Question

A ceiling fan with 16-in. blades rotates at 45 rpm.

Answer 1

A. We're told the fan completes 45 full revolutions every minute. One full revolution is $2\pi\,\mathrm{rad}$, so the fan has an angular speed of

$\dfrac{2\pi\,\mathrm{rad}}{1\,\mathrm{rev}}\cdot\dfrac{45\,\mathrm{rev}}{\mathrm{min}}=90\pi\dfrac{\rm{rad}}{\rm{min}}$

B. Throughout one full revolution of the fan, a point at the tip of one of the fan's blades travels the circumference of a circle with radius 16 in. The circumference is then $2\pi(16\,\mathrm{in})=32\pi\,\mathrm{in}$. This means the linear speed is

$\dfrac{32\pi\,\mathrm{in}}{1\,\mathrm{rev}}\cdot\dfrac{45\,\mathrm{rev}}{\mathrm{min}}=1440\pi\,\dfrac{\rm{in}}{\rm{min}}$