general, the number of degrees of freedom is equal to the number of coordinates
required to completely specify the state of an object. If each of the coordinates is
independent of the others, the coordinates are shown as holonomic coordinates.

4. CONVENTIONS OF REPRESENTATION
For a given particle, position, velocity, and acceleration can be specified in three
primary forms: vector form, rectangular coordinate form, and unit vector form.

   The rectangular coordinate form: Quantity F at (x, y, z)

   The vector form:        F=Fxi+Fyj+Fzk

                                    xi  yj  zk
   Unit vector form:       FF
                                     x y z
                                      2    2       2
5. LINEAR PARTICLE MOTION

A linear or rectilinear system is one in which particle move only in straight lines. The
relationship among position, velocity, and acceleration for a linear system are given by

                                             s(t )   v(t )dt    a(t )dt dt

                                                           ds (t )
                                             v(t )                  a(t )dt
                                                            dt
                                                                       2
                                         dv(t ) d s(t )
                                 a(t )                                   2
                                          dt     dt
 The average velocity and acceleration over a period from t to                   1
                                                                                     t2   are
                          2

                           v(t )dt          s s
                   v     1
                                             2        1

                              t t           t t
                    ave
                               2     1        2    1



                          2

                           a(t )dt          v v
                   a     1
                                             2    1

                              t t           t t
                    ave
                              2      1        2    1
6. DISTANCE AND SPEED

displacement (or linear displacement) is the net change in a particle’s position, as
determined from the position function. Distance is a scalar quantity, equal to the
magnitude of displacement. When specifying distance, the direction is not
considered.
              Displacement = s(t )  s(t )
                                  2      1




Similarly velocity and speed have different meanings: velocity is a vector quantity;
Speed is a scalar quantity and is equal to the magnitude of velocity.

7. UNIFORM MOTION
The term uniform motion means uniform velocity. The velocity is constant and the
acceleration is zero. For a constant velocity system, the position function varies
linearly with time.

               s(t )  s  vt
                       0

               v(t )  v
               a(t )  0
8. UNIFORM ACCELERATION

The acceleration is constant in many cases. (Gravitational acceleration, where a  g , is
a notable example.) If the acceleration is constant, we have

               a(t )  a

               v(t )  a  dt  v  at
                                0

                                           1
                s(t )  a  dt  s  v t  at
                            2
                                    0    0
                                                 2
                                                     43.13
                                           2

9. LINEAR ACCELERATION

Linear acceleration means that the acceleration increases uniformly with time.

10. PROJECTILE MOTION

A projectile is placed into motion by an initial impulse.

Consider a general projectile set into motion at an angle of  (from the horizontal
plane) and initial velocity v . Its range is R, the maximum altitude attained is H, and
                            0
the total flight time is T. In the absence of air drag, the following rules apply to the
case of a level target.

     . The trajectory is parabolic.

     . The impact velocity is equal to the initial launch angle,  .

     . The range is maximum when   45 . 0




     . The time for the projectile to travel from the launch point to the apex is
       equal to the time to travel from apex to impact point.

     . The time for the projectile to travel from the apex of its flight path to
       impact is the same time an initially stationary object would take to fall a
       distance H.
11. ROTATIONAL PARTICLE MOTION

Rotational particle motion (also known as angular motion and circular motion) is
motion of a particle around a circular path.

Rotating particle is defined by: angular position, , angular velocity,  , and angular
acceleration,  , functions. These variables are analogous to the s (t), v (t), and a (t)
functions for linear systems.


                (t )   w(t )dt  a(t )dt   2




                         d (t )
               w(t )               (t )dt
                          dt

                        d (t ) d  (t )
                                      2

                (t )                   2
                         dt      dt

The average velocity and acceleration are
                                    2

                                      (t )dt          
                                  1
                                                       2         1

                                        t t           t t
                      ave
                                        2      1        2     1




                            2

                             (t)dt
                                  2  1
                ave       1
                               
                       t 2  t1 t 2  t1


12. RELATION BETWEEN LINEAR AND ROTATIONAL VARIABLES

In general, the linear variables can be obtained by multiplying the rotational
variables by the path radius r.

