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)`.)

Answer

2

Explanation

If we twist (rotate) the disc through a small clockwise angle`theta` the spring will be deformed (compressed) by a distance`x=theta` Hence, the spring force `F_(s)=kx=k(Rtheta)`will produce a restoring torque.
Restoring torque : `tau=-F_(s)R` where `F_(s)=kRtheta`
This gives `tau=-kR^(2)theta`
It means after removing the external (applied) torque, the restoring torque rotates the disc with an angular acceleration which will bring 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_(c)a` where
`tau=-kR^(2)theta`
This gives `a=(kR^(2)theta)/(I_(c))` where `I_(c)=(mR^(2))/(2)`
Then `a=-(2k)/(m)theta`
Comparing the above equation with `a=-omega^(2)theta`, we have `omega=sqrt((2K)/(m))`
After susbstitiuting the values we get `omega=2 rad//s`.