Help With Fairly Easy Rotational Motion Problem?

Question

A ceiling fan with 83-cm-diameter blades is turning at 64 rpm. Suppose the fan coasts to a stop 28 sec after being turned off.

1) What is the speed of the tip of a blade 10 s after the fan is turned off?

2) Through how many revolutions does the fan turn while stopping?

Answer 1

r = 0.83/2 m = 0.415 m (radius)

Initial conditions:

omega(0) = 64 * 2 * pi / 60 = 6.70 rad/s (angular speed)

Assume angular acceleration is constant:

alpha = change in omega / time = -6.70 / 28 = -0.239 rad/s^2

In 10 sec:

omega(10) = omega(0) + alpha * 10

omega(10) = 6.70 - 0.239*10 = 4.31 rad/sec

velocity(10) = omega(10) * r = 4.31 * 0.415 = 1.79 m/s

theta = 1/2 alpha * t^2 = 1/2 (0.239) * 28^2 = 93.69 radians

number of revolutions = theta / 2/pi = 14.9