A disc of mass `m` hanged by a string is attached at `P` and a spring of stiffness `k` is attached at `O`. Find the frequency of small angular oscillation of the disc if the string does not slide over the pulley. Assume `l_(0) = Ml` of the disc about `O`.

Answer

`(1)/(2 pi) sqrt ((k(R + r)^(2))/(I_(0) mr^(2)))`

Explanation

Taking torque of spring force force about P for a small displacement of O, we have
`Kx (R + r) = I_(p) alpha , where x = (R + r) theta`
or `k(R - r) 2 theta = (mr^(2) + I_(0)) omega^(2) theta`
`omega = sqrt((k(R + r)^(2))/(mr^(2) + I_(0)))`
`f = (omega)/(2 pi) = (1)/(2 pi) sqrt ((k(R + r)^(2))/(I_(0) mr^(2)))`