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 $=V_{\text{solid sphere }}+V_{\text{removed part }}$

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