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Gravitational field intensity

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=F
Component of force F sin   get cancelled net force 2Fcos θ . 
For the entire ring- GM4a2m2×2cosθ


Alternative 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

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