Question

From a solid sphere of mass $M$ and radius $R$, a spherical portion of radius $\frac{R}{2}$ is removed, as shown in the figure. Taking gravitational potential $V = 0$ at $r = \infty$, the potential at the centre of the cavity thus formed is
($G =$ gravitational constant)

• A. $-\frac{2GM}{3R}$
• B. $\frac{-2GM}{R}$
• C. $\frac{-GM}{2R}$
• D. $\frac{-GM}{R}$

Answer 1

Answer: (D)

Solid sphere is of mass $M$, radius $R$.
Spherical portion removed have radius $R / 2$, therefore its mass is $M / 8$.
Potential at the centre of cavity $= V _{\text {solid sphere }}+ V _{\text {removed part }}$

$=\frac{-G M}{2 R^{3}}\left[3 R^{2}-\left(\frac{R}{2}\right)^{2}\right]+\frac{3 G(M / 8)}{2(R / 2)} $
$=\frac{-G M}{R}$