                      v  r
                       t




             v  v cos   r cos 
               t ,x             t




             v  v sin   r sin 
               t,y              t
                       dv
               a 
                 t
                           r
                            t


                       dt

If the path radius is constant, as it would be in rotational motion, the linear distance
(i. e., the arc length) traveled is

                     s  r

13. NORMAL ACCELERATION

In general a restraining force will be directed toward the center of rotation.
Whenever a mass experiences a force, acceleration is acting. Since the inward
acceleration is perpendicular to the tangential velocity and tangential acceleration.
It is known as normal acceleration, a .     n
                        2
                    v
                a   r  v 
                 n
                        t           2
                                        t
                     r

The resultant acceleration, a, is the vector sum of the tangential and normal
Acceleration. The magnitude of the resultant acceleration is

               a a a      2
                            t
                                2
                                n
The x- and y- components of the resultant acceleration are

               a  a sin   a cos 
                x    n        t




               a  a cos   a sin 
                y    n            t




The normal and tangential accelerations can be expressed in terms of the x- and y-
Components of the resultant acceleration.

               a  a sin   a cos 
                n    x         y




               a  a cos   a sin 
                t    n            y




14. CORIOLIS ACCELERATION

Consider a particle moving with linear radial velocity v away from the center of a
                                                          r

flat disc rotating with constant velocity  . Since v  r , the particle’s tangential
                                                     t

velocity will increase as it moves away from the center of rotation. This increase is
said to be produced by the tangential coriolis acceleration, a .
                                                              c




               a  2v 
                c        r
15. RELATIVE MOTION

The term relative motion is used when motion of a particle is described with respect
to something else in motion. The particle’s position, velocity, and acceleration may
be specified with respect to another moving particle or with respect to a moving
frame of reference.

Consider two particles, A and B, are moving with different velocities along a
straight line. The separation between the two particles at any specific instant is the
Relative position, s , of B with respect to A. calculate as the difference between
                         B A


their two absolute positions.

               s   B A
                         s s B   A




Similarly, the relative velocity and relative acceleration of B with respect to A are
the differences between the two absolute velocities and absolute accelerations,
respectively.

               v   B A
                         v v B   A

               a   BA
                         a a B   A
16. DEPENDENT MOTION

When the position of one particle in a multiple-particle system depends on the
position of one or more other particles, the motions are said to be dependent. A
block-and-pulley system with one fixed rope end is a dependent system.


   Since the length of the rope is constant, the sum of the rope segments
    representing distances between the blocks and pulleys is constant.
    By convention, the distances are measured from the top of the block to the
    Support point. NOTE: There are two ropes supporting block A, two
    Ropes supporting block B, and one rope supporting block C,

         2s  2s  s  cons tan t
            A     B   c




   The relative relationships between the block’s velocities or accelerations are the
    same as the relationships between the block’s positions.

                2v  2v  v  0
                  A       B       c




                2a  2a  a  0
                  A       B   c
                                        KINETICS

1. INTRODUCTION TO KINETICS

Kinetics is the study of motion and the forces causing motion.

2. RIGID BODY MOTION

The most general type of motion is rigid body motion.

     • Pure translation: the orientation of the object is unchanged as it’s position
     changes. (Motion can be in straight or curved paths)

     • rotation about a fixed axis: All particles with in the body move in concentric
     circles about he axis of rotation.

     • general plane motion: The motion can be represented in two dimensions.
     • motion about a fixed point: This describes any three dimensional motion with one
     fixed point, such as a spinning top or a truck mounted crane. The distance from a
     fixed point to any particle in the body is constant.

     • general motion: This is any motion not falling into one of the other four categories.
3. STABILITY OF EQUILIBRIUM POSITIONS:

Stability is defined in terms of a body’s relationship with an equilibrium position. Stable
equilibrium exists if a body returns to original equilibrium position after experiencing a
displacement. Unstable equilibrium exists if the body moves away from the equilibrium
position.

4. CONSTANT FORCES

Forces that do not vary with time are constant forces.

External forces are responsible for the external motion of the body. Internal forces are
the forces that hold together parts of a rigid body.

