Question

A ceiling fan blade is rotating at 1275 rpm (revolutions per minute) . What is its angular velocity in rad/s

Answer 1

Explanations & Calculations

• When it is asked in units of radians per second we need to calculate how much of an angle does the object cover in a second.

• If we calculate how many rotations the fan completes within a second, it will be

$\qquad\qquad \begin{aligned} &=\small \frac{1275\,rotations}{60s}\\ &=\small 21.25\,rps \end{aligned}$

• As you know in a complete rotation there is an angle of 360 degrees or $\small 2\pi$ radians covered.
• Then it's all about calculating how many radians are there in 21.25 rotation.
• It is then,

$\qquad\qquad \begin{aligned} &=\small 21.25\,rotation\times2\pi\,rad/rotation\\ &=\small 133.52\,rad \end{aligned}$

• Then the needed answer is

$\qquad\qquad \begin{aligned} \small \omega&=\small \bold{133.52\,rads^{-1}} \end{aligned}$

• Above is for your understanding & the same is obtained in one go by the shortcut method

$\qquad\qquad \begin{aligned} \small \omega &=\small 2\pi\frac{n}{60} \end{aligned}$ where n is the frequency given in rpm

• Then,

$\qquad\qquad \begin{aligned} \small \omega&=\small 2\pi\times\frac{1275}{60}\\ &=\small 133.52\,rads^{-1} \end{aligned}$