In the chapter on Uniform Circular Motion and Gravitation, we examined the case of motion along a circular path at constant speed—this is known as uniform circular motion. In this case, the angular velocity [latex]\omega[/latex] remains constant. Angular velocity is defined as the time rate of change of the rotation angle [latex]\theta[/latex]:

Here, [latex]\theta[/latex] is measured in radians and represents the angle of rotation, as illustrated in Figure 67.1. The relationship between angular velocity [latex]\omega[/latex] and linear velocity [latex]v[/latex] is given by:

or equivalently,

where [latex]r[/latex] is the radius of curvature. In the standard sign convention, angular quantities that proceed counterclockwise are positive, while clockwise rotations are negative.

However, in many real-world situations, angular velocity is not constant. For instance, a figure skater pulling in their arms spins faster, a merry-go-round speeds up as a child pushes it, and a hard disk decelerates when powered off. In each of these cases, angular velocity changes over time—this is angular acceleration.

Angular acceleration, symbolized as [latex]\alpha[/latex], is defined as the rate of change of angular velocity over time:

Here, [latex]\Delta \omega[/latex] is the change in angular velocity and [latex]\Delta t[/latex] is the time interval over which the change occurs. The SI unit of angular acceleration is radians per second squared ([latex]\text{rad/s}^2[/latex]). A positive value of [latex]\alpha[/latex] indicates increasing angular velocity (speeding up), while a negative value indicates decreasing angular velocity (slowing down).

Imagine if the bicycle from the previous example had been resting on its wheels rather than upside-down. It would have accelerated forward, rolling along the ground before coming to a stop. This example reveals an important connection between circular motion and linear motion. Specifically, how are linear acceleration and angular acceleration related?

In circular motion, linear acceleration is directed tangent to the circle at the point of interest. As shown in Figure 67.2, this kind of acceleration is called tangential acceleration and is denoted [latex]a_{\text{t}}[/latex].

Tangential acceleration [latex]a_{\text{t}}[/latex] refers to the rate of change of speed (magnitude of velocity) along the circular path, but it does not affect the direction of motion. Meanwhile, centripetal acceleration, denoted [latex]a_{\text{c}}[/latex], changes the direction of the velocity vector, not its magnitude. This acceleration always points inward toward the center of the circle, keeping the object moving along a curved path, as seen in Figure 67.3.

We can now express the relationship between tangential (linear) acceleration [latex]a_{\text{t}}[/latex] and angular acceleration [latex]\alpha[/latex]. Recall from linear kinematics that:

In circular motion, since [latex]v = r\omega[/latex], we substitute:

Because [latex]r[/latex] is constant for a given path, we factor it out:

But the rate of change of angular velocity is angular acceleration [latex]\alpha[/latex], so we have:

This formula directly links angular and linear acceleration. We can also rearrange it to express angular acceleration in terms of linear acceleration:

These equations show that tangential acceleration is proportional to angular acceleration. The larger the angular acceleration, the larger the tangential acceleration. The radius [latex]r[/latex] also matters: for a given [latex]\alpha[/latex], a larger radius leads to greater linear acceleration, and vice versa.

Example: Consider the drive wheels of a car. If the angular acceleration of the wheels increases, so does the car’s linear acceleration. However, a car with smaller wheels would experience smaller tangential acceleration for the same angular acceleration.

Example 67.2: Calculating the Angular Acceleration of a Motorcycle Wheel

A powerful motorcycle can accelerate from 0 to 30.0 m/s (about 108 km/h) in 4.20 s. What is the angular acceleration of its 0.320-m-radius wheels? (See Figure 67.4.)

Strategy

We are given information about the linear velocities of the motorcycle. Thus, we can find its linear acceleration [latex]{a}_{\text{t}}[/latex]. Then, the expression [latex]\alpha =\frac{{a}_{\text{t}}}{r}[/latex] can be used to find the angular acceleration.

Solution

The linear acceleration is

[latex]\begin{array}{lll}{a}_{\text{t}}& =& \frac{\Delta v}{\Delta t}\\ & =& \frac{\text{30.0 m/s}}{\text{4.20 s}}\\ & =& \text{7.14}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}.\end{array}[/latex]

We also know the radius of the wheels. Entering the values for [latex]{a}_{\text{t}}[/latex] and [latex]r[/latex] into

[latex]\alpha =\frac{{a}_{\text{t}}}{r}[/latex], we get

[latex]\begin{array}{lll}\alpha & =& \frac{{a}_{\text{t}}}{r}\\ & =& \frac{\text{7.14}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}}{\text{0.320 m}}\\ & =& \text{22.3}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}.\end{array}[/latex]

Discussion

Units of radians are dimensionless and appear in any relationship between angular and linear quantities.

So far, we have introduced three key rotational kinematic quantities:

• Angular displacement [latex]\theta[/latex]
• Angular velocity [latex]\omega[/latex]
• Angular acceleration [latex]\alpha[/latex]

These quantities are rotational analogs of the familiar translational motion quantities:

• Displacement [latex]x[/latex]
• Velocity [latex]v[/latex]
• Acceleration [latex]a[/latex]

Table 67.1 summarizes the correspondence between rotational and translational quantities, along with their defining relationships. This analogy provides the foundation for analyzing rotational motion using tools and concepts similar to those used in linear motion.