Physics - Gravitation Question with Solution | TestHub

According to Kepler's third law of planetary motion, the square of the time period of revolution of a planet around the sun is directly proportional to the cube of the semi-major axis of its elliptical orbit.

For a circular orbit, the semi-major axis is equal to the radius of the orbit $R$.

Therefore, $T^2\propto R^3$

Taking the square root on both sides, we get:

$T\propto R^{3/2}$

Alternatively, equating the gravitational force to the required centripetal force:

$\frac{GMm}{R^2}=\frac{m{v}^2}{R}$

$v=\sqrt{\frac{GM}{R}}$

The time period $T$ is given by:

$T=\frac{2\pi R}{v}=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi }{\sqrt{GM}}{R}^{3/2}$

Thus, $T\propto R^{3/2}$.

Answer: ${\text{R}}^{3/2}$