Question

A particle of mass $m$ is moved from the surface of the earth to a height $h$. The work done by an external agency to do this is (1) $m g h$ for $h < < R$ (3) $\frac{1}{2} m g h$ for $h=R$ (2) $m g h$ for all $h$ (4) $-\frac{1}{2} m g h$ for $h=R$

• A. 1, 2 and 3 are correct
• B. 1 and 2 are correct
• C. 2 and 4 are correct
• D. 1 and 3 are correct

Answer 1

The Correct Option is D

$U_{i}=-\frac{G M m}{R} $ $U_{f}=-\frac{G M m}{(R+h)}$ Work done $=$ change in gravitational potential energy $E=U_{f}-U_{i} $ $=-G M m\left[\frac{1}{R+h}-\frac{1}{R}\right] $ $W=-\frac{G M m}{R}\left[\left(1+\frac{h}{R}\right)^{-1}-1\right] $ $W=-\frac{G M m}{R}\left[1-\frac{h}{R}-1\right] $ $W=\frac{G M m h}{R^{2}} $ $W=m g h $ for $h \leq R $ Also $U_{i}=-\frac{G M m}{R}$ and on the surface $h=R$ $U_{f}=-\frac{G M m}{R+R}$ $=-\frac{G M m}{2 R}$ Work done $(W)=U_{f}-U_{i}$ $=\frac{G M m}{2 R} $ $W=\frac{1}{2} m g R $ $\left[\because g=\frac{G M}{R^{2}}\right]$