X = 24

$\therefore$ $4 = \frac{x}{2}$

$4 = r \theta$

$\therefore$ $x = 2r \theta$

$\frac{4}{r} = \theta$

$\frac{1}{2} M_1 x^2 + \frac{1}{2} my^2 + \frac{1}{2} I_G \theta ^2$

$\frac{1}{2} M_1 4r^2 \theta ^2 + \frac{1}{2} m r^2 \theta ^2 + \frac{1}{2} (\frac{mr^2}{2}) \theta^2$

$K E = 2m_1 r^2 \theta^2 + \frac{1}{2} mr^2 \theta^2 + \frac{1}{4} mr^2 \theta^2$

$\rightarrow$ $PE = (PE)_k = \frac{1}{2} ky^2 = \frac{1}{2} kr^2 \theta^2$

By energy method,

$2M_1 r^2 2 \theta \theta + \frac{1}{2} mr^2 2 \theta \theta + \frac{1}{4} mr^2 2 \theta \theta + \frac{1}{2} kr^2 2 \theta \theta = 0$

$4r M_1 r^2 \theta + mr^2 \theta + \frac{mr^2}{2} \theta + kr^2 \theta = 0$

$4M_1 r^2 \theta + \frac{3mr^2}{2} \theta + kr^2 \theta = 0$

$[ 4m_1 r^2 + \frac{3mr^2}{2}] \theta + kr^2 \theta = 0$

$\theta + \frac{kr^2}{r2 [4m_1 + \frac{3m}{2}]} = 0$

$\theta + \frac{k}{(\frac{4m_1 + 3m}{2})} \theta = 0$

$\therefore$ $W_n = \sqrt{ \frac{k}{ (4m_1 + \frac{3m}{2})}}$ rad/sec