5a. LINEAR MOMENTUM

The vector linear momentum (usually just momentum) is defined by


                     p  mv [SI]
                            mv
                       p      [U.S]
                            gc




Momentum has direction as the velocity vector. Momentum has the units of force        time
(e.g., lbf-sec or N.s).

Momentum is conserved when no external forces act on a particle. If no forces act on the
particle, the velocity and direction of the particle are unchanged. The law of conservation
of momentum states that the linear momentum is unchanged if no unbalanced forces act
on the particle.


This does not prohibit the mass and velocity from changing, however, only the product of

               m v  m v
                   o   o         f   f
5B. ENERGY AND WORK

The energy of a mass represents the capacity of the mass to do work

Recall from Thermo and Fluids:

     d        V 2
                                 p V    2
                                                     p V   2
                                                                
Q  W    u     gz     u     gz     u     gz 
       dt       2           out
                                    2           
                                                 in
                                                        2      

For dynamics we have work principle energy or

       V            V     
            2              2

W   m  gz    m   gz 
        2
        2
                      2
                       1
                             

note:
                       2
                   V
Kinetic Energy  m
                    2
Potential Energy  mgh

                                1
For a spring: Potential Energy  kx   2


                                2
6. THE BALLISTIC PENDULAM

Ballistic pendulum: A projectile of known mass but unknown velocity is fired into a
hanging target (the pendulum). The projectile is captured by the pendulum which moves
forward and upward. Kinetic energy is not conserved during impact, because some of the
kinetic energy is transformed into heat. However momentum is conserved during impact.

Note: No external forces act on the block during impact and momentum of the system is
conserved.

                    Pbeforeimpa
                              ct
                                       P         t
                                         afterimpac




                    m v proj    proj
                                        (m  m ) v
                                               proj              pend   pend




In terms of kinetic energy we have:


                    1
                      (m  m ) v
                               proj         pend
                                                      2
                                                          pend    (m  m )gh
                                                                        proj   pend
                    2
                          v   pend
                                      2gh



7. ANGULAR MOMENTUM

The vector angular momentum (also known as moment of momentum) taken about a point
O is the moment of the linear momentum vector. Angular momentum has units distance 
force  time (e.g., ft-lbf-sec or N.m.s ). It has the same direction as the rotation vector and
can be determined from the vectors by the use of the right hand rule.

                          h  r  mv
                              O
                                              [SI]

                                     r  mv
                          h  O
                                              [US]
                                        g
                                        c




For rigid body rotating about an axis passing through its center of gravity located at point
O, the scalar value of angular momentum is given by

                          h  I [SI]
                              O
                               I
                         h 
                           O
                                  [U.S]
                               gc




8. NEWTON’S FIRST LAW OF MOTION

First Law: A particle will remain in a state of rest or will continue to move with constant
velocity unless an unbalanced external force acts on it.

Law of conservation of momentum form: If the resultant external force acting on a
particle is zero, then the linear momentum P of the particle is constant.

9. NEWTON’S SECOND LAW OF MOTION

Newton’s Second Law: The acceleration of a particle is directly proportional to the force
acting on it and inversely proportional to the particle mass. The direction of acceleration
is same as the force of direction.

This law can be stated in terms of the force vector required to cause a change in
momentum. The resultant force is equal to the rate of change of linear momentum.
                               dP
                          F
                               dt



If the mass is constant with respect to time, The scalar form F is

                                    dv
                          Fm           ma   [SI]
                                    dt

                               m dv ma
                          F                 [U.S]
                               g dt g
                                c         c




For a rotating body, the torque T, required to change the angular momentum is

                               dh
                          T         o


                                dt

If the moment of inertia is constant, the scalar form is
                                    d
                          TI           I
                                    dt

                                 I d I
                          T         
                                g dt g
                                    c         c




10. CENTRIPETAL FORCE

Newton’s second law says that there is a force for everything that a body experiences. For
a body moving around a curved path, the total acceleration can be separated into
tangential and normal components. The force associated with the normal acceleration is
known as the centripetal force.
                                        2
                                mv
                     F  ma 
                      c         n
                                        t
                                          [SI]
                                 r
                                2
                          mv
                     F 
                      c
                                t
                                   [U.S.]
                           gr
                            c




The so-called centrifugal force is an apparent force on the body directed away from the
center of rotation. The centripetal and the centrifugal forces are equal in magnitude and
opposite in direction.
11. NEWTON’S THIRD LAW OF MOTION

Third law: For every acting force between two bodies, there is an equal but opposite
reacting force on the same line of action.

