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

Options

A. `3:2`
B. `4:3`
C. `5:3`
D. `5:6`

Answer

C

Explanation

`M_(A)=sigma5piR_(A)^(2), M_(B)=sigma4piR_(B)^(2)`
where `sigma` is surface density
`implies V_(A)=(-GMA)/(R_(A)), V_(B)=(-GM_(B))/(R_(B))`
`(V_(A))/(V_(B))=(M_(A))/(M_(B)) (R_(B))/(R_(A))=(sigma4piR_(A)^(2))/(sigma4piR_(B)^(2)) (R_(B))/(R_(A))=(R_(A))/(R_(B))`
Given `(V_(A))/(V_(B))=(M_(A))/(M_(B))=3/4`
Then `R_(B)=4/3R_(A)`
For new shell of mass `M` and radius `R`
`M=M_(A)+M_(B)=sigma4piR_(A)^(2)+sigma4piR_(B)^(2)`
`sigmapiR^(2)=4pi(R_(A)^(2)+R_(B)^(2))`
Then `V/(V_(A))=(M R_(A))/(RM_(A))=(sigma4pi[R_(A)^(2)+R_(B)^(2)])/((R_(A)^(2)+R_(B)^(2))^(1/2)) (R_(A))/(sigma4piR_(A)^(2))=sqrt((R_(A)^(2)+R_(B)^(2)))/(R_(A))=5/3`