Find the natural frequency of the system as shown in the figure. The pulleys are massless and frictionless.
(Take `k=9pi^(2)N//m` and `m=4kg`)

Explanation

To find frequency we have to first calculate the acceleration of the block as a function of displacement. In such type of complicated systems we can find this by using the equation of energy conservation and then differentiating it. One important point is to be noted is that the spring is already stretched by some amount and beyond this extension we can not take energy to be equal to `(1//2)kx^(2)` so we have to first find out the initial extension.
`mg=T_(1)`, `T_(1)=2T_(2)`
`T_(2)=2T_(3)`, `T_(3)kx_(0)`
`rArr x_(0)=(mg)/(4k)`
If suppose the block is further pulley down by `x`, spring stretches by `4x`
`U`(Total energy of system) `=(1)/(2)k(x_(0)+4x)^(2)-mgx+(1)/(2)mv^(2)`
`(dU)/(dt)=0`
`k(x_(0)+4x)4(dx)/(dt)-mg"(dx)/(dt)+mv(dv)/(dt)=0`
`(dv)/(dt)=(d^(2)x)/(dt^(2))=-16(k)/(m)x`
`rArr f=(1)/(2pi)sqrt((16k)/(m))rArrf=3`.