First slide

Gravitaional potential energy

Question

A spacecraft of mass m describes a circular orbit of radius r₁, around the earth of mass M. Calculate the additional energy to be imparted to the spacecraft to transfer it to a circular orbit of larger radius r₂,

Difficult

A

$\frac{GMm (r_{2} - r_{1})}{2 r_{1} r_{2}}$
B

$\frac{GMm (r_{1} - r_{2})}{2 r_{1} r_{2}}$
C

$\frac{{GMmr}_{1} r_{2}}{(r_{2} - r_{1})}$
D

$\frac{{GMmr}_{1} r_{2}}{(r_{2} + r_{1})}$

Solution

For a satellite, orbiting in a circular orbit or radius r,
We have,
$\frac{GMm}{r^{2}} = \frac{{mv}^{2}}{r} or {mv}^{2} = \frac{GMm}{r}$
$K.E. = \frac{1}{2} \frac{GMm}{r} and P.E. = - \frac{GMm}{r}$
Total energy E = $- \frac{GMm}{2 r}$
Total energy of the satellite orbiting in circular orbit of radius r₁ is given by
$E_{1} = - \frac{GMm}{2 r_{1}}$
Similarly, for an orbit or radius r₂
$E_{2} = - \frac{GMm}{2 r_{2}}$
So, additional energy imparted will be
$\begin{array}{l} E_{2} - E_{1} = - \frac{GMm}{2 r_{2}} - (- \frac{GMm}{2 r_{1}}) \\ = \frac{GMm}{2} [\frac{1}{r_{1}} - \frac{1}{r_{2}}] = \frac{GMm (r_{2} - r_{1})}{2 r_{1} r_{2}} \end{array}$

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