Physics - Gravitation Question with Solution | TestHub

Gravitational force$=\frac{GMm}{d^2}$(Newton's law)

But in this case,$F=\frac{GMm}{R^n}$and it provides the necessary centripetal force$\frac{m{v}^2}{R}$

$\therefore$$\frac{m{v}^2}{R}=\frac{GMm}{R^n}$
$\therefore$$v=\sqrt{\frac{GM}{R^{n-1}}}$and the period$T$of a planet,$T=\frac{2\pi R}{v}$

$\therefore$$T=2\pi R\sqrt{\frac{R^{n-1}}{GM}}=2\pi \sqrt{\frac{R^{n+1}}{GM}}=\frac{2\pi }{\sqrt{GM}}\cdot R^{\frac{n+1}{2}}$

$\therefore$$T\propto R^{\frac{n+1}{2}}$