PHYS101: Introduction to Mechanics

How fast is an object rotating? We define angular velocity $\omega$ as the rate of change of an angle. In symbols, this is

$\omega=\dfrac{\Delta \theta}{\Delta t}$

where an angular rotation $\Delta \theta$ takes place in a time $\Delta t .$ The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity $\omega$ is analogous to linear velocity $v$. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length $\Delta s$ in a time $\Delta t$, and so it has a linear velocity

$v=\dfrac{\Delta s}{\Delta t}$

From $\Delta \theta=\dfrac{\Delta s}{r}$ we see that $\Delta s=r \Delta \theta .$ Substituting this into the expression for $v$ gives

$v=\dfrac{r \Delta \theta}{\Delta t}=r \omega$

We write this relationship in two different ways and gain two different insights:

$v=r \omega \text { or } \omega=\dfrac{v}{r}.$

The first relationship in $v=r \omega$ or $\omega=\dfrac{v}{r}$ states that the linear velocity $v$ is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest $r$ ), as you might expect. We can also call this linear speed $v$ of a point on the rim the tangential speed. The second relationship in $v=r \omega$ or $\omega=\dfrac{v}{r}$ can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed $v$ of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins-large $v$ means a large $\omega$, because $v=r \omega .$ Similarly, a larger-radius tire rotating at the same angular velocity $(\omega)$ will produce a greater linear speed $(v)$ for the car.

Figure 6.5 A car moving at a velocity $v$ to the right has a tire rotating with an angular velocity $\omega$. The speed of the tread of the tire relative to the axle is $v$, the same as if the car were jacked up. Thus the car moves forward at linear velocity $v=r \omega$, where $r$ is the tire radius. $A$ larger angular velocity for the tire means a greater velocity for the car.

Both $\omega$ and $v$ have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation -it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 6.6.

Figure 6.6 As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.