           F reacting
                         F
                           acting




12. DYNAMIC EQUILIBRIUM

An accelerating body is not in static equilibrium. Accordingly, the familiar equations of
statics (  F  0 and  M  0 ) do not apply. However, if the so-called inertial force ma, is
included in the static equilibrium equation, the body is said to be in dynamic equilibrium.

                                        F  ma  0 [SI]

                                          ma
                                    F      0       [U.S.]
                                          g c

13. FLAT FRICTION
Friction is a force that always resists motion or impending motion. It always acts parallel
to the contacting surfaces. The frictional force, F , exerts on a stationary body is known
                                                   f


as static friction. Coulomb friction, and fluid friction. If the body is moving, the friction
is known as dynamic friction and is less than the static friction.

The actual magnitude of the frictional force depends on the normal force, N, and the
coefficient of friction, f, between the body and the surface. For a body resting on a
horizontal surface, the normal force is the weight of the body.

                              F  N

17. BELT FRICTON

Friction between a belt, rope, or band wrapped around a pulley or a sheave is responsible
for the transfer of torque. The basic relationship between these belt tensions and the
coefficient of friction neglects centrifugal effects and is given by the equation

                     F
                       e
                       m ax   f


                     F m in




the net transfer torque is
                     T  (F  F )r
                                max        min




The power transmitted by the belt running at tangential velocity v is given by
                                                                     t




                     p  ( Fm ax  Fm in )vt




18. ROLLING RESISTANCE

Rolling resistance opposes motion, but it is not friction. Rather it is due to the
deformation of the rolling body and the supporting surface. Rolling resistance is
characterized by a coefficient of rolling resistance, a, which has units of length. Since this
deformation is very small, the rolling resistance in the direction of motion is

                             mga
                     F 
                      r
                                 [SI]
                              r
                            mga w  a
                     Fr                        [U.S.]
                            rg c   r


19. MOTION OF RIGID BODIES
When a rigid body experiences pure translation, its position changes without any change
in orientation. At any instant, all points on the body have same displacement, velocity
and acceleration. The behavior of a rigid body in translation is given by Eqs. below,

                    F  ma [SI]
                      x        x                 44.36
                    F  ma [SI]
                      y        y                 44.37
When the torque acts on a rigid body we have:

                   T  I           [SI]

                          I
                   T               [U.S]
                          gc




20. IMPULSE

Impulse, Imp, is a vector quantity equal to the change in momentum. Units of linear
impulse are the same as for linear momentum: lbf-sec and N.s. Units of lbf-ft-sec and
N.m.s are used for angular impulse. The scalar magnitudes of linear impulse and the
angular impulse is
                                   t2

                      Im p   Fdt [linear]
                                   t1
                                   t2

                       Im p   Tdt [angular]
                                   t1




If the applied force or torque is constant, impulse is easily calculated. Large force acting
for a very short period of time is known as an impulsive force.

                       Im p  F( t  t ) [linear]
                                         2     1




                       Im p  T (t 2  t1 )   [angular]

If the impulse is known, the average force acting over the duration of the impulse is

                                Im p
                       I 
                                 t
                        avg




21. IMPULSE-MOMENTUM PRINCIPLE

The change in momentum is equal to the applied impulse. This is known as the impulse-
momentum principle. For a linear system with constant force and mass,

                       Im p  p
                      F( t  t )  m( v  v )
                         2       1      2       1
                                                        [SI]

                                 m( v  v )
                      F( t  t ) 
                         2       1
                                        2
                                                [U.S.]
                                                1


                                      g     c

For an angular system with constant torque and moment of inertia, the analogous
equations are

                      T( t  t )  I( w  w )
                           2     1      2       1
                                                        [SI]

