Question

A ceiling fan rotates at an angular speed of 30.0 revolution per minute. Find
(a) the angular speed of the fan in rad/s.
(b) the speed of the tip of the fan blade which is 0.60 m from the axis of rotation.
(c) the angular displacement of a blade of the fan in 5.0 s.

Answer 1

a) The angular speed is: $f = 30rpm = 0.5Hz$. Then the angular speed in rad/s is:

$$\omega = 2\pi f = 2\pi\cdot 0.5 = \pi\space rad/s$$

b) The linear speed of points $r = 0.6m$ from the axis of rotation is given by:

$$v = \omega r = \pi\cdot 0.6 = 0.6\pi\space m/s$$

c) The angular displacement of a blade is given by:

$$\varphi = \omega t = \pi\space rad/s \cdot 5s = 5\pi \space rad$$

Answer. a) $\pi\space rad/s$, b) $0.6\pi\space m/s$, c) $5\pi \space rad$.