Question

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance, then the time period of a planet in circular orbit of radius $R$, around the sun will be proportional to:

• A. $R^{\frac{n+1}{2}}$
• B. $R^{\frac{n-2}{2}}$
• C. $R^n$
• D. $R^{\frac{n-1}{2}}$

Answer 1

The correct option is A $R^{\frac{n+1}{2}}$

Given Gravitational Force $F=\frac{k}{R^n}$
Force on a body in circular motion $F=mR{\omega }^2$
Hence , angular velocity $\omega =\sqrt{\frac{k}{m{R}^{n+1}}}$
Time Period $T=\frac{2\pi }{\omega }$
Hence , $T\propto R^{\frac{n+1}{2}}$