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Question

A small sphere of radius r and mass m oscillates back and forth on a smooth concave surface of radius R as shown in figure. Find the time period of small oscillations.


A
2π7R5g
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B
2π3g7R
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C
2π7(Rr)5g
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D
2π5(Rr)7g
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Solution

The correct option is C 2π7(Rr)5g
When you displace the sphere through a small angle θ.


Total energy of system
TE=KE of sphere + PE of sphere
=(12mv2+12Iω2)+mgh
[Taking mean point as a reference point]
=[12mv2+12(25mr2)(vr)2]+mg(Rr)(1cos θ)
TE=710mv2+mg(Rr)(1cos θ)
For SHM, ddt(TE)=0
710m.2v×dvdt+mg(Rr)[0+sin θ]dθdt=0
75mva+mg(Rr)sin θ×ω0=0
[dθdt=ω0=vRr & a=α(Rr)]
75mω0(Rr)(Rr)α+mg(Rr)θω0=0
[sinθθ for small angle]
α=5gθ7(Rr)[α=aRr]
So ω2=5g7(Rr) [α=ω2θ for SHM]
So, T=2π1ω2=2π7(Rr)5g

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