Rotational Motion

1
Rotational Motion
2
Angular Acceleration

• In uniform circular motion there is a constant
  radial acceleration.
• ar v2 / r rw2
• If the angular velocity changes there is
  acceleration tangent to the circle as well as
  radially.

The angular acceleration is a
3
The Effect of Torque

• A tangential force on a mass creates an
  acceleration.
• Tangential force Ft m at
• Tangential acceleration at ra
• The force is associated with a torque.
• Torque t r Ft

Ft
r
m
4
Rotational Law of Acceleration

• The force law can be combined with rotational
  motion.
• Torque t r Ft r m at m r2 a
• If torque replaces force, and angular
  acceleration replaces acceleration, this looks
  like the law of acceleration.

5
Torque and Work

• A force does work on an object acting over a
  distance.
• A torque does work on an object rotating through
  an angle.

r
Dq
6
Rotational Work-Energy

• The net work done by forces on an object equals
  the change in kinetic energy.
• The net work done by torques on an object equals
  the change in rotational kinetic energy.

7
Rotational Power

• As with translational motion, power is the rate
  of work done.

• The earth is slowing due to the tides.
• About 28 s / century
• 1 part in 108
• The kinetic energy is changing.
• The power dissipation is large
• About 7 billion hp

8
Rotation and Translation

• A rolling wheel is moving forward with kinetic
  energy.
• The velocity is measured at the center of mass.
• KCM ½ m v2

• A rolling wheel is rotating with kinetic energy.
• The axis of rotation is at the center of mass.
• Krot ½ I w2

w
v
9
Rolling Energy

• The energy of a rolling wheel is due to both the
  translation and rotation.
• The velocity is linked to the angular velocity.
• The effective energy is the same as a wheel
  rotating about a point on its edge.
• Parallel axis theorem

10
Energy Conserved

• A change in kinetic energy is due to work done on
  the wheel.
• Work is from a force
• Force acts as a torque
• Rolling down an incline the force is from
  gravity.
• Pivot at the point of contact
• The potential energy is converted to kinetic
  energy.

v
R
F mg
q