Rotational Energy

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.

ri
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.

vCM
?
4
Rotational Motion

• Kinematic equations with constant linear
  acceleration were defined.
• v v0 at
• x x0 v0 t ½at2
• v2 v0 2 2a(x - x0 )

• Kinematic equations with constant angular
  acceleration are similar.
• w w0 at
• q q0 w0 t ½at2
• w2 w0 2 2a (q - q0 )

5
Compact Disc

• A CD spins so that the tangential speed is
  constant.
• The constant speed is 1.3 m/s
• The inner radius is at 0.023 m
• The outer radius is at 0.058 m
• The 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 w v/r.
• Inner w 1.3 m/s / 0.023 m 57 rad/s.
• Outer w 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 v2
• The velocity can be converted to angular
  quantities.
• K ½ m (r w)2
• K ½ (m r2) w2
• The term (m r2) 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.
• KCM ½ m v2
• Krot ½ I w2

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 R2
• The kinetic energy for the earth Krot 1/5 M R2
  w2
• With values K 2.56 x 1029 J

The energy is equivalent to 250 million times the
worlds nuclear arsenal
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