Question

Find the natural frequency of the system shown in figure. The pulleys are smooth and
massless.

Answer 1

$4T=Mg$

$T=\frac{Mg}{4}$

Each spring $x_0$ at mean position

$k{x}_0=T=\frac{Mg}{4}$

Let block displace x displacement then spring stretched $2x$ each.

Now,$Mg-4T^{\prime }=Ma$

$Mg-4T^{\prime }=Ma$

$Mg-4x(x_0+2x)=Ma$

$Mg-4k{x}_0-8kx=Ma$

$-8kx=Ma$

$-\frac{8k}{M}x=a$

Comparing with $-omeg{a}^{2}x=a$

$\omega =\sqrt{\frac{8k}{M}}$