9.1 Angular acceleration

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 ${\text{rad/s}}^2$ . (b) If she now slams on the brakes, causing an angular acceleration of $–87.3{\text{rad/s}}^2$ , how long does it take the wheel to stop?

Strategy for (a)

The angular acceleration can be found directly from its definition in $\alpha =\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}$ because the final angular velocity and time are given. We see that $\mathrm{\Delta }\omega$ is 250 rpm and $\mathrm{\Delta }t$ is 5.00 s.

Solution for (a)

Entering known information into the definition of angular acceleration, we get

Because $\mathrm{\Delta }\omega$ is in revolutions per minute (rpm) and we want the standard units of ${\text{rad/s}}^2$ for angular acceleration, we need to convert $\mathrm{\Delta }\omega$ from rpm to rad/s:

Entering this quantity into the expression for $\alpha$ , we get

Strategy for (b)

In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for $\mathrm{\Delta }t$ , yielding

Solution for (b)

Here the angular velocity decreases from $\text{26.2 rad/s}$ (250 rpm) to zero, so that $\mathrm{\Delta }\omega$ is $–\text{26.2 rad/s}$ , and $\alpha$ is given to be $–\text{87.3}{\text{rad/s}}^2$ . Thus,

Discussion

Note that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall—the velocity change is large in a short time interval.