Question

If the gravitational force were to vary inversely as $m^{th}$ power of the distance, then the time period of a planet in circular orbit of radius r around the sun will be proportional to

• A. $r^{-m/2}$
• B. $r^{m/2}$
• C. $r^m$
• D. $r^{\left(\frac{m+1}{2}\right)}$

Answer 1

The correct option is D $r^{\left(\frac{m+1}{2}\right)}$

For motion of planet in circular orbit Centripetal Force $=$ Gravitational Force

$\therefore$ $mR{w}^2=\frac{{GM}_m}{R^n}$ or $w=\sqrt{\frac{GM}{R^{n+1}}}$

$\therefore$ $T=\frac{2\pi }{w}=2\pi \sqrt{\frac{R^{m+1}}{GM}}=\frac{2\pi }{\sqrt{GM}}{R}^{\left(\frac{m+1}{2}\right)}$

$T$ is proportional to $R^{\left(\frac{m+1}{2}\right)}$