Q. The work that must be done in lifting a body of weight $P$ from the surface of the earth to a height $h$ is
A) $\frac{PRh}{R - h}$
B) $\frac{R + h}{PRh}$
C) $\frac{PRh}{R + h}$
D) $\frac{R - h}{PRh}$
Explanation
(c) Force on the body $= \frac{GMm}{x^{2}}$
To move it by a small distance $dx$,
Work done $= F\ dx = \frac{GMm}{x^{2}}dx$
Total work done $= GMm\ \int_{R}^{R + h}\frac{dx}{x^{2}} = \left\lbrack \frac{- GMm}{x} \right\rbrack_{R}^{R + h}$
$= GMm\left\lbrack \frac{1}{R} - \frac{1}{R + h} \right\rbrack$
$= \left\lbrack \frac{(R + h) - R}{R(R + h)} \right\rbrack = \frac{GMmh}{R(R + h)}$
$\frac{GM}{R^{3}} \times \frac{mhR}{R + h} = \frac{gmhR}{R + h} = \frac{PRh}{R + h}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.