A clothes dryer is spinning at 75 rpm.A student opens its door and it comes to rest after rotating through four revolutions.Assume constant acceleration.What was the magnitude of the dryer's angular acceleration?

Question

Quizplus · Accepted Answer

The angular acceleration can be found using the equation $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$, where $\theta$ is the angular displacement (in radians), $\omega_0$ is the initial angular velocity, $\alpha$ is the angular acceleration, and $t$ is the time. However, since time is not given, we use another approach involving the initial and final angular velocities and the angular displacement. The relevant equation is $\omega^2 = \omega_0^2 + 2\alpha\theta$, where $\omega$ is the final angular velocity (0 rad/s since it comes to rest), $\omega_0$ is the initial angular velocity in rad/s (which needs to be converted from rpm), and $\theta$ is the angular displacement in radians.First, convert the initial angular velocity from rpm to rad/s: $75 \, \text{rpm} = 75 \times \frac{2\pi}{60} = 7.85 \, \text{rad/s}$.The angular displacement is given as four revolutions, which in radians is $4 \times 2\pi = 8\pi$.Substitute these values into the equation and solve for $\alpha$:$0 = (7.85)^2 + 2\alpha(8\pi)$,Solving for $\alpha$ gives $\alpha = -1.23 \, \text{rad/s}^2$. The negative sign indicates deceleration, but the question asks for the magnitude, so the answer is $1.23 \, \text{rad/s}^2$.

A Clothes Dryer Is Spinning at 75 Rpm