                                     I( w  w )
                      T( t  t ) 
                           2     1
                                        2           1
                                                        [U.S.]
                                         g  c




22. IMPULSE-MOMENTUM PRINCIPLE IN OPEN SYSTEMS

Note: The impulse-momentum principle can be used to determine the forces acting on
flowing fluids. In terms of a mass flow rate we have

                               mv
                      F            mv [SI]                  44
                                t
                               mv mv
                       F              [U.S.]                                     44.55(b)
                               g t
                                c
                                      g     c




23. IMPACTS

In an impact or collision, Contact is very brief, and the effect of external force is
insignificant. Therefore, momentum is conserved, even though energy may be lost.
Consider two particles, initially moving with velocities v and v on a collision path. The
                                                                               1       2
                                                                                              1
conservation of momentum equation can be used to find the velocities after impact, v          1
      1
and v .
      2

                     m v m v m v m v
                       1   1        2   2   1
                                                1
                                                1    2
                                                         1
                                                         2




The impact is said to be an inelastic impact if kinetic energy is lost. The impact is said to
be perfectly inelastic or perfectly plastic if the two particles stick together and move on
with the same final velocity.

The impact is said to be an elastic impact only if kinetic energy is conserved.

                    m v m v m v m v
                       1
                           2
                           1        2
                                        2
                                        2   1
                                                12
                                                1    2
                                                         12
                                                         2    elasticim pact




24. COEFFICIENT OF RESTITUTION
A simple way of determining whether the impact is elastic or inelastic is by calculating
the coefficient of restitution, e. The collision is inelastic if the e <1.0, perfectly inelastic if
e=0, and elastic if e=1.0. The coefficient of restitution is a ratio

                 relativeseperationve locity v  v         1    1

              e                                          1    2


                  relativeapproachvelocity v  v           2    1

26. NEWTON’S LAW OF GRAVITATION

Newton’s law of gravitation also known as Newton’s law of universal gravitation, is

                                      Gm m
                                 F       1
                                          2
                                              2


                                        r


                        VIBRATING SYSTEMS

FREE VIBRATION

The simple mass and ideal spring is an example of free vibration. After the mass is
displaced and released, it will oscillate up and down. Since there is no friction, (i.e., the
vibration is undamped). This system is described by
                                    F  0;ma  kx  0
                                      2
                                  dx
                               m      kx [SI]
                                          2
                                  dt
w is known as the natural frequency of vibration or angular frequency. It has units of
radians per second. It is not the same as the linear frequency, f, which has units of hertz
(formerly known as cycles per second). The period of oscillation, T, is the reciprocal of
the linear frequency.

                          k
                    w      [SI]
                          m

                          kg
                    w         c
                                          [U.S.]
                           m

                         w 1
                    f     
                         2 T

FREE ROTATION
The so called torsional pendulum is similar to the spring mass combination, ignoring the
mass and moment of inertia of the shaft, the differential equation is

                                       d 2

                                k I
                                   r          2
                                                  [SI]
                                       dt

                                                      GJ d G
                                                           4

  kr   is The torsional spring constant, and      k 
                                                   r
                                                        
                                                      L   32L

DAMPED FREE VIBRATIONS

When friction resists the oscillatory motion, the system is said to be damped. This third
type of friction is known as viscous damping.

The viscous damping force can be a function of v. With linear damping, the damping
force is proportional to velocity, C is known as the coefficient of viscous damping.

                                dx
                      F  Cv  C
                                dt
The differential equation of motion is
                        2
                      dx           dx
                    m     kx  C
                            2
                      dt           dt

UNDAMPED, FORCED VIBRATIONS

When an external disturbing force, F(t), acts on the system, the system is said to be
forced.

Consider the sinusoidal periodic force with a forcing frequency  and maximum value F .
                                                               f                     0




                    F( t )  F cos  t
                                0   f




The differential equation of motion is
                        2
                      dx
                    m     kx  F cos  t
                            2            0   f
                      dt

14. DAMPED, FORCED VIBRATIONS

If a viscous damping force is added to a sinusoid ally forced system, the differential
equation of motion is
   2
  dx          dx
m     kx  C  F cos  t
       2           0    f
  dt          dt