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Question

A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.

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Solution

Let the angular velocity of the system about the point is suspension at any times be ω.

Sp ve=(Rr) ω

Again ve=rω

[where ω1= rotaional velocity of the sphere ]

l=ver=(Rrr)ω ...(1)

By energy method, total energy in SHM is constant.

So, mg (Rr) (1cos θ)+12mv2c+12Iω2= constant

mg(Rr) (1cos θ)+12m(Rr)2 ω2+12mr2(Rrr)ω2= constant

mg(Rr)(1cos θ)+12m(Rr)2 ω2+15mr2(Rrr)ω2= constant

g (Rr)(1cos θ)+(Rr)ω2 [12+15]= constant

Taking derivative,

g (Rr)sinθdθdt=710(Rr)2 2ωdωdt

g sin θ=2×(710)(Rr)α

g sin θ=(75)(Rr)α

α=5g sin θ7 (Rr)

=5gθ7 (Rr)

αθ=ω2

=5g7 (Rr)= constant

So the motion is S.H.M. again

ω=5g7 (Rr)

T=2π 7 (Rr)5g


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