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A rod of length L is supported by two ideal strings of length l such that the system hangs in a vertical plane.Case 1: Rod is kept horizontal, displaced slightly perpendicular to the plane and allowed to oscillate.Case 2: Rod is given a small twist about central axis and then allowed to oscillate.Let T1,T2 be the periods of oscillations in the two respective cases.

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a

T1= 2πlg

b

T1= 2πl2g

c

T2= 2πL6g

d

T2= 2πl3g

answer is A.

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Detailed Solution

Case 1 : Similar to general case of simple pendulum.T1=2πlgCase 2 : Let θ be the angle that the rod rotate about its central axis and ϕ be the angle made by string with the vertical.In vertical , 2Tcosϕ=mg---(1)In horizontal plane, −(2Tsinϕ)L2=Iα⇒−(2Tsinϕ)L2=mL212d2θdt2⇒−(Tϕ)=mL12d2θdt2⇒-mgϕ2=mL12d2θdt2⇒d2θdt2=−6gLϕα=−3glθ∴L2θ=lϕcomparing with α=-ω2θ⇒T2=2πl3g

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