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					      Rigid-Body Rotation

            rotating and revolving


§ 9.1–9.2
       Fixed-Axis Rotation
• Much easier than changing-axis
• Hard enough on its own
              Radians
A dimensionless angle measure

                          arc length
                      distance from axis



          length
                 = dimensionless!
          length
      Radian Measurements
• Complete cycle = 2pr

                         r




• Complete cycle = 2p radians
• 1 radian = 57.3°
         Periodic Processes
• You will often encounter radians and
  angular speed for repeating processes
• Not restricted to rotation or circular motion
               Poll Question
What is the equivalent of 180° in radians?
A. p/4.
B. p/3.
C. p/2.
D. p.
               Poll Question
What is the equivalent of 45° in radians?
A. p/4.
B. p/3.
C. p/2.
D. p.
    Angular Position
           2

               s

     r             1
q


         • Arc length s
         • Radius r
                      s
         • Angle q = r
           Angular Speed
Rate of change of position

 • Angular speed w

           dq   d s    1 ds = vT
        w=    =      = r
           dt   dt r     dt   r

 • vT = tangential speed
                Group Work
A particle moves in a circular path of radius r.
a) What is its angular displacement q after 2.0
   complete rotations?
b) What is its path length s after 2.0 complete
   rotations?
c) If it takes time t to complete 2.0 rotations, what
   is its average tangential speed v?
d) If it takes time t to complete 2.0 rotations, what
   is its average angular speed w?
           Angular Velocity
What is the direction of angular motion?
Right-hand rule:
• Curl right-hand fingers in the direction of
  rotation.
• Extended right thumb points in the
  direction of w.
• Rotation Axis || w.
        Angular Acceleration
Rate of change of angular velocity

         dw d2 s   1 d2s a||
      a=   = 2 r = r   2= r
         dt dt       dt

• a|| = tangential acceleration
• Valid for a fixed axis of rotation
  (acceleration about the w axis)
   Angular Kinematic Formulas
Constant a, a || w

      w = w0 + at
      q = q0 + w0t + 1/2 at2
      w2 = w02 + 2a(q – q0)

Note the similarity to the linear kinematic
formulas!
                 Poll Question
A ladybug sits at the outer edge of a merry-go-round, and a
gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution
once each second. The gentlemen bug's angular speed is


A.   half the ladybug's
B.   the same as the ladybug's
C.   twice the ladybug's
D.   wicked fast
E.   impossible to determine
                 Poll Question
A ladybug sits at the outer edge of a merry-go-round, and a
gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution
once each second. The gentlemen bug's linear speed is


A.   half the ladybug's
B.   the same as the ladybug's
C.   twice the ladybug's
D.   wicked fast
E.   impossible to determine
                  Poll Question
A ladybug sits at the outer edge of a merry-go-round that is
turning and slowing down. At the instant shown, the radial
component of the ladybug's (Cartesian) acceleration is


A.   in the +x direction
B.   in the –x direction
C.   in the +z direction
D.   in the –z direction
E.   in the +y direction
F.   in the –y direction
                  Poll Question
A ladybug sits at the outer edge of a merry-go-round that is
turning and slowing down. At the instant shown, the
tangential component of the ladybug's (Cartesian)
acceleration is
A.   in the +x direction
B.   in the –x direction
C.   in the +z direction
D.   in the –z direction
E.   in the +y direction
F.   in the –y direction
                  Poll Question
A ladybug sits at the outer edge of a merry-go-round that is
turning and slowing down. At the instant shown, the vector
expressing her angular velocity is


A.   in the +x direction
B.   in the –x direction
C.   in the +z direction
D.   in the –z direction
E.   in the +y direction
F.   in the –y direction
                  Poll Question
A ladybug sits at the outer edge of a merry-go-round that is
turning and slowing down. At the instant shown, the vector
expressing her angular acceleration is


A.   in the +x direction
B.   in the –x direction
C.   in the +z direction
D.   in the –z direction
E.   in the +y direction
F.   in the –y direction
      Rigid-Body Rotation

            moments of inertia


§ 9.3–9.4
Rolling without slipping




Center-of-mass (axis) speed
           v = rw
   Rolling without slipping




Center-of-mass (axis) acceleration
             a|| = ra
Rolling without slipping




Rim centripetal acceleration
      a = v2/r = w2r
                Poll Question
Which has the greatest kinetic energy?
A.   A bar rotating at speed w about its long
     axis.
B. A bar rotating at speed w about its
   middle, perpendicular to its long axis.
C. A bar rotating at speed w about its end.
D. All of these have the same kinetic energy.
E. Cannot be determined.
         Rotating Kinetic Energy
                K = 1/2 Iw2

I = moment of inertia
  (rotational analogue of mass)

units?
          Moment of Inertia
Of a particle of mass m, distance r from axis

               1/2 Iw2 = 1/2 mv2

• What is I?
             Poll Question
Two cylindrical objects with equal mass and
radius are rotated about their axes. Which
has the greater moment of inertia?

A. A solid cylinder.

B. A hollow cylinder.

C. Their moments are the same.
         Moments of Inertia
Usually expressed in the form I = CMR2

C depends on the shape (mass distribution)
 of the object
                Moments of Inertia




Source: Young and Freedman, Table 9-2 (p. 291).

				
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