The pulley shown in figure has a moment of inertias I about its xis and mss m. find the tikme period of vertical oscillastion of its centre of mass. The spring has spring constant k and the string does not slip over the pulley.

Explanation

Let in equilibrium extension of spring is `x_0`, tension in spring is `T=kx_0`

`mg=2T=2kx_0`
`x_0=(mg)/(2k)`
After equilibrium position, pulley goes down by x, spring extends by `2x`.
The energy of the system is
`E=1/2mv^2+1/2Iomega^2-mgx+1/2k(x_0+2x)^2`
`=1/2mv^2+1/2I(v^2)/(r^2)-mgx+1/2k(x_0+2x)^2`
Since total energy is constant
`(dE)/(dt)=0`
`0=1/2(m+I/r^2)2v(dv)/(dt)-mg((dx)/(dt))`
`+1/2k*2(x_0+2x)2(dx)/(dt)`
`=(m+I/r^2)va-mgv+2k(x_0+2x)v`
`=(m+I/r^2)a-mg+2kx_0+4kx`
`(m+I/r^2)a=-4kx`
`a=-(4k)/((m+I//r^2))x=-omega^2x`
`T=2pisqrt((m+I//r^2)/(4k))`