A uniform disc of mass `m` and radius `R` is pivoted smoothly at its centre of mass. A light spring of stiffness `k` is attached with the dics tangentially as shown in the Fig. Find the angular frequency in `(rad)/(s)` of torsional oscillation of the disc. (Take `m=5kg` and `K=10(N)/(m)`.)

Explanation

If we twist (rotate) the disc through small clockwise anghe `theta`, the spring will be deformed (compressed) by a distant `x=Rtheta`, hence the spring force `F_S=kx=k(Rtheta)` will produce a restoring torque.
Restoring torque: `tau=-F_SR` where `F_S=kRtheta`
This gives `tau=-kR^2theta`
It means after removing the external (applied) torque the restoring torque rotates the disc with an angular acceleration `alpha` which will bering the spring-disc system back to its original state.
Newton's law of rotation (or torque equation): Applying Newton's second law of rotation we have.
`tau=I_Calpha`
where `tau==kR^2theta`
This gives `alpha=-(kR^2theta)/(I_C)` where `I_C=(mR^2)/(2)`
Then `a=-(2k)/(m)theta`
Comparing the above equation with `alpha=-omega^2theta`, we have
`omega=sqrt((2k)/(m))`
After sustituting the values we get `omega=2(rad)/(s)`.