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Physics LibreTexts

10.1: Angular Acceleration

Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity \(\omega\) was defined as the time rate of change of angle \(\theta\).

\[ \omega = \dfrac{\Delta \theta}{\Delta t},\]

where \(\theta\) is the angle of rotation as seen in Figure. The relationship between angular velocity \(\omega\) and linear velocity \(v\)  was also defined in Rotation Angle and Angular Velocity as \[v = r \omega\] or \[\omega = \dfrac{v}{r}, where \(r\)  is the radius of curvature, also seen in Figure. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative

The given figure shows counterclockwise circular motion with a horizontal line, depicting radius r, drawn from the center of the circle to the right side on its circumference and another line is drawn in such a manner that it makes an acute angle delta theta with the horizontal line. Tangential velocity vectors are indicated at the end of the two lines. At the bottom right side of the figure, the formula for angular velocity is given as v upon r.

Figure 10.2.1. This figure shows uniform circular motion and some of its defined quantities.

Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer’s hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration, in which \(\omega\)  changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration \(\alpha\) is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows: \[\alpha = \dfrac{\Delta \omega}{\Delta t},\] where


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