Work done in raising a body of mass $m$ from the surface of the earth to a height $\frac{R}{3}$ is .........…
Work done in raising a body of mass $m$ from the surface of the earth to a height $\frac{R}{3}$ is ......... (where, $R$ is radius of earth, $M$ is mass of the earth and $G$ is gravitational constant)
$\frac{G M m}{4 R}$
$\frac{4 G M m}{R}$
$\frac{3 G M m}{4 R}$
$\frac{G M m}{3 R}$
Solution
At surface, gravitational potential energy,
$
U_1=\frac{-G M m}{R}
$
At height $n R, U_2=\frac{-G M m}{R+n R}$
$
\begin{aligned}
\left|U_2-U_1\right| & =\left|\left(\frac{-G M m}{R+n R}\right)-\left(\frac{-G M m}{R}\right)\right| \\
& =\frac{G M m}{R}-\frac{G M m}{R+n R} \\
& =\frac{G M m}{R}\left(1-\frac{1}{n+1}\right)=\left(\frac{n}{n+1}\right) \frac{G M m}{R}
\end{aligned}
$
For height $x=\frac{R}{3}, n=\frac{1}{3}$
Work done $=\left|U_2-U_1\right|=\left(\frac{\frac{1}{3}}{\frac{1}{3}+1}\right) \frac{G M m}{R}=\frac{G M m}{4 R}$