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Question

A small object of uniform density rolls up a curved surface without slipping, with an initial velocity v m/s. It reaches up to a maximum height of v2g m with respect to the initial position. The object is

A
Ring
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B
Solid sphere
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C
Hollow sphere
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D
Disc
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Solution

The correct option is A Ring
Let the mass of object be m kg and radius be r m
Initially,
vi=v, ωi=ω
For pure rollingv=rω ...(i)
When object reaches maximum height (h=v2g), it stops completely, so vf=0, ωf=0
Applying mechanical energy conservation from initial to final position:
Gain in PEgravitational=Loss in KEtrans+ Loss in KErot
mgh=12mv2+12ICMω2 ...(ii)
From Eq. (i) and (ii),
mg(v2g)=12mv2+12ICM×(v2r2)

m=m2+ICM2r2
m2=ICM2r2

ICM=mr2
The moment of inertia of object passing through its centre or centre of mass matches with that of a ring.
Object in question is a ring.

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