Question

Find the angular and linear speed for a wheel that rotates 32 rpm

Answer 1

One revolution corresponds to an angular displacement of $2\pi$, so its angular speed is

$32\,\mathrm{rpm}=\left(32\dfrac{\rm rev}{\rm min}\right)\left(2\pi\dfrac{\rm rad}{\rm rev}\right)=64\pi\dfrac{\rm rad}{\rm min}$

$\pi$ radians is equivalent to 180 degrees, so the angular speed could also be

$\left(64\pi\dfrac{\rm rad}{\rm min}\right)\left(\dfrac{180}\pi\dfrac{\rm deg}{\rm rad}\right)=11,520\dfrac{\rm deg}{\rm min}$

The linear speed depends on the wheel's radius. Suppose its radius is $r$. Then the wheel has circumference $2\pi r$ units. A point on the edge of the wheel travels this distance in one revolution, so its linear speed is

$\left(32\dfrac{\rm rev}{\rm min}\right)\left(2\pi r\dfrac{\rm u}{\rm rev}\right)=64\pi r\dfrac{\rm u}{\rm min}$

(where $u$ stands for units of length)