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Mass density of a solid sphere is $\rho$. Radius of sphere is R. The gravitational field at a distance $r$ from the center of the sphere inside it is

$(a)\;\frac{4 \rho G \pi r}{3} \quad (b)\;\frac{4 \rho G \pi r^2}{3} \quad (c)\;\frac{4 \rho G \pi R^3}{3r^2} \quad (d)\;\frac{ \rho G R^3}{\pi r}$

Gravitational Potential energy at a distance r from center where $r \leq R$ is
$E(r) =\large\frac{GM}{R^3} r$
$\qquad =\large\frac{Gr}{R^3}\bigg(\frac{4}{3}$$\pi R^3 \rho\bigg)$
$\qquad =\large\frac{4 G r \pi \rho}{3}$
Hence a is the correct answer.

edited Feb 18, 2014 by meena.p