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Satellites

An artificial...

Question

An artificial satellite is moving around earth in a circular orbit with speed equal to one fourth the escape speed of a body from the surface of earth. The height of satellite above earth's surface is (R is radius of earth)

A

3R

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B

5R

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C

7R

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D

8R

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Solution

The correct option is C 7R
Escape velocity $= \sqrt{2 g R}$
Speed of satellite= $\frac{\sqrt{2 g R}}{4}$
By force balance,
$\frac{G M m}{x^{2}} = \frac{m (\frac{\sqrt{2 g R}}{4})^{2}}{x}$
$x = \frac{G M}{2 g R} \times 16 = 8 \frac{G M}{g R} = 8 R$
Height of satellite above earth's surface is $8 R - R = 7 R$

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