Question

Starting from rest, a particle rotates in a circle of radius $R=\sqrt{2}m$ with an angular acceleration $\alpha =\frac{\pi }{4}rad/s^2$. Average velocity $\left(\text{in}m/s\right)$ of the particle over the time it rotates quarter circle.

Answer 1

Step 1: Draw a rough diagram.
Given $\alpha =\frac{\pi }{4},R=\sqrt{2}m$
Angular displacement $=\theta =\frac{\pi }{2}rad$

Step 2: Find time taken by the particle to go from P to Q.

Formula used: $\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$

$\theta =\frac{\pi }{2}={\omega }_{0}t+\frac{1}{2}\alpha t^2$

$\frac{\pi }{2}=0+\frac{1}{2}\times \frac{\pi }{4}\times t^2$

$t=2s$

Step 3: Find Average velocity of particle.
Formula Used:
$\text{Average velocity}=\frac{\text{Displacement}}{\text{Time}}$

$v_{avg}=\frac{R\sqrt{2}}{t}$

$=\frac{2}{2}=1m/s$

Final answer: $1m/s$