Question

In the figure shown, there is a hollow hemisphere of radius $R$. It has a uniform mass distribution having total mass $m$. The gravitational potentials at points $A,D$ and $B$ are $V_A,V_D$ and $V_B$ respectively. Distance of $D$ and $B$ from the centre $C$ are $\frac{R}{2}$ and $2R$ respectively. The points $C,D$ and $B$ are lying on a radial line of the hollow hemisphere.

• A. $V_A=-\frac{Gm}{R}$
• B. $V_D=-\frac{Gm}{R}$
• C. $V_B=-\frac{Gm}{2R}$
• D. $V_A=V_D<V_B$

Answer 1

Answer: (A), (B), (C), (D)

$V_A=V_D<V_B$

Consider another identical hemisphere to complete a hollow spherical shell.


The potential at a point D due to half shell

$V_D=\frac{1}{2}\times$ potential due to complete shell at D (due to symmetry) $=\frac{1}{2}\times (-\frac{G\cdot 2m}{R})=-\frac{Gm}{R}$

$V_A=\frac{1}{2}\times$ potential due to complete shell at $A=\frac{1}{2}\times (-\frac{G\cdot 2m}{R})=-\frac{Gm}{R}$

$V_B=\frac{1}{2}\times$ potential due to complete shell at $B=\frac{1}{2}\times (-\frac{G\cdot 2m}{2R})=-\frac{Gm}{2R}$

Ans. $v_A=v_D=-\frac{Gm}{R},v_B=-\frac{Gm}{2R}$