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# A Spherical hole of radius $\large\frac{R}{2}$ is made in a solid sphere of radius R. The mass of the sphere before hollowing was M. The gravitational field at the center of the hole due to the remaining mass is

$a)\; zero \\b)\;\frac{GM}{2 R^2} \\c)\;\large\frac{GM}{8R^2}\\ d)\;\frac{GM}{R^2}$

The energy at center of hole is energy due to empty part $(E_1)$ plus energy due to remaining mass $(E_2)$
The energy due to empty mass =0 at the center of the empty portion
$E=E_1 +E_2$
$E_2= E-E_1$
$\quad= \large\frac{1}{8} \frac{GM}{\bigg(\Large\frac{R}{2} \bigg)^2}$
$E= \large\frac{GM}{2 R^2}$
Hence b is the correct answer.

edited Feb 17, 2014 by meena.p