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A uniform ring of mass 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 3a from the centre of the sphere. The gravitational force (F) exerted by the sphere on the ring is

A
3GMm8a2
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B
2GMm3a2
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C
7GMm2a2
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D
3GMma2
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Solution

The correct option is D 3GMm8a2
Given data :
Mass of the ring - m
Radius of the ring - a
Mass of the sphere - M
Radius of sphere - a
Distance between center of ring and center of sphere - r=3a
The gravitational field, F of the sphere is given by:
F=GMr(r2+a2)32 ........(1)
where, G is gravitational constant.
Using the value of r in equation (1), we get:
F=GM(3a)((3a)2+a2)32

F=3GMa(3a2+a2)32

F=3GMa(4a2)32

F=3GMa(23)(a3)

F=3GM8(a2)

Now, the force, F exerted by the sphere on the ring is,
F=F×m

F=(3GM8(a2))m

F=3GMm8(a2)

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