A uniform ring of mass m is lying at a distance 3a form the centre of a sphere of mass M just over the sphere (where a is the radius of the ring as well as that of the sphere). Then magnitude of gravitational force between them is

For a small mass dm, distance from the center of sphere is 2a. force along ‘dm’  G,M,dm4a2=FComponent of force F sin   get cancelled net force 2Fcos θ . For the entire ring- GM4a2m2×2cosθGMm4a23a2a=3GMm8a2Alternative solution:-field intensity due the ring at the centre of the sphere is g=GMx(R2+x2)32=GMa(a2+3a2)32=3GM8a2 F=mg=3GMm8a2