First slide

Gravitational potential and potential energy

Question

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 earth)

Moderate

A

$m g R \frac{n}{n - 1}$
B

$n m g R$
C

$m g R \frac{n^{2}}{n^{2} + 1}$
D

$m g R \frac{n}{n + 1}$

Solution

$I m e t h o d Δ U = \frac{m g h}{1 + \frac{h}{R}} = \frac{m g n R}{1 + \frac{n R}{R}} = \frac{n m g R}{n + 1} I I m e t h o d r_{1} = R, r_{2} = R + h a n d U = - \frac{G m M}{R} Δ U = U_{2 -} U_{1} = G m M (\frac{1}{r_{1}} - \frac{1}{r_{2}}) = G m M (\frac{1}{R} - \frac{1}{R + h}) = G m M (\frac{1}{R} - \frac{1}{R + n R}) = \frac{G m M}{R} (1 - \frac{1}{1 + n}) = \frac{m g R^{2}}{R} (\frac{1 + n - 1}{1 + n}) (∵ g = \frac{G M}{R^{2}} \Rightarrow G M = g R^{2}) = \frac{n m g R}{(n + 1)}$

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