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Variation of G Due to Height and Depth

A satellite o...

Question

A satellite of mass $m$ , revolving in a circular orbit of radius $r$ around the earth of mass $M$ has magnitude of total energy $E$ . Then its angular momentum will be

\sqrt{2 E m r^{2}}

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\sqrt{2 E m r}

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\sqrt{E m r^{2}}

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\sqrt{4 E m r^{2}}

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Solution

The correct option is A $\sqrt{2 E m r^{2}}$

The orbital velocity is $V_{0} = \sqrt{\frac{G M}{r}}$
The magnitude of total energy $E = \frac{G M m}{2 r}$
The angular momentum is $L = m V_{0} r$
$= m \sqrt{\frac{G m}{r}} r$
$= \sqrt{\frac{G M m^{2} r^{2}}{r}}$
$= \sqrt{2 m r^{2} (\frac{G M m}{2 r})}$
$= \sqrt{2 E m r^{2}}$

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