Question

A particle of mass $m$ is made to move with uniform speed $v$ along the perimeter of a regular polygon of $2n$ sides. The magnitude of impulse applied at each corner of the polygon is:

• A. $2mv\sin \frac{\pi }{2n}$
• B. $mv\sin \frac{\pi }{2n}$
• C. $2mv\cos \left(\frac{\pi }{2n}\right)$
• D. $mv\cos \frac{\pi }{2n}$

Answer 1

The correct option is A $2mv\sin \frac{\pi }{2n}$

Considering $n=2$
The polygon is a square
$p_i=mv(\uparrow )$
$p_f=mv(-\uparrow )$
Impulse $=\overrightarrow{J}=\Delta p=p_f-p_i=mv(-\uparrow )-mv(\uparrow )$
$|\overrightarrow{J}|=\sqrt{2}mv=2mv\sin \left(\frac{\pi }{4}\right)$ or $2mv\cos \left(\frac{\pi }{4}\right)=2mv\sin \left(\frac{\pi }{2n}\right)$
Doing the same for $n=3$
Therefore the magnitude of impulse applied at each corner of the polygon is $2mv\sin \left(\frac{\pi }{2n}\right)$