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Q.

An object of mass m lying at rest at depth of ' $R / 2$ ' from the Earth’s surface, is to made an Earth’s satellite at a height of ' $R / 2$ ' from the Earth’s surface. The work required to do this is... (M-mass of the Earth, R-radius of the Earth and neglect the rotation of earth.)

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a

$\frac{15 G M m}{2 R}$

b

$\frac{17 G M m}{24 R}$

c

$\frac{G M m}{R}$

d

$\frac{25 G M m}{24 R}$

answer is B.

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Detailed Solution

The potential energy at depth is given by $U_{1} = V_{1} m = - \frac{G M}{2 R^{3}} (3 R^{2} - \frac{R^{2}}{4}) m = - \frac{11 G M m}{8 R}$
The energy of satellite at a height R/2 is given by $E_{2} = - \frac{G M m}{3 R}$
The work required $= E_{2} - U_{1} = \frac{25 G M m}{24 R}$

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