Question

A body moves in a circular orbit of radius R under the action of a central force. Potential due to the central force is given by $V\left(r\right)=kr$(k is a positive constant). Period of revolution of the body is proportional to.

• A. $R^{1/2}$
• B. $R^{-1/2}$
• C. $R^{-3/2}$
• D. $R^{-5/2}$

Answer 1

The correct option is A $R^{1/2}$

Potential due to central force $V\left(r\right)=kr$

Thus central force acting the body $|F|=\frac{dV}{dr}$

$⟹$ $|F|=k$

This central force balances the centripetal force acting on the body revolving in a circular orbit of radius $R$.

$\therefore$ $\frac{m{v}^2}{R}=k$

We get $v=\sqrt{\frac{kR}{m}}$

Time period of revolution $T=\frac{2\pi R}{v}$

Or $T=\frac{2\pi R}{\sqrt{\frac{kR}{m}}}$

$⟹$ $T\propto R^{1/2}$