Q.

A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the  sphere at a distance r3 as shown in the figure. The gravitational force exerted by the sphere on the ring will be

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a

GMm8r2

b

GMm4r2

c

3GMm8r2

d

GMm8r33

answer is C.

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Detailed Solution

dF=GMdm4r2F=ΣdFcos⁡θ=ΣGMdm4r2cos⁡θ=GM4r2×3r2rΣdm=3GMm8r2Alternative solution:The gravitational field due to the ring at a distance 3r is given byE=Gm(3r)r2+(3r)23/2 or E=3Gm8r2The required force is EM, i.e., (3Gm)M/8r2
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A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the  sphere at a distance r3 as shown in the figure. The gravitational force exerted by the sphere on the ring will be