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Rotational Kinematics

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					Rotational Kinematics

      Tipler Ch. 9
                   Position
• In translational motion,               x=3
                                                         x
  position is represented by a       0          5
  point, such as x.
                                                linear
                                     p/2
• In rotational motion,
  position is                            r
  represented by an                        q
                                 p                  0
  angle, such as q, and
  a radius, r.
                                               angular
                                     3p/2
              Displacement
                               Dx = - 4
• Linear displacement is
  represented by the                                x
  vector Dx.                    0          5
                                           linear
                                 p/2
• Angular displacement
  is represented by Dq,
  which is not a vector,        Dq
  but behaves like one     p                   0
  for small values. The
  right hand rule
  determines direction.                   angular
                                3p/2
         Tangential and angular
             displacement
• A particle that rotates
  through an angle Dq                 s
  also translates through
  a distance s, which is                       r
  the length of the arc                   Dq
  defining its path.
• This distance s is related to the
  angular displacement Dq by the
  equation s = rDq.
             Speed and velocity
 • The instantaneous velocity
   has magnitude vT = ds/dt             s    vT
   and is tangent to the circle.

• The same particle rotates                       r
  with an angular velocity w =              Dq
  dq/dt.                           vT   w is outward
• The direction of the angular          according to
  velocity is given by the right            RHR
  hand rule.
• Tangential and angular
  speeds are related by the
  equation v = r w.
                 Acceleration
• Tangential acceleration is
  given by aT = dvT/dt.                 s     vT
• This acceleration is parallel
  or anti-parallel to the
  velocity.                                 Dq     r
• Angular acceleration of this    vT     w is outward
  particle is given by a =               according to
  dw/dt.                                     RHR
• Angular acceleration is
  parallel or anti-parallel to
  the angular velocity.                Don’t forget
• Tangential and angular                centripetal
  accelerations are related by         acceleration.
  the equation a = r a.
   Problem: Assume the particle is
            speeding up.
a) What is the direction of the           What changes if
   instantaneous velocity, v?
                                           the particle is
b) What is the direction of the
   angular velocity, w?                   slowing down?
c) What is the direction of the
   tangential acceleration, aT?
d) What is the direction of the
   angular acceleration a?
e) What is the direction of the
   centripetal acceleration, ac?
f) What is the direction of the overall
   acceleration, a, of the particle?
     First Kinematic Equation
• v = vo + at (linear form)
  – Substitute angular velocity for velocity.
  – Substitute angular acceleration for
    acceleration.
• w = wo + at (angular form)
   Second Kinematic Equation
• x = xo + vot + ½ at2 (linear form)
  – Substitute angle for position.
  – Substitute angular velocity for velocity.
  – Substitute angular acceleration for
    acceleration.
q = qo + wot + ½ at2 (angular form)
     Third Kinematic Equation
• v2 = vo2 + 2a(x - xo)
  – Substitute angle for position.
  – Substitute angular velocity for velocity.
  – Substitute angular acceleration for
    acceleration.
• w2 = wo2 + 2a(q - qo)
 Torque and Newton’s 2nd Law
• Rewrite SF = ma for rotating systems
  – Substitute torque for force.
  – Substitute rotational inertia for mass.
  – Substitute angular acceleration for
    acceleration.
S = I a
  : torque
  I: rotational inertia
  a: angular acceleration
                           Torque
Torque is the rotational analog of force that
causes rotation to begin.
Consider a force F on the beam that is applied a distance r from
the hinge on a beam. (Define r as a vector having its tail on the
hinge and its head at the point of application of the force.)

                                   Hinge (rotates)
A rotation occurs due to the
combination of r and F. In                           r
this case, the direction is
clockwise.                            Direction of rotation
What do you think is the
                                                                F
                                       Direction of torque is
direction of the torque?
                                       INTO THE SCREEN.
           Calculating Torque
• The magnitude of the torque is proportional to
  that of the force and moment arm, and torque is
  at right angles to plane established by the force
  and moment arm vectors. What does that sound
  like?
• =rF
  –  : torque
  – r: moment arm (from point of rotation to point of
    application of force)
  – F: force
                                                           Practice Problem
A 1.0-kg wheel of 25-cm radius is at rest on a fixed axis. A force of 0.45 N is applied
tangent to the rim of the wheel for 5 seconds.
a) After this time, what is the angular velocity of the wheel?
b)Through what angle does the wheel rotate during this 5 second period?
Rotational Inertia Calculations

  • I = Smr2 (for a system of particles)
  • I =  dm r2 (for a solid object)
  • I = Icm + m h2 (parallel axis theorem)
    – I: rotational inertia about center of mass
    – m: mass of body
    – h: distance between axis in question and
      axis through center of mass
                                                         Sample Problem
• A 2.0-kg mass and a 3.0-kg mass are mounted on opposite ends a 2.0-m
  long rod of negligible mass. What is the rotational inertia about the center of
  the rod and about each mass, assuming the axes of rotation are
  perpendicular to the rod?
                                                  Practice Problem

