A ring having non-uniform distribution of mass `M` and radius `R` is being considered. A point mass `m_(0)` is taken slowly towards the ring. In doing so, work done by the external force against the gravitational force exerted by ring is

Options

A. `(GMm_(0))/(sqrt(2)R)`
B. `(GMm_(0))/R[1/sqrt(2)-1/(sqrt(5))]`
C. `(GMm_(0))/R[1/sqrt(5)-1/(sqrt(2))]`
D. It is not possible to find the required work as the nature of distribution of mass is not known.

Answer

B

Explanation

Even though the distribution of the mass is unknown, we can find the potential due to the ring on any axial point because from any axial point the entire mass is at the same distance (whatever woud be the nature of distribution).
Potential at `A` due to the ring is `V_(A)=(GM)/sqrt((2R)`
Potential at `B` due to the ring is `V_(B)=(GM)/(sqrt(5)R)`
`d_(U)=U_(f)-U_(i)-U_(i)=U_(B)-U_(A)=m_(0)(V_(B)-V_(A))`
`(GMm_(0))/R[-1/(sqrt(5))+1/(sqrt(2))]`
`W_(gr)=-W_(ext)`
`W_(gr)=dU=-W_(ext)`
`W_(ext)=dU=(GMm_(0))/R[1/(sqrt(2))-1/(sqrt(5))]`