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 to $R^{-5 / 2}$, then $T$ would be proportional to

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

Answer 1

Answer: (D)

According to the question, the gravitational force between the planet and the star is $F \propto \frac{1}{ R ^{5 / 2}}$
$ \therefore F =\frac{ GMm }{ R ^{5 / 2}} $
Where $M$ and $M$ be masses of star and planet respectively.
For motion of planet in circular orbit,
$ \begin{array}{l} mR \omega^2=\frac{ GMm }{ R ^{5 / 2}} \\ mR \left(\frac{2 \pi}{ T }\right)^2=\frac{ GMm }{ R ^{5 / 2}} \\ \left(\therefore \omega=\frac{2 \pi}{ T }\right) \\ \frac{4 \pi^2}{ T ^2}=\frac{ GM }{ R ^{7 / 2}} \Rightarrow T ^2=\frac{4 \pi^2}{ GM } R ^{7 / 2} \\ T ^2 \propto R ^{7 / 2} \text { or } T ^2 \propto R ^{7 / 4} \end{array} $