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Question

# 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 r√3 as shown in the figure. The gravitational force exerted by the sphere on the ring will be.

A
GMm8r2
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B
GMm4r2
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C
3GMm8r2
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D
GMm8r33
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Solution

## The correct option is C √3GMm8r2R.E.F imageOn a differential partof ring , the gravitationalforce (dF) by sphere =GMdm(2)→r(2R2)(2R)⇒d→F=4Mdm→r4R2(R)=GMdm→r4R3Total gravitational (F)=∫m0GMdm→r4R3→r is resolved into vectors arewith magnitude (√3→r2) directed perpendicular to the plane of ring and other radially with magnitude (|→r|2)Let them be r⊥ and →rc receptively.Then,→r=→r⊥+→rcSo,dF=GMdm→r4R3=GMdm4R3(→r⊥+→rc)→F=∫m0→df=∫m0GMdm(→r⊥)4R3+∫m0GMdm→rc4R3→F=∫m0GMdm4R3(√32)R^r⊥+∫m0GMdm4R3(R2)^rc→F=√3GM8R2∫m0dm^r⊥+GM8R2∫m0dm^rcAt every point ^r⊥ is same but^rc changes accordingly such that∫m0dm^r⊥=M^r⊥ and ∫m0^rc=0So, →F=√3GM8R2m^r⊥option C is correct.

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