$\frac{1}{2} Kx ^{2}+\frac{1}{2} mv ^{2}+\frac{1}{2} I \omega^{2}= C$

$\frac{1}{2} K (2 x ) \frac{ dx }{ dt }+\frac{1}{2} m (2 v ) \frac{ dv }{ dt }+\frac{1}{2} I \frac{2 v }{ r ^{2}} \frac{ d v }{ dt }=0$

$Kvx +\frac{ m }{4} va +\frac{ m }{2} va =0$

$- Kx =\frac{3 ma }{4}$

Simplify further.

$a =-\frac{4 K }{3 m } x =-\omega^{2} x$

$\omega^{2}=\frac{4 K }{3 m }$

$\omega=\sqrt{\frac{4 K }{3 m }}$

Substitute the values as,

$\omega=\sqrt{\frac{4(100)}{3(1)}}$

$=\frac{20}{\sqrt{3}} rad / sec$