Question

A fan blade is initially rotating an angular speed of 4.6 rpm. It slows down and eventually comes to rest in a time of 32 seconds after turning through a total of 8.8 revolution. Find a.) The angular speed in rad/s. b.) The average angular velocity ang c.) The average angular acceleration.

Answer 1

Initial angular velocity is $\omega_o=4.6\times \frac{2\pi}{60}=0.48\:rad/s$

Final angular velocity is $\omega_f=0\:rad/s$

Final time $t=t_f=32s$ and angle traversed is $\theta=8.8\times2\pi=55.29 \:rad$

(a).As we have not provided at which time we need to find the angular speed, thus

we assume time t is arbitrary, hence

$$\omega=\omega_o+(-\alpha) t\implies \omega=\omega_o-\alpha t\:rad/s$$

As angular acceleration is decreasing.

(b). Since, average angular velocity is given by

$$\omega_{av}=\frac{\Delta \theta}{\Delta t}$$

Thus, on plugin the value we get,

$$\omega_{av}=\frac{55.29-0}{32-0}=1.72 \: rad/s$$

(c). Since,

$$\alpha_{av}=\frac{\Delta \omega}{\Delta t}=\frac{\omega_f-\omega_o}{t_f-t_o}$$

Hence, on plugin the value we get

$$\alpha_{av}=\frac{0-0.48}{32-0}=-0.015 \:rad/s^2$$