Rotational Motion Angular Acceleration In uniform circular motion

Rotational Motion

Angular Acceleration ] In uniform circular motion there is a constant radial acceleration. • ar = v 2 / r = rw 2 ] The angular acceleration is a If the angular velocity changes there is acceleration tangent to the circle as well as radially.

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 m

Rotational Law of Acceleration ] The force law can be combined with rotational motion. • Torque: t = r Ft = r m at = m r 2 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. ] A rolling wheel is rotating with kinetic energy. ] The velocity is measured at the center of mass. ] The axis of rotation is at the center of mass. • Krot = ½ I w 2 • KCM = ½ m v 2 v w

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