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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.
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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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