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

• A. $ 2mgR $
• B. $ mgR $
• C. $ \frac{1}{2}\,mgR $
• D. $ \frac{1}{4}\,mgR $

Answer 1

The Correct Option is C

The potential energy of an object at the surface of the earth
$U_1=-\frac{GMm}{R} $ .....(i)
The potential energy of the object at a height
$h= R$ from the surface of the earth
$U_2=-\frac{GMm}{R+ h}$
$=-\frac{GMm}{R+R} $ .....(ii)
Hence, the gain in potential energy of the object
$\Delta U=U_2-U_1 $
$\Delta U=-\frac{GMm}{R+R}+\frac{GMm}{R}$
$\Delta U=-\frac{GMm}{2R}+\frac{GMm}{R}$
$\Delta U=\frac{1}{2}\frac{GMm}{R}$
But we know that $GM =gR^{2}$
Hence, $ \Delta U=\frac{1}{2}\frac{gR^2m}{R}$
or $\Delta U=\frac{1}{2} g R m$
or $ \Delta U=\frac{1}{2}mgR$