Question

The gravitational potential of two homogeneous spherical shells $A$ and $B,$ having the same surface mass density, at their respective centres are in the ratio $3:4$ . If the two shells coalesce into single one such that the surface mass density remains the same, then the ratio of potential at an internal point of the new shell and A is equal to

• A. $3:2$
• B. $4:3$
• C. $5:3$
• D. $5:6$

Answer 1

Answer: (C)

$M_{A}=\sigma 4\pi R_{A}^{2}, \, M_{B}=\sigma 4\pi \, R_{B}^{2}$
where $\sigma $ is surface density
$V_{A}=\frac{- G M_{A}}{R_{A}},V_{B}=\frac{- G M_{B}}{R_{B}}$
$\frac{V_{A}}{V_{B}}=\frac{M_{A}}{M_{B}}\frac{R_{B}}{R_{A}}=\frac{\sigma 4 \pi R_{A}^{2}}{\sigma 4 \pi R_{B}^{2}}\frac{R_{B}}{R_{A}}=\frac{R_{A}}{R_{B}}$
Given $\frac{V_{A}}{V_{B}}=\frac{R_{A}}{R_{B}}=\frac{3}{4}$
then $R_{B}=\frac{4}{3} \, R_{A}$
For new shell of mass $M$ and Radius $R$ -
$\frac{V}{V_{A}}=\frac{R}{R_{A}}=\frac{\sqrt{\left[R_{A}^{2} + R_{B}^{2}\right]}}{R_{A}}=\frac{5}{3}$