Chapter 7: Angular Kinematics
7.2 Angular Acceleration
Paul Peter Urone and Roger Hinrichs
Learning Objectives
By the end of this section, you will be able to:
- Describe uniform circular motion.
- Explain non-uniform circular motion.
- Calculate angular acceleration of an object.
- Observe the link between linear and angular acceleration.
Other sections in this chapter (Rotational Velocity, Centripetal Force and Acceleration, and Fictitious Forces) discuss only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity [latex]\omega[/latex] was defined as the time rate of change of angle [latex]\theta[/latex]:
where [latex]\theta[/latex] is the angle of rotation as seen in the figure below. The relationship between angular velocity [latex]\omega[/latex] and linear velocity [latex]v[/latex] was also defined in Chapter 7.1: Rotation Angle and Angular Velocity as
or
where [latex]r[/latex] is the radius of curvature, also seen in the figure below. 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]\omega[/latex] changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration [latex]\alpha[/latex] is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:
where [latex]\Delta \omega[/latex] is the change in angular velocity and [latex]\Delta t[/latex] is the change in time. The units of angular acceleration are [latex]\left(\text{rad/s}\right)\text{/s}[/latex], or [latex]{\text{rad/s}}^{2}[/latex]. If [latex]\omega[/latex] increases, then [latex]\alpha[/latex] is positive. If [latex]\omega[/latex] decreases, then [latex]\alpha[/latex] is negative.
Example: Calculating the Angular Acceleration and Deceleration of a Bike Wheel
Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s.
(a) Calculate the angular acceleration in [latex]{\text{rad/s}}^{2}[/latex].
(b) If she now slams on the brakes, causing an angular acceleration of [latex]–87.3\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}[/latex], how long does it take the wheel to stop?
Strategy for (a)
The angular acceleration can be found directly from its definition in [latex]\alpha =\frac{\Delta \omega }{\Delta t}[/latex] because the final angular velocity and time are given. We see that [latex]\Delta \omega[/latex] is 250 rpm and [latex]\Delta t[/latex] is 5.00 s.
Solution for (a)
Entering known information into the definition of angular acceleration, we get
Because [latex]\Delta \omega[/latex] is in revolutions