First slide

Gravitaional potential energy

Question

A satellite of mass m is orbiting the earth (of radius R ) at a height h from its surface. The total energy of the satellite in

Moderate

A

$\frac{m g_{0} R^{2}}{2 (R + h)}$
B

$- \frac{m g_{0} R^{2}}{2 (R + h)}$
C

$\frac{2 m g_{0} R^{2}}{R + h}$
D

$- \frac{2 m g_{0} R^{2}}{R + h}$

Solution

Total energy of satellite at height h from the earth surface,
$E = P . E . + K . E .$
$= - \frac{G M m}{(R + h)} + \frac{1}{2} m v^{2}$ ………..(1)
$Also, \frac{m v^{2}}{(R + h)} = \frac{G M m}{(R + h)^{2}} or, v^{2} = \frac{G M}{R + h}$ …..(2)
From eqns. (1) and (2),
$\begin{array}{l} E = - \frac{G M m}{(R + h)} + \frac{1}{2} \frac{G M m}{(R + h)} = - \frac{1}{2} \frac{G M m}{(R + h)} \\ = - \frac{1}{2} \frac{G M}{R^{2}} \times \frac{m R^{2}}{(R + h)} = - \frac{m g_{0} R^{2}}{2 (R + h)} (∵ g_{0} = \frac{G M}{R^{2}}) \end{array}$

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