A ring of mass m and radius a is connected to an inextensible string which passes over a frictionless pulley. The other end of string is connected to upper end of a massless spring of spring constant k. The lower end of the spring is fixed. The ring can rotate in the vertical plane about hinge without any friction. If horizontal position of ring is equilibrium position then find time period of small in oscillations of the ring.

Explanation

When displacement of the ring is `theta`, then extension in spring `= (2a theta + x_(0))`
Energy of system
`E = (1)/(2) k (2a theta + x_(0))^(2) - mg a theta + (1)/(2)l ((d theta)/(dt))^(2)`
`E = (1)/(2) k (2a theta + x_(0))^(2) - mg a theta + (1)/(2) ((1)/(2) ma^(2) + ma^(2)) ((d theta)/(dt))^(2)`
`rArr (dE)/(dt) = k (2a theta + x_(0)) 2a (d theta)/(dt) - mg a (d theta)/(dt) + (3)/(2) ma^(2) ((d theta)/(dt)) (d^(2) theta)/(dt^(2))`
As `(dE)/(dt) = 0, k (2 a theta + x_(0)) 2a - mg a = - (3)/(2) ma^(2) (d^(2) theta)/(dt^(2))`
`4ak theta + 2a k x_(0) - mg a = - (3)/(2) ma^(2) (d^(2) theta)/(dt^(2))`
`:. (8)/(3) (k)/(m) theta = - (d^(2) theta)/(dt^(2))`
`:. omega = sqrt((8k)/(3m))`
`:. T = 2pi sqrt((3m)/(8k))`