A cavity of radius R/2 is made inside a solid sphere of radius R. The centre of the cavity is located at a distance R/2 from the centre of the sphere. The gravitational force on a particle of mass m at a distance R/2 from the centre of the sphere on the line joining both the centres of the sphere and the cavity is (opposite to the centre of the cavity) [Here g = (GM)/R², where M is the mass of the sphere]

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none of these

$\frac{mg}{2}$

$\frac{mg}{16}$

$\frac{3 mg}{8}$

answer is B.

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Detailed Solution

Mass of sphere $M \propto R^{3}$ . If mass of complete sphere is M, then, the mass of cavity will be M/8.
Using the figure, F = force due to whole sphere - force due to cavity.

$\begin{array}{l} F = \frac{GMm}{R^{3}} (\frac{R}{2}) - \frac{G (M / 8) m}{(R)^{2}} \\ = \frac{GMm}{R^{2}} \frac{1}{2} - \frac{G (M / 8) m}{R^{2}} \\ = \frac{GMm}{R^{2}} [\frac{4}{8} - \frac{1}{8}] = \frac{3}{8} \frac{GMm}{R^{2}} = \frac{3}{8} mg \end{array}$

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