## Angular Acceleration

We define angular acceleration as the change in angular velocity with respect to time. The equation is $\alpha = \frac{\Delta \omega}{\Delta t}$, where $\alpha$ represents angular acceleration.

As you read, pay attention to Example 10.1, which shows how to calculate the angular acceleration of a bike wheel. In the first part of the problem, we calculate the angular acceleration of the wheel given the change in angular velocity and time. In the second part of the problem, we calculate the time needed to stop an already spinning wheel given angular deceleration as initial velocity, using the same angular acceleration equation. See a diagram of a rotating object showing the relationship between linear and angular velocity in Figure 10.3.

### Check Your Understanding

Angular acceleration is a vector, having both magnitude and direction. How do we denote its magnitude and direction? Illustrate with an example.

##### Solution

The magnitude of angular acceleration is $\alpha$ and its most common units are $\mathrm{rad} / \mathrm{s}^{2}$. The direction of angular acceleration along a fixed axis is denoted by $\mathrm{a}+$ or $\mathrm{a}$ - sign, just as the direction of linear acceleration in one dimension is denoted by a + or a - sign. For example, consider a gymnast doing a forward flip. Her angular momentum would be parallel to the mat and to her left. The magnitude of her angular acceleration would be proportional to her angular velocity (spin rate) and her moment of inertia about her spin axis.