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. $R^{(n+1)/2}$
• B. $R^{(n-1)/2}$
• C. $R^n$
• D. $R^{(n-2)/2}$

Answer 1

The correct option is A $R^{(n+1)/2}$

The necessary centripetal force required for a planet to move around the Sun = Gravitational force exerted on it,
$\frac{{mv}^2}{R}=\frac{G{M}_{e}m}{R^n}$
or $v={\left(\frac{G{M}_e}{R^{n-1}}\right)}^{1/2}$
as $T=\frac{2\pi R}{v}=2\pi R\times {\left(\frac{R^{n-1}}{G{M}_e}\right)}^{1/2}$
$T=2\pi \left[\frac{R^{\frac{(n+1)}{2}}}{{\left(G{M}_e\right)}^{1/2}}\right]$
$\therefore T\propto R^{(n+1)/2}$