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 $R$ -
$\frac{V}{V_A}=\frac{R}{R_A}=\frac{\sqrt{\left[R_A^2+R_B^2\right]}}{R_A}=\frac{5}{3}$