Chapter 6 Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity[latex]\boldsymbol{\omega}[/latex]was defined as the time rate of change of angle[latex]\boldsymbol{\theta}:[/latex]

where[latex]\boldsymbol{\theta}[/latex]is the angle of rotation as seen in Figure 1. The relationship between angular velocity[latex]\boldsymbol{\omega}[/latex]and linear velocity[latex]\boldsymbol{v}[/latex]was also defined in Chapter 6.1 Rotation Angle and Angular Velocity as

or

where[latex]\boldsymbol{r}[/latex]is the radius of curvature, also seen in Figure 1. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative

Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer’s hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration, in which[latex]\boldsymbol{\omega}[/latex]changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration[latex]\boldsymbol{\alpha}[/latex]is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:

where[latex]\boldsymbol{\Delta\omega}[/latex]is the change in angular velocity and[latex]\boldsymbol{\Delta{t}}[/latex]is the change in time. The units of angular acceleration are[latex]\textbf{(rad/s)/s},[/latex]or[latex]\boldsymbol{\textbf{rad/s}^2}.[/latex]If[latex]\boldsymbol{\omega}[/latex]increases, then[latex]\boldsymbol{\alpha}[/latex]is positive. If[latex]\boldsymbol{\omega}[/latex]decreases, then[latex]\boldsymbol{\alpha}[/latex]is negative.

If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is tangent to the circle at the point of interest, as seen in Figure 2. Thus, linear acceleration is called tangential acceleration[latex]\boldsymbol{a_{\textbf{t}}}.[/latex]

Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from Chapter 6 Uniform Circular Motion and Gravitation that in circular motion centripetal acceleration,[latex]\boldsymbol{a_{\textbf{c}}},[/latex]refers to changes in the direction of the velocity but not its magnitude. An object undergoing circular motion experiences centripetal acceleration, as seen in Figure 3. Thus,[latex]\boldsymbol{a_{\textbf{t}}}[/latex]and[latex]\boldsymbol{a_{\textbf{c}}}[/latex]are perpendicular and independent of one another. Tangential acceleration[latex]\boldsymbol{a_{\textbf{t}}}[/latex]is directly related to the angular acceleration[latex]\boldsymbol{\alpha}[/latex]and is linked to an increase or decrease in the velocity, but not its direction.

Now we can find the exact relationship between linear acceleration[latex]\boldsymbol{a_{\textbf{t}}}[/latex]and angular acceleration[latex]\boldsymbol{\alpha}.[/latex]Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in Chapter 2 One-Dimensional Kinematics) to be

For circular motion, note that[latex]\boldsymbol{v=r\omega},[/latex]so that

The radius[latex]\boldsymbol{r}[/latex]is constant for circular motion, and so[latex]\boldsymbol{\Delta(r\omega)=r(\Delta\omega)}.[/latex]Thus,

By definition,[latex]\boldsymbol{\alpha=\frac{\Delta\omega}{\Delta{t}}}.[/latex]Thus,

or

These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car’s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration[latex]\boldsymbol{\alpha}.[/latex]

Example 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 4.)

Strategy

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

Solution

The linear acceleration is

[latex]\begin{array}{lcl} \boldsymbol{a_{\textbf{t}}} & \boldsymbol{=} & \boldsymbol{\frac{\Delta{v}}{\Delta{t}}} \\ {} & \boldsymbol{=} & \boldsymbol{\frac{30.0\textbf{ m/s}}{4.20\textbf{ s}}} \\ {} & \boldsymbol{=} & \boldsymbol{7.14\textbf{ m/s}^2.} \end{array}[/latex]

We also know the radius of the wheels. Entering the values for[latex]\boldsymbol{a_{\textbf{t}}}[/latex]and[latex]\boldsymbol{r}[/latex]into[latex]\boldsymbol{\alpha=\frac{a_{\textbf{t}}}{r}},[/latex]we get

[latex]\begin{array}{lcl} \boldsymbol{\alpha} & \boldsymbol{=} & \boldsymbol{\frac{a_{\textbf{t}}}{r}} \\ {} & \boldsymbol{=} & \boldsymbol{\frac{7.14\textbf{ m/s}}{20.320\textbf{ m}}} \\ {} & \boldsymbol{=} & \boldsymbol{22.3\textbf{ 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 defined three rotational quantities— [latex]\boldsymbol{\theta,\:\omega},[/latex]and[latex]\boldsymbol{\alpha}.[/latex]These quantities are analogous to the translational quantities[latex]\boldsymbol{x},\:\boldsymbol{v},[/latex]and[latex]\boldsymbol{a}.[/latex]Table 1 displays rotational quantities, the analogous translational quantities, and the relationships between them.

Check Your Understanding

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

PHET EXPLORATIONS: LADYBUG REVOLUTION

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug’s x,y position, velocity, and acceleration using vectors or graphs.