If g is the acceleration due to gravity on earth's surface, the gain of 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 e

Question

If g is the acceleration due to gravity on earth's surface, the gain of 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

TestHub · Accepted Answer

The potential energy of an object at the surface of the earth

$U_{1} = - \frac{G M m}{R} \dots . (i)$

The potential energy of the subject at a height $h = R$ from the surface of the earth

$U_{2} = - \frac{G M m}{R + h} = - \frac{G M m}{R + R} \dots (i i)$

Hence, the gain in potential energy of the object

$∆ U = U_{2} - U_{1}$

$∆ U = - \frac{G M m}{R + R} + \frac{G M m}{R}$

$∆ U = - \frac{G M m}{2 R} + \frac{G M m}{R}$

$∆ U = \frac{1}{2} \frac{G M m}{R}$

But we know that $G M = {g R}^{2}$

$H e n c e, ∆ U = \frac{1}{2} \frac{{g R}^{2} m}{R}$

$o r ∆ U = \frac{1}{2} m g R$

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