1. Rotational Energy

2. Rigid Body  Real objects have mass at points other than the center of mass.  Each point in an object can be measured from an origin at the center of mass.  If the positions are fixed compared to the center of mass it is a rigid body. riri

3. Translation and Rotation  The motion of a rigid body includes the motion of its center of mass.  This is translational motion  A rigid body can also move while its center of mass is fixed.  This is rotational motion.  v CM

4. Rotational Motion  Kinematic equations with constant linear acceleration were defined. v = v 0 + atv = v 0 + at x = x 0 + v 0 t + ½at 2x = x 0 + v 0 t + ½at 2 v 2 = v 0 2 + 2a(x - x 0 )v 2 = v 0 2 + 2a(x - x 0 )  Kinematic equations with constant angular acceleration are similar.  =  0 +  t  =  0 +  0 t + ½  t 2  2 =  0 2 + 2  (  -  0 )

5. Compact Disc  A CD spins so that the tangential speed is constant. The constant speed is 1.3 m/sThe constant speed is 1.3 m/s The inner radius is at 0.023 mThe inner radius is at 0.023 m The outer radius is at 0.058 mThe outer radius is at 0.058 m The total time is 74 min, 33 s = 4500 sThe total time is 74 min, 33 s = 4500 s  Find the angular velocity at the beginning (inner) and end.  Find the constant angular acceleration.  Angular velocity is  = v/r. Inner:  = 1.3 m/s / 0.023 m = 57 rad/s. Outer:  = 1.3 m/s /0.058 m = 22 rad/s.  Angular acceleration is

6. Circular Energy  Objects in circular motion have kinetic energy. K = ½ m v 2K = ½ m v 2  The velocity can be converted to angular quantities. K = ½ m (r  ) 2K = ½ m (r  ) 2 K = ½ (m r 2 )  2K = ½ (m r 2 )  2  The term (m r 2 ) is the moment of inertia of a particle.

7. Integrating Mass  The kinetic energy is due to the kinetic energy of the individual pieces.  The form is similar to linear kinetic energy. K CM = ½ m v 2 K rot = ½ I  2

8. Spinning Earth  How much energy is stored in the spinning earth?  The earth spins about its axis. The moment of inertia for a sphere: I = 2/5 M R 2 The kinetic energy for the earth: K rot = 1/5 M R 2  2 With values: K = 2.56 x 10 29 J The energy is equivalent to 250 million times the world’s nuclear arsenal next