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# The radius of the earth $R_e$ and the acceleration due to gravity at its surface is g. The work required to rising a body of mass m to height h from the surface of the earth will be

$(a)\;\frac{mgh}{\bigg(1+\large\frac{h}{R_e}\bigg)} \quad (b)\;\frac{mgh}{\bigg(1+\large\frac{h}{R_e}\bigg)^2} \quad (c)\;\frac{mgh}{\bigg(1-\large\frac{h}{R_e}\bigg)} \quad (d)\;\frac{mg}{\bigg(1+\large\frac{h}{R_e}\bigg)}$

Work done = Change in potential energy
$W=GMe_m \bigg[\large\frac{1}{R_e}-\frac{1}{R_{e+h}}\bigg]$
$\quad=\large\frac{GM_emh}{R_e (R_e +h)}$
$\quad=\large\frac{gR_e^2mh}{R_e (R_e+h)}\qquad$$[GM_e=gR_e^2]$
$\quad= \large\frac{mgh}{\bigg(1+\Large\frac{h}{R_e}\bigg)}$
Hence a is the correct answer.

edited Feb 17, 2014 by meena.p