Q. A solid sphere of mass M and radius R has a spherical gravity of radius $\frac{R}{2}$ such that the centre of cavity is at distance R/ 2 from the centre of the sphere. A point mass $m$ is placed inside the cavity at a distance R/4 from the centre of sphere. The gravitational pull between the sphere and the point mass $m$ is
A) $\frac{11GMm}{R^{2}}$
B) $\frac{14GMm}{R^{2}}$
C) $\frac{GMm}{{2R}^{2}}$
D) $\frac{GMm}{R^{2}}$
Explanation
(b) To calculate the force of attraction on the point mass $m$ we should calculate the force due to the solid sphere and subtract from this the force which the mass of the hollow sphere would have exerted on $m\ ie,\ $
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$\ \ \ \ \ F = \frac{GmM}{x^{2}} - \frac{GmM^{''}}{y^{2}}$
$\ \ \ \ \lbrack x = R/4,\ x + y = R/2\rbrack$
$\ \ \ \ \ M = \left( \frac{4}{3} \right)\pi R^{3}\rho$
$and\ \ M^{''} = \frac{4}{3}\pi\left( \frac{R}{2} \right)^{3}\rho = \frac{M}{8}$
$\ \ \ \ \ \ \ \ \ \ \ \ F = \frac{GMm}{(R/4)^{2}} - \frac{Gm(M/8)}{(R/4)^{2}} = \frac{14GmM}{R^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.