First slide

Gravitaional potential energy

Question

A satellite of mass m is orbiting the earth at a height H from its surface. If M be the mass of earth and R, its radius, then the energy required to pull the satellite from earth's gravitational field will be

Moderate

A

$\frac{GmM}{2 (R + h)}$
B

$\frac{2 GmM}{(R + h)}$
C

$\frac{GmM}{2 (R + h)^{2}}$
D

$\frac{4 GmM}{(R + h)^{2}}$

Solution

Distance of the satellite from the centre of earth
$r = (R + h)$
$∴ \frac{{mv}^{2}}{(R + h)} = \frac{GmM}{(R + h)^{2}} or \frac{1}{2} {mv}^{2} = \frac{GMm}{2 (R + h)}$
or $K . E . = \frac{GMm}{2 (R + h)}$
$P.E. = - \frac{GmM}{r} = - \frac{GmM}{(R + h)}$
Total energy = K.E. + P.E.
$= \frac{GmM}{2 (R + h)} - \frac{GmM}{(R + h)} = - \frac{GmM}{2 (R + h)}$
So, energy needed = $[GmM / 2 (R + h)]$

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