Angular Acceleration

Learning Objectives

• Be able to define angular acceleration.
• Be able to calculate angular acceleration from a change in angular velocity.
• Be able to calculate tangential acceleration from angular acceleration and radius.

Key Terms

	Definition	Symbol and Equation	SI Units
Angular Acceleration	Angular acceleration is a change in angular velocity divided by time.	\(\alpha =\displaystyle \frac {\Delta\omega}{t} \)	\(\displaystyle \frac {rad}{s^2} \)
Tangential Acceleration	Tangential acceleration is the rate at which the tangential velocity changes.	\( a =\displaystyle \frac {\Delta v}{t} \)	\(\displaystyle \frac {m}{s^2} \)

Translational acceleration tells how fast a velocity is changing. It is measured in \( \frac {m}{s^2} \). This might mean an object starts moving faster or slower. Angular acceleration tells us how fast an angular velocity is changing. It is measured in\(\frac {rad}{s^2} \). This would happen if an object starts spinning faster or slower. The ice skater above starts spinning slow and increases her rate of rotation. This is an example of angular acceleration.

Tangential Acceleration

Tangential Acceleration

\(a=\displaystyle \alpha\, r\)

Tangential acceleration is the rate of change of the tangential velocity. If we measure our angle in radians we can say that \(\\a=\displaystyle \alpha\, r\). The tangential velocity equals the angular velocity times the radius. Similar to arc length and tangential velocity, there are some instances when the tangential acceleration and the acceleration of the center of mass are linked. A rolling object that is not slipping is one example. In these cases the tangential acceleration is the same as the acceleration for the center of mass or the translational acceleration.

Sample Problem

A car has tires with a radius of .3m and goes from rest to 30m/s in 10s. What is the angular acceleration of the tires?

If an object is rolling with out slipping then its translational acceleration is the same as the tangential acceleration. If we can calculate the translational acceleration, then we will know the tangential acceleration.

\(\\a =\displaystyle \frac {\Delta v}{t} = \displaystyle \frac {30m/s}{10s}=3m/s^2\)

\(\\a=\displaystyle \alpha\, r\, \longrightarrow \alpha=\frac{a}{r} = \frac{3m/s^2}{.3m}=10rad/s^2\)