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 were proportional of $R^{\frac{-5}{2}}$, 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

$R^{\frac{7}{4}}$

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

$mR{\omega }^2=\frac{k}{R^{\frac{5}{2}}}$($k$ 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, $T\propto R^{\frac{7}{4}}$