Figure shows a system consisting of pulley having radius R, a spring of force constant k and a block of mass m. Find the period of its vertical oscillation.

Explanation

The following steps are usually followed in this method:
Step 1. Find the mean position. In following figure point A shows mean position.
Step 2. Write down the mean position force relation from figure.
mg = `kx_(0)`
Step 3. Assume that particle is performing SHM with
amplitude A. Then displace the particle from its mean position.
Step 4. Find the total mechanical energy (E) in the displaced position since, mechanical energy in SHM remains constant `(dE)/(dt)=0`
`E= (1)/(2) mv^(2) + (1)/(2) Iomega^(2)+(1)/(2)k (x+x_(0))^(2)` - mgx
`E= (1)/(2) mv^(2) + (1)/(2)1(v^(2))/(R^(2))+(1)/(2)k (x+x_(0))^(2)`-mgx
`(dE)/(dt)= (2mv)/(2) (dv)/(dt)+(2Iv)/(2R^(2))(dv)/(dt)+ (2k(x+x_(0)))/(2) (dx)/(dt)`
`-mg"" (dx)/(dt) " " cdots (1)`
Put `(dx)/(dt) = v " and " (dv)/(dt) = (d^(2)x)/(dt^(2))`
in eq. (1) put
`(dE)/(dt) =0`
`implies mv (d^(2)x)/(dt^(2))+ (Iv)/(R^(2)) (d^(2)x)/(dt^(2))+kxv + kx_(0)v- mgv =0`
Which gives `(m+(1)/(R^(2)))(d^(2)x)/(dt^(2))+kx =0`
`(d^(2)x)/(dt^(2))+(k)/((m+(1)/(R^(2))))x=0 " " cdots (2)`
compare eq. (2) with S.H.M. eq. the
`omega^(2)= (k)/((m+(1)/(R^(2))))implies T = 2pisqrt(((m+1)//R^(2))/(k))`