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Principle of Superposition

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Question

Figure shows a spherical cavity inside a lead sphere. The surface of the cavity passes through the centre of the sphere and touches the right side of the sphere.The mass of the sphere before hollowing was M. With what gravitational force does the hollowed out lead sphere attract a particle of mass m that lies at a distance d from the centre of the lead sphere on the straight line connecting the centres of the spheres and of the cavity.

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Solution

Radius of hollow sphere is $\frac{R}{2}$ , so mass in this hollow portion would had been, $\frac{M}{8}$ .
Now net force on m due to whole sphere=force due to remaining mass+force due to cavity mass.
$∴$ Force due to remaining mass=force due to whole sphere-force due to cavity mass
$= \frac{G M m}{d^{2}} - \frac{G M m}{8 {(d - R / 2)}^{2}}$
$= \frac{G M m}{d^{2}} ⎡ ⎢ ⎢ ⎢ ⎣ 1 - \frac{1}{8 {(1 - \frac{R}{2 d})}^{2}} ⎤ ⎥ ⎥ ⎥ ⎦$

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