The work done to raise a mass from the surface of the earth to a height , which is equal to the radius of the earth, is [2019]

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Solution

Q.
The work done to raise a mass $m$ from the surface of the earth to a height $h$ , which is equal to the radius of the earth, is [2019]

1 $\frac{3}{2} m g R$

2 $m g R$

3 $2 m g R$

4 $\frac{1}{2} m g R$

Ans.
(4)

Work done = Change in potential energy

$= u_{f} - u_{i} = \frac{- G M m}{(R + h)} - (\frac{- G M m}{R})$

where $M$ is the mass of Earth and $R$ is the radius of Earth.

$∴$ $W = G M m [\frac{1}{R} - \frac{1}{(R + h)}]$

Now, $h = R$

$∴$ $W = G M m [\frac{1}{R} - \frac{1}{2 R}] = \frac{G M m}{2 R}$

$\Rightarrow W = \frac{m g R}{2}$ $[∵ g = \frac{G M}{R^{2}}]$

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