10.1 Rotational variables  (Page 3/11)

Since $\overrightarrow{r}$ is constant, the term $\overrightarrow{\theta }\times \frac{d\overrightarrow{r}}{dt}=0$ . Since $\overrightarrow{v}=\frac{d\overrightarrow{s}}{dt}$ is the tangential velocity and $\overrightarrow{\omega }=\frac{d\overrightarrow{\theta }}{dt}$ is the angular velocity, we have

That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in [link] . From part (a) of this figure, we see that with the angular velocity in the positive z -direction, the rotation in the xy -plane is counterclockwise. In part (b), the angular velocity is in the negative z- direction, giving a clockwise rotation in the xy- plane.

Angular acceleration

We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define an angular acceleration    for describing situations where $\omega$ changes. The faster the change in $\omega$ , the greater the angular acceleration. We define the instantaneous angular acceleration     $\alpha$ as the derivative of angular velocity with respect to time: