Question

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

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

Answer 1

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

The necessary centripetal force required for a planet to move around the sun = Gravitational force exerted on it

So, $\frac{m{v}^2}{R}=\frac{G{M}_{e}m}{R^n}$

or, $v={\left(\frac{G{M}_e}{R^{n-1}}\right)}^{\frac{1}{2}}$

As $T=\frac{2\pi R}{v}$

$T=2\pi R\times {\left(\frac{R^{n-1}}{G{M}_e}\right)}^{\frac{1}{2}}$

So, $T=2\pi \left(\frac{R^{\left(\frac{n+1}{2}\right)}}{\sqrt{G{M}_e}}\right)$

$\therefore T\propto R^{\left(\frac{n+1}{2}\right)}$