Question

A body of mass $m$ rises to a height $5 R$ from the surface of earth. If $g$ is the acceleration due to gravity at the surface of earth, the increase in potential energy is $(R=$ radius of earth $)$

• A. $\frac{4}{5}\, m g R$
• B. $\frac{5}{6} \,m g R$
• C. $\frac{6}{7}\, m g R$
• D. $m g R$

Answer 1

Answer: (B)

Gravitational potential energy at any point at a distance $r$ from the centre of the earth is $U=-\frac{G M m}{r}$ where $M$ and $m$ are the masses of earth and body respectively. At the surface of the earth, $r=R \therefore U_{1}=-\frac{G M m}{R}$ At a height $h$ from surface, $r=R+h=R+5 R=6 R$
$\therefore U_{2}=-\frac{G M m}{6 R}$
Increase in potential energy is
$\Delta U=U_{2}-U_{1}=-\frac{G M m}{6 R}-\left(-\frac{G M m}{R}\right)$
$=\frac{5 G M m}{6 R}$
$=\frac{5}{6} m g R$
$\left(\because g=\frac{G M}{R^{2}}\right)$