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Question

# The pulley shown in figure has a MOI I about its axis and mass m.Find the time period of vertical oscillations of its COM. The spring has a spring constant k and the string does not slip over the pulley

A
2πI/r2+m4k
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B
1πI/r2+m4k
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C
6πI/r2+m4k
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D
None of these
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Solution

## The correct option is A 2π√I/r2+m4k2T=mg or 2ky=mg or y=Mg2kWriting energyU=12Iω2+12mv2−mgx+12k[mg2k+2x]2=12[Ir2+m]v2+m2g28k+2kx2Put dUdt=0 as the system is conservative,giving0=[Ir2+m]vdvdt+4kxvor dvdt=a=−4kx(Ir2+m)∴ T=2π√I/r2+m4k

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