Question

The potential energy of interaction between the semicircular ring of mass $M$ and radius $R$, and the particle of mass$M$ placed at the centre of curvature of the semicircular arc is

• A. $-\frac{G{M}^2}{R}$
  
• B. $-\frac{2G{M}^2}{R}$
• C. $-\frac{G{M}^2}{\pi R}$
• D. none of these

Answer 1

The correct option is A $-\frac{G{M}^2}{R}$


$v=\frac{-GMM}{R}=-\frac{G{M}^2}{R}$

Considering a small element mass $dm$ on the ring of thickness $dt$.
Thus,
$dm=\frac{M}{\pi R}.dt$
Interaction energy between small mass and point mass $M$,
$dU=-\frac{GMdm}{R}$
$dU=-\frac{GM}{R}.\frac{M}{\pi R}.dt$
Total interaction energy,
$U=\int dU={\int }_0^{\pi R}-\frac{G{M}^2}{\pi R^2}.dt$
$U=-\frac{G{M}^2}{\pi R^2}[t{]}_0^{\pi R}=-\frac{G{M}^2}{R}$