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Q.

The change in potential energy when a body of mass m is raised to a height nR from the earth’s surface is: (R=Radius of the earth)

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a

$m g R [\frac{n}{(n - 1)}]$

b

$\frac{m g R}{n}$

c

$m g R [\frac{n}{(n + 1)}]$

d

$m g R [\frac{n^{2}}{(n^{2} + 1)}]$

answer is C.

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Detailed Solution

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change in PE,
$\begin{array}{l} Δ U = U_{2} - U_{1} \\ ∴ Δ U = \frac{- G M m}{(R + n R)} + \frac{G M m}{R} \\ ⇒ Δ U = \frac{- G M m}{R (1 + n)} + \frac{G M m}{R} = \frac{G M m}{R} [\frac{- 1}{1 + n} + 1] \\ \Rightarrow Δ U = (\frac{(R^{2} g) m}{R} \times \frac{n}{1 + n}) [∴ g = \frac{G M}{R^{2}}] \\ \Rightarrow Δ U = m g R (\frac{n}{1 + n}) \end{array}$

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