A uniform ring of mass M and radius R is placed with its centre at distance √3R from the centre of a uniform sphere of mass m and radius R, along the common axis. Find the gravitational force exerted by the ring on the sphere.

Question

G1 Physics · Accepted Answer

Step 1: By the shell theorem the external field of the uniform sphere is the same as that of point mass m at its centre. Every element of the ring is at distance √[(√3R)²+R²]=2R from the sphere centre. Step 2: Force magnitude on ring element dm is Gm dm/(4R²). Only axial components survive; cosα=√3R/(2R)=√3/2. Step 3: Integrating over the ring givₑ F = [GmM/(4R²)](√3/2)=√3GmM/(8R²), along the common axis. This is source option D.