Rotational Kinematics Physics Presentations

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					          Rotational Kinematics




8/30/09                     11
          The analogies between translational and rotational motion
                       Linear        Rotational
                       Motion        Motion
     position          x                              angular
                                                      position
     velocity          v = dx/dt                      angular
                                                      velocity
     acceleration      a = dv/dt                      angular
                                                      acceleration
     mass              m                              moment of
                                                      inertia
     linear            p=mv                           angular
     momentum                                         momentum
     force             F = ma                         torque
     work                                             work

     power             P = Fv                         power
     kinetic energy                                   kinetic energy

8/30/09                                                          22
     Angular Variables
     Consider a planar object rotating about an axis perpendicular to its
     plane. The position is described as a point on the object by the
     coordinates r and Á, where Á is the angle measured with respect to the
     axis. When the object moves through an angle Á, the point moves a
     distance s along the arc. We define the angle Á in radians as
                                      s=rÁ




8/30/09                                                         33
  The linear velocity in meters per second of a point as moves around a circle is
  called the tangential velocity


  We define the angular velocity ! in radians per second as ! = d Á/dt. Thus
  v=r!
  If a point is accelarating along its path with tangential acceleration ®, then



  We define the angular acceleration ® in radians per second,
  Thus


  It is common to describe rotating objects by their frequency of revolution in
  revolutions per second. Since 2 rev is 2¼ rad, then




8/30/09                                                              44
     Problems
     1.) An electric drill rotates at 1600 rev/min. Through what angle does it
     turn in 4 ms? If it reaches this speed from rest in 0.32 s, what is its average
     angular acceleration?




     2.) The moon goes around the earth in about 27.3 days. What is its angular
     velocity?
     a.) Convert 27.3 days to seconds



     b.) Plug in the numbers


8/30/09                                                            55
    Moment of Inertia
    The moment of inertia represents the effort you need to get
    something to turn. Consider a continuous object to be composed
    of many small pieces of mass dm. We can write:




    If the mass is spread throughout the volume with density ½, the
    mass in a volume dV is dm = ½ dV. For a surface mass density
    ¾, dm = ¾ dA. For a line mass density ¸, dm = ¸ dx. In these
    cases the moment of inertia can be written:

                      or                   or




8/30/09                                                         66
Derivation of the Moment of Inertia for a uniform
round disk of thickness b and radius R.




So the moment of inertia is:




 8/30/09                                77
    Derivation of the Moment of Inertia for a
    uniform rod of dimensions of length l and radius
    r
                          axis

                                 x
                      0

                                     l–h

                  h


8/30/09                                    88
  We have a rod of uniform composition, same density everywhere, and it
  has a mass m and length l. We place the axis of rotation at O, a distance
  h from one of the ends. Simple enough, now we pick an element of
  volume of a short segment of length dx and cross-sectional area A, a
  distance x from O. That means:



  Since we need to find the total rotational inertia of the entire rod, we
  need to integrate from x = -h to x = l - h:




8/30/09                                                       99
 If the axis of rotation is at the end of the rod, h =
 0, which simplifies the equation greatly.
 Thin Rod About Axis Through One End
 Perpendicular to Length




If the axis of rotation is at the center, h = l/2,
which also simplifies the equation greatly.




 8/30/09                                         1010
Thin Rod About Axis Through Center Perpendicular
to Length




 8/30/09                              1111
          Moment of Inertia                          I
          hoop or cylindrical shell about its axis


          solid cylinder or disk

          rod about perpendicular axis through end


          rectangular plate         about
          perpendicular axis through center

          solid sphere

          spherical shell


8/30/09                                                  1212
  Problems
  1.) Calculate the moment of inertia of a solid cylinder of mass M and radius
  R about its axis.

  Solution

                      and




8/30/09                                                           1313
 Angular Momentum and Torque
 The angular momentum with respect to the origin of a particle with
 position r and momentum p = mv is defined as


 because ! is perpendicular to r ,         . If the angle between r and
 p is µ, then the magnitude of L is


 The time rate of change of the of the angular momentum is




  The cross-product of velocity and momentum is zero, because these vectors are
  parallel. The magnitude of the torque is




8/30/09                                                         1414
    Problems
    1.) You push a merry-go-round at its edge, perpendicular to the radius. If the
    merry-go-round has a diameter of 4.0 m and you push with a force of 160 N,
    what torque are you applying?
    Solution


    2.) You apply force on a wrench to loosen a pipe. If the wrench is 22 cm long
    and you apply 120 N at an angle of     with respect to the wrench, what
    torque are you applying?
    Solution


    3.) Because the force of gravity is directed along the line joining the centers
    of the sun and the earth, the force is negligible. The earth revolves in a slightly
    elliptical orbit around the sun. It is 1.47 £ 108 km away from and travelling at
    a speed of 30.3 km/s when it is nearest to the sun (perihelion). The earth’s farthest
    distance from the sun (aphelion) is 1.52 £ 108 km. How fast is the earth moving at
    its aphelion?
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      Solution




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