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 ' \(\mathrm{R}{ }^{-5 / 2}\), then ' T ', would be proportional to

For the planet to orbit around the star, the centripetal force must be provided by gravitational force. Hence, \(\mathrm{F}_{\mathrm{G}}=\mathrm{F}_{\mathrm{a}}\)
\(\mathrm{F}_{\mathrm{a}} \propto-\mathrm{R}^{-5 / 2}\)
....(Given)
Here, -ve sign indicates force is towards the centre of orbit.
\(\begin{array}{ll}
& \Rightarrow \mathrm{a} \propto-\mathrm{R}^{-5 / 2} \\
\therefore \quad & -\omega^2 \mathrm{R} \propto-\mathrm{R}^{-5 / 2} \\
\therefore \quad & \omega^2 \propto \mathrm{R}^{-(5+2) / 2} \\
\therefore & \frac{4 \pi^2}{\mathrm{~T}^2} \propto \mathrm{R}^{-7 / 2} \\
\therefore \quad & \mathrm{~T}^2 \propto \mathrm{R}^{7 / 2} \\
& \Rightarrow \mathrm{~T} \propto \mathrm{R}^{7 / 4}
\end{array}\)