First slide

Gravitaional potential energy

Question

If g is the 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 the earth to a height equal to the radius R of the earth is

Moderate

A

2mgR
B

mg(R/2)
C

mg(R/4)
D

MgR

Solution

The gravitational potential energy of the body on earth's surface
$U_{R} = \frac{GMm}{R}$
At a height h from the earth's surface
$U_{R + h} = - \frac{GMm}{(R + h)} = - \frac{GMm}{2 R} (∵ h = R)$
Therefore, gain in potential energy
$\begin{array}{l} = U_{R + h} - U \\ = - \frac{GMm}{2 R} - (- \frac{GMm}{R}) = \frac{GMm}{2 R} \\ = \frac{{gmR}^{2}}{2 R} = \frac{1}{2} mgR \end{array}$

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