First slide

Gravitational potential and potential energy

Question

If g is acceleration due to gravity on the Earth's surface, the gain in the potential energy of an object of mass m raised from the surface of Earth to a height equal to the radius R of the earth is

Moderate

A

$\frac{1}{2} mgR$
B

$2 mgR$
C

$mgR$
D

$\frac{1}{4} mgR$

Solution

At a distance x from the centre of earth, the gravitational force
$\begin{array}{l} F = \frac{GMm}{x^{2}} \\ dW = Fdx = \frac{GMm}{x^{2}} dx \\ We get W = \int_{R}^{R + h} \frac{GMm}{x^{2}} dx = GMm [\frac{1}{R} - \frac{1}{R + h}] \\ = {gR}^{2} m [\frac{1}{R} - \frac{1}{R + h}] \end{array}$
Therefore, the gain in potential energy is
${gR}^{2} m [\frac{1}{R} - \frac{1}{2 R}] = \frac{mgR}{2}$

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