    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 2π√7(R−r)5gWhen 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(R−r)(1−cos θ) ⇒TE=710mv2+mg(R−r)(1−cos θ) For SHM, ddt(TE)=0 ⇒710m.2v×dvdt+mg(R−r)[0+sin θ]dθdt=0 ⇒75mva+mg(R−r)sin θ×ω0=0‘ [∵dθdt=ω0=vR−r & a=α(R−r)] ⇒75mω0(R−r)(R−r)α+mg(R−r)θω0=0 [∵sinθ≈θ for small angle] ⇒α=−5gθ7(R−r)[∵α=aR−r] So ω2=5g7(R−r) [∵α=−ω2θ for SHM] So, T=2π√1ω2=2π√7(R−r)5g  Suggest Corrections  0      Similar questions  Explore more