FIGURE 4.23 shows a particle moving in a circle from an initial angular position ${\theta }_{\text{i}}$ at time $t_{\text{i}}$ to a final angular position ${\theta }_{\text{f}}$ at a later time $t_{\text{f}}$. The change $\text{Δ}\theta ={\theta }_{\text{f}}-{\theta }_i$ is called the angular displacement. We can measure the particle’s circular motion in terms of the rate of change of $\theta$, just as we measured the particle’s linear motion in terms of the rate of change of its position s.

FIGURE 4.23 A particle moves with angular velocity $\omega$.

In analogy with linear motion, let’s define the average angular velocity to be

As the time interval $\text{Δ}t$ becomes very small, $\text{Δ}t\to 0$, we arrive at the definition of the instantaneous angular velocity:

The symbol $\omega$ is a lowercase Greek omega, not an ordinary w. The SI unit of angular velocity is $\text{rad/s,}$ but $\text{°}\text{/}\text{s,}$ $\text{rev/s,}$ and $\text{rev}\text{/}\text{min}$ are also common units. Revolutions per minute is abbreviated rpm.

Angular velocity is the rate at which a particle’s angular position is changing as it moves around a circle. A particle that starts from $\theta =0$ rad with an angular velocity of $0.5\text{rad}\text{/}\text{s}$ will be at angle $\theta =0.5$ rad after 1 s, at $\theta =1.0$ rad after 2 s, at $\theta =1.5$ rad after 3 s, and so on. Its angular position is increasing at the rate of 0.5 radian per second. A particle moves with uniform circular motion if and only if its angular velocity $\omega$ is constant and unchanging.

Angular velocity, like the velocity $v_s$ of one-dimensional motion, can be positive or negative. The signs shown in FIGURE 4.24 are based on the fact that $\theta$ was defined to be positive for a counterclockwise rotation. Because the definition $\omega =d\theta \text{/}dt$ for circular motion parallels the definition $v_s=ds\text{/}dt$ for linear motion, the graphical relationships we found between $v_s$ and s in Chapter 2 apply equally well to $\omega$ and $\theta :$

FIGURE 4.24 Positive and negative angular velocities.

You will see many more instances where circular motion is analogous to linear motion with angular variables replacing linear variables. Thus much of what you learned about linear kinematics carries over to circular motion.

EXAMPLE 4.10 A graphical representation of circular motion

FIGURE 4.25 shows the angular position of a painted dot on the edge of a rotating wheel. Describe the wheel’s motion and draw an $\omega \text{-}\text{versus}\text{-}t$ graph.

FIGURE 4.25 Angular position graph for the wheel of Example 4.10.

FIGURE 4.25 Full Alternative Text

SOLVE Although circular motion seems to “start over” every revolution (every $2\pi \text{rad})$, the angular position $\theta$ continues to increase. $\theta =6\pi \text{rad}$ corresponds to three revolutions. This wheel makes 3 ccw rev (because $\theta$ is getting more positive) in 3 s, immediately reverses direction and makes 1 cw rev in 2 s, then stops at $t=5\text{s}$ and holds the position $\theta =4\pi \text{rad}$. The angular velocity is found by measuring the slope of the graph:

$$\begin{array}{ll}t=0-3\text{ s}\hfill & \text{s}\text{l}\text{o}\text{p}\text{e}=\text{Δ}\theta \text{/}\text{Δ}t=6\pi \text{ rad/}3\text{ s}=2\pi \text{ rad}\text{/}\text{s}\hfill \\ t=3-5\text{ s}\hfill & \text{ slope}=\text{Δ}\theta \text{/}\text{Δ}t=-2\pi \text{ rad/}2\text{ s}=-\pi \text{ rad}\text{/}\text{s}\hfill \\ t>5\text{ s}\hfill & \text{ slope}=\text{Δ}\theta \text{/}\text{Δ}t=0\text{ rad}\text{/}\text{s}\hfill \end{array}$$

These results are shown as an $\omega \text{-}\text{versus}\text{-}t$ graph in FIGURE 4.26. For the first 3 s, the motion is uniform circular motion with $\omega =$ $2\pi \text{rad/s}$. The wheel then changes to a different uniform circular motion with $\omega =-\pi \text{rad}\text{/}\text{s}$ for 2 s, then stops.

FIGURE 4.26 $\omega$-versus-t graph for the wheel of Example 4.10.

FIGURE 4.26 Full Alternative Text

Not surprisingly, the angular velocity $\omega$ is closely related to the period and speed of the motion. As a particle goes around a circle one time, its angular displacement is $\text{Δ}\theta =2\pi \text{rad}$ during the interval $\text{Δ}t=T$. Thus, using the definition of angular velocity, we find

The period alone gives only the absolute value of $\left|\omega \right|$, which is the angular speed. You need to know the direction of motion to determine the sign of $\omega$.

As Figure 4.21 showed, the velocity vector $\vec{v}$ is always tangent to the circle. In other words, the velocity vector has only a tangential component, which we will designate $v_t$. The tangential velocity is positive for ccw motion, negative for cw motion.

Combining $v=2\pi r/T$ for the speed with $\omega =2\pi /T$ for the angular velocity—but keeping the sign of $\omega$ to indicate the direction of motion—we see that the tangential velocity and the angular velocity are related by

Because $v_t$ is the only nonzero component of $\vec{v}$, the particle’s speed is $v=\left|v_t\right|=\left|\omega \right|r$. We’ll sometimes write this as $v=\omega r$ if there’s no ambiguity about the sign of $\omega$.

As a simple example, a particle moving cw at $2.0\text{m}\text{/}\text{s}$ in a circle of radius 40 cm has angular velocity

where $v_t$ and $\omega$ are negative because the motion is clockwise. Notice the units. Velocity divided by distance has units of $\text{s}{}^{-1}$. But because the division, in this case, gives us an angular quantity, we’ve inserted the dimensionless unit rad to give $\omega$ the appropriate units of $\text{rad}\text{/}\text{s}$.