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Satellites

ENERGY OF AN ...

Question

ENERGY OF AN ORBITING SATELLITE
The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius $R_{1}$ to another of radius $R_{2} (R_{2} > R_{1})$ is:

G m M (\frac{1}{R_{1}^{2}} - \frac{1}{R_{1}^{2}})

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G m M (\frac{1}{R_{1}} - \frac{1}{R_{2}})

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2 G m M (\frac{1}{R_{1}} - \frac{1}{R_{2}})

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\frac{1}{2} G m M (\frac{1}{R_{1}} - \frac{1}{R_{2}})

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Solution

The correct option is D $\frac{1}{2} G m M (\frac{1}{R_{1}} - \frac{1}{R_{2}})$
Total energy of the orbiting satellite of mass m having orbital radius r is $E = - \frac{G M m}{2 r}$ where M is the mass of the planet.
Additional kinetic energy required to transfer the satellite from a circular orbit of radius $R_{1}$ to another radius $R_{2}$ is $= E_{2} - E_{1}$
$= - \frac{G M m}{2 R_{2}} - (- \frac{G M m}{2 R_{1}}) = - \frac{G M m}{2 R_{2}} + \frac{G M m}{2 R_{1}} = \frac{G M m}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

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