A solid sphere of mass $M$ and radius $R$ has a spherical cavity of radius $R/2$ carved out from it. Taking gravitational potential $V = 0$ at $r = \infty$, the potential at the centre of the cavity is:
Using the superposition principle: Potential at cavity centre = (Potential due to full sphere at that point) $-$ (Potential due to removed small sphere at its own centre).
The cavity centre is at distance $R/2$ from the big sphere centre.
Potential due to full sphere at $r = R/2$ (inside):
$V_{full} = -\frac{GM}{2R^3}\left(3R^2 - \frac{R^2}{4}\right) = -\frac{11GM}{8R}$
Potential due to removed sphere (mass $M/8$, radius $R/2$) at its own centre:
$V_{small} = -\frac{3G(M/8)}{2(R/2)} = -\frac{3GM}{8R}$
$V_{cavity} = V_{full} - V_{small} = -\frac{11GM}{8R} + \frac{3GM}{8R} = -\frac{GM}{R}$
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