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)

$R^{7/4}$

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,
$mR{\omega }^2=\frac{GMm}{R^{5/2}}$
$mR{\left(\frac{2\pi }{T}\right)}^2=\frac{GMm}{R^{5/2}}$

$(\therefore \omega =\frac{2\pi }{T})$

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

$T^2\propto R^{7/2}$ or $T^2\propto R^{7/4}$