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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.
Solution
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 \[T = 2\pi\sqrt{\frac{R_2^2}{g R_1^2}}\] , where g is the acceleration due to gravity at the surface of the planet.
\[\text { Now }, T^2 = 4 \pi^2 \frac{R_2^2}{g R_1^2}\]
\[ \Rightarrow g = \frac{4 \pi^2}{T^2}\frac{R_2^2}{R_1^2}\]
∴ Acceleration due to gravity of the planet = \[\frac{4 \pi^2}{T^2}\frac{R_2^2}{R_1^2}\]
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