Derive the rotational inertia of a long thin rod of length L and mass M
     about the center.
                                                  Practice Problem

Derive the rotational inertia of a long thin rod of length L and mass M
     about a point 1/3 from one end
a)   using integration of I =  r2 dm
b)   using the parallel axis theorem and the rotational inertia of a rod
     about the center.
                                                      Practice Problem
Derive the rotational inertia of a ring of mass M and radius R about the
center using the formula I =  r2 dm.
        Angular Momentum
• Angular momentum is a quantity that tells
  us how hard it is to change the rotational
  motion of a particular spinning body.
• Objects with lots of angular momentum
  are hard to stop spinning, or to turn.
• Objects with lots of angular momentum
  have great orientational stability.
Angular Momentum of a particle

• For a single particle with known
  momentum, the angular momentum can
  be calculated with this relationship:
• L=rp
  – L: angular momentum for a single particle
  – r: distance from particle to point of rotation
  – p: linear momentum
                                                          Practice Problem
   Determine the angular momentum for the revolution of
a)  the earth about the sun.
b)  the moon about the earth.
                                                      Practice Problem
Determine the angular momentum for the 2 kg particle shown
a) about the origin.                              y (m)
b) about x = 2.0.
                                               5.0




                                                                   5.0   x (m)



                                              -5.0
                                                     v = 3.0 m/s
    Angular Momentum - solid
             object
• For a solid object, angular momentum is
  analogous to linear momentum of a solid object.
• P = mv (linear momentum)
  – Replace momentum with angular momentum.
  – Replace mass with rotational inertia.
  – Replace velocity with angular velocity.
• L = I w (angular momentum)
  – L: angular momentum
  – I: rotational inertia
  – w: angular velocity
                                                          Practice Problem
    Set up the calculation of the angular momentum for the rotation of the
earth on its axis.
Law of Conservation of Angular Momentum
 • The Law of Conservation of Momentum states
   that the momentum of a system will not change
   unless an external force is applied. How would
   you change this statement to create the Law of
   Conservation of Angular Momentum?
 • Angular momentum of a system will not change
   unless an external torque is applied to the
   system.
 • LB = LA (momentum before = momentum after)
                                                          Practice Problem

A figure skater is spinning at angular velocity wo. He brings his arms and legs
closer to his body and reduces his rotational inertia to ½ its original value.
What happens to his angular velocity?
                                                                    Practice Problem
A planet of mass m revolves around a star of mass M in a highly elliptical orbit. At
point A, the planet is 3 times farther away from the star than it is at point B. How does
the speed v of the planet at point A compare to the speed at point B?
                                                                   Practice Problem
A 50.0 kg child runs toward a 150-kg merry-go-round of radius 1.5 m, and jumps
aboard such that the child’s velocity prior to landing is 3.0 m/s directed tangent to the
circumference of the merry-go-round. What will be the angular velocity of the merry-
go-round if the child lands right on its edge?
 Angular momentum and torque
• In translational systems, remember that
  Newton’s 2nd Law can be written in terms of
  momentum.
• F = dP/dt
  – Substitute force for torque.
  – Substitute angular momentum for momentum.
 = dL/dt
  – t: torque
  – L: angular momentum
  – t: time
   So how does torque affect angular
            momentum?
• If  = dL/dt, then torque changes L with respect
  to time.
• Torque increases angular momentum when the
  two vectors are parallel.
• Torque decreases angular momentum when the
  two vectors are anti-parallel.
• Torque changes the direction of the angular
  momentum vector in all other situations. This
  results in what is called the precession of
  spinning tops.
If torque and angular momentum are
              parallel…

   Consider a disk rotating
   as shown. In what
   direction is the angular
   momentum?

   Consider a force applied
   as shown. In what                   r
   direction is the torque?
                                   F
   The torque vector is parallel to
   the angular momentum vector.        L is out
   Since  = dL/dt, L will increase
   with time as the rotation speeds.
                                        is out
If torque and angular momentum are
            anti-parallel…

   Consider a disk rotating
   as shown. In what
   direction is the angular
   momentum?

   Consider a force applied
   as shown. In what                    r
   direction is the torque?
                                   F
   The torque vector is anti-parallel
   to the angular momentum vector.      L is in
   Since  = dL/dt, L will decrease
   with time as the rotation slows.
                                         is out
If the torque and angular momentum
           are not aligned…

• For this spinning
  top, angular
  momentum and
  torque interact in a
  more complex way.      r   L
• Torque changes                   = r  Fg
  the direction of the
  angular                              
  momentum.                  Fg
• This causes the                  = dL/dt
  characteristic
  precession of a                      DL
  spinning top.