Question

The radius of a planet is R₁ and a satellite revolves round it in a circle of radius R₂. The time period of revolution is T. Find the acceleration due to the gravitation of the planet at its surface.

Answer 1

The time period of revolution of the satellite around a planet in terms of the radius of the planet and radius of the orbit of the satellite is given by $T=2\mathrm{\pi }\sqrt{\frac{R_2^2}{g{R}_1^2}}$, where g is the acceleration due to gravity at the surface of the planet.
$$\begin{gathered} \mathrm{Now},T^2=4{\mathrm{\pi }}^2\frac{R_2^2}{g{R}_1^2} \\ \Rightarrow g=\frac{4{\mathrm{\pi }}^2}{T^2}\frac{R_2^2}{R_1^2} \end{gathered}$$
∴ Acceleration due to gravity of the planet = $\frac{4{\mathrm{\pi }}^2}{T^2}\frac{R_2^2}{R_1^2}$