Question

A small 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 between the planet and the star is proportional to $R^{\frac{-5}{2}}$, then $T$ would be proportional to

• A. $R^{\frac{3}{2}}$
• B. $R^{\frac{3}{5}}$
• C. $R^{\frac{7}{2}}$
• D. $R^{\frac{7}{4}}$

Answer 1

The correct option is D $R^{\frac{7}{4}}$

Given, the gravitational force between the planet and the star is
$F\propto \frac{1}{R^{\frac{5}{2}}}$

For motion of planet in a circular orbit,
$mR{\omega }^2=\frac{k}{R^{\frac{5}{2}}}$($k$ is constant)

$mR{\left(\frac{2\pi }{T}\right)}^2=\frac{k}{R^{\frac{5}{2}}}(\because \omega =\frac{2\pi }{T})$

$\frac{4{\pi }^2}{T^2}=\frac{k/m}{R^{\frac{7}{2}}}\Rightarrow T^2=\frac{4{\pi }^2}{k/m}{R}^{\frac{7}{2}}$

$T^2\propto R^{\frac{7}{2}}$

or, $T\propto R^{\frac{7}{4}}$