A uniform ring of mas m and radius a is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is at a distance `sqrt3` a from the centre of the sphere. Find the gravitational force exerted by the sphere on the ring.

Explanation

The gravitational field at any point on the ring due to the sphere is equal to the field due to single particle of mass M placed at the centre of the sphere. Thus, the force on the ring due to the sphere is also equal to the force on it by particle of mass M placed at this point. By Newton's third law it is equal to the force on the particle by the ring. Now the gravitational field due to the ring at a distnce `d = sqrt(3)` a on its axis is given as
`g = (Gmd)/((a^(2)+d^(2))^(3//2))=(sqrt(3)Gm)/(8a^(2))`

The force on sphere of mass M placed here is
`F = Mg`
`= (sqrt(3)GMm)/(8a^(2))`