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Question

A solid cylinder is attached to a horizontal massless spring so that it can roll without slipping on a horizontal surface. The spring constant k is 3N/m. If the system is released from rest, find the time period of small oscillation of centre of mass of the cylinder
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A
2πmk
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B
2π3m2k
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C
2π2m3k
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D
2πm2k
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Solution

The correct option is B 2π3m2k
E=12kx2+12Icmω2+12mv2cmE=12kx2+12(12mR2)(vcmR2)+12mv2cm[ccm=Rω]E=12kx2+14mv2cm+12mv2cmwetakethattheenergyconstantsothederivativeoftheenergywithrespecttotimeiszerodEdt=122kxdxdt+323mvcmdvcmdtdEdt=122kxvcm+322mvcmdvcmdt0=vcm(kx+32md2xdt2)0=kx+32md2xdt232md2xdt2+kx=0d2xdt2+2k3mx=0Lettheangularfrequencybeω=2k3mtheangularfrequencyisequaltoω=2πfandtimeperiodT=1fThenω=2π1TT=2π1ωT=2π12k3mT=2π3m2k.
Hence, the option B is the correct answer.

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