A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance `r sqrt(3)` as shown in the figure. The gravitational force exerted by the sphere on the ring will be

Options

A. `(GMm)/(8r^(2))`
B. `(GMm)/(4r^(2))`
C. `sqrt3(GMm)/(8r^(2))`
D. `(GMm)/(8r^(3)sqrt(3))`

Answer

C

Explanation

`dF=G(Mdm)/(4r^(2))`
`F=SigmadFcostheta`
`=Sigma(GMdm)/(4r^(2))costheta`
`=(GM)/(4r^(2))xx(sqrt(3)r)/(2r)Sigmadm`
`=(sqrt(3)GMm)/(8r^(2))`
Alternative solution:
The gravitational field due to the ring at a distance `sqrt(3)r` is given by
`E=(Gm(sqrt(3)r))/([r^(2)+(sqrt(3)r)^(2)]^(3/2))` or `E=(sqrt(3)Gm)/(8r^(2))`
The required force is `EM` i.e., `(sqrt(3) Gm)M//(8r^(2))`