Rotational Motion

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

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

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.

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

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.

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

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

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

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