Question

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

• A. $T^2$ is proportional to $R^3$
• B. $T^2$ is proportional to $R^{\frac{7}{2}}$
• C. $T^2$ is proportional to $R^{\frac{3}{2}}$
• D. $T^2$ is proportional to $R^{3.75}$

Answer 1

The correct option is B

$T^2$ is proportional to $R^{\frac{7}{2}}$

If a planet of mass $m$ is revolving with velocity $v$ in an orbit of radius $R$, we have
$\frac{m{v}^2}{R}=\frac{K}{R^{\frac{5}{2}}}$ where $K=GmM$ is a constant.
$\Rightarrow v=\sqrt{\frac{K}{m{R}^{\frac{3}{2}}}}$
Time period $T=\frac{2\pi R}{v}=2\pi \sqrt{\frac{m}{K}{R}^{\frac{7}{2}}}$
$\Rightarrow T^2\propto R^{\frac{7}{2}}$