Inside a uniform sphere of mass M ( M is mass of complete sphere) and radius R , a cavity of radius $R /3$ is made in the sphere as shown :-

Question

Inside a uniform sphere of mass M ( M is mass of complete sphere) and radius R , a cavity of radius R /3 is made in the sphere as shown :-

TestHub · Accepted Answer

Use superposition: cavity field = field of complete sphere − field of the removed sphere of radius R/3.

Both fields are proportional to the position vector from their own centres; their difference is a constant vector, so the gravitational field inside the cavity is uniform → (A) correct, (B) incorrect.

Total potential energy of a particle at the centre P of the cavity (complete sphere + negative cavity):

$U = - \frac{GM m}{R} [\frac{3}{2} - \frac{( 2/3 R ) ^{2}}{2 R ^{2}}] - \frac{3 GM m}{2 \times R /3} \times \frac{( R /3 ) ^{2}}{R ^{2}} \Rightarrow \frac{1}{2} m v_{p}^{2} = \frac{20 GM m}{18 R}$

$v_{p} = \frac{20 GM}{9 R}$ → (D) correct, (C) incorrect.

Hence (A) and (D).

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