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

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

Answer 1

Answer: (B)

Gravitational potential energy on the earth surface, $U_r=\frac{-GMm}{R}$

Gravitational potential energy at a height h above the earth's surface, $U_h=\frac{-GMm}{R+h}$

$U_h=\frac{-GMm}{R+R}=\frac{-GMm}{2R}$

Gain in gravitational potential energy $=U_h-U_r$

$=\frac{-GMm}{2R}-\left(\frac{-GMm}{R}\right)=\frac{GMm}{R}-\frac{GMm}{2R}$

$=\frac{GMm}{2R}=\frac{1}{2}mgR$