Imagine a light planet revolving around a very massive star in a circular orbit of radius R with period T. If the gravitational force of attraction between the planet and the star is proportional to ${\text{R}}^{\frac{-5}{2}}$ , then${\text{T}}^{\text{2}}$ is proportional to

The mass of light planet = m

Radius in which it is revolving = R

Orbital velocity $={\text{v}}_{\text{0}}$

${\text{v}}_{\text{0}}=\frac{2\pi \text{R}}{\text{T}}$

Gravitational force between massive star & planet

$\text{F}\propto {\text{R}}^{-5/2}$

$\text{F}={\text{kR}}^{-5/2}$  (k is constant)

We can have for orbiting planet

$\frac{{\text{mv}}_{\text{0}}^{\text{2}}}{\text{R}}=\text{F}$

$\Rightarrow \frac{{\text{mv}}_{\text{0}}^{\text{2}}}{\text{R}}=\text{k}{\text{.R}}^{-5/2}$

$\Rightarrow {\text{v}}_{\text{0}}^{\text{2}}=\frac{\text{k}}{\text{m}}{\text{R}}^{-3/2}\because {\text{v}}_{\text{0}}=\frac{2\pi \text{R}}{\text{T}}$

$\Rightarrow \frac{4{\pi }^2{\text{R}}^{\text{2}}}{{\text{T}}^{\text{2}}}=\frac{\text{k}}{\text{m}}{\text{R}}^{-3/2}$

$\frac{{\text{R}}^{2+3/2}}{{\text{T}}^{\text{2}}}=\frac{\text{k}}{\text{m}4{\pi }^2}$

$\therefore \frac{{\text{R}}^{\text{7/2}}}{{\text{T}}^{\text{2}}}=\text{constant}$

$\text{i}\text{.e}\text{.,}{\text{T}}^2\propto {\text{R}}^{\text{7/2}}\text{or}{\text{T}}^{\text{2}}\propto {\text{R}}^{3.5}$