Question

A particle is moving in a circle and its angular velocity is given as $\omega =\pi t$ where $\omega$ is in radians and $t$ is in seconds. Find the average angular velocity at the end of one half circle.

• A. $\sqrt{100\pi }\text{rad/s}$
• B. $\sqrt{90\pi }\text{rad/s}$
• C. $\frac{\pi }{2}\text{rad/s}$
• D. $\frac{\pi }{\sqrt{2}}\text{rad/s}$

Answer 1

The correct option is D $\frac{\pi }{\sqrt{2}}\text{rad/s}$

We have, $\omega =\pi t$
i.e $\frac{d\theta }{dt}=\pi t$
$\Rightarrow d\theta =\pi tdt$

Integrating above equation with proper limits (Here to complete half circle, $\theta$ varies from $0$ to $\pi$ )
$\Rightarrow {\int }_0^{\pi }d\theta ={\int }_0^t\pi tdt$
$\Rightarrow [\theta {]}_0^{\pi }=\pi {\left[\frac{t^2}{2}\right]}_0^t$
$\Rightarrow \pi =\frac{\pi }{2}[t^2-0]$
$\Rightarrow 2=t^2$
$\Rightarrow t=\sqrt{2}\text{s}$

Average angular velocity $=\frac{\text{Total angular displacement}}{\text{time taken}}$
$=\frac{\pi }{\sqrt{2}}\text{rad/s}$