In the system shown in Fig, a disc pivoted at its centre has a spring attached at a distance 'a' form centre and `a` mass 'm' that cen rotate the disc. For small angular displacement calculate the frequency of oscillation

Explanation

In the equilibrium position stretch of spring is `x_(0) = a theta_(0)`. Torque fo spring force and weight balance each other,
i.e., `Ka^(2) theta_(0) = mgr …..(i)`
In a slightly displacement position equations of motion are `mg -T = ma = mr alpha ....(ii)`
and `Tr - Ka^(2) (theta_(0) + theta) = T alpha ....(iii)`
Substituting the value of `T` from Eqn.(ii) into Eqn. (iii), we get
`m(g -r alpha) r - Ka^(2) (theta_(0) +theta) =I alpha`
or `alpha = - (ka^(2))/((I +mr^(2))) theta`
Compering above Eqn. with necessary consition of `SHM`, i.e.,
`alpha =- omega^(2) theta , omega = sqrt((Ka^(2))/(I+mr^(2)))`
and `f = (1)/(2) sqrt((Ka^(2))/(I+mr^(2)))`