A ring of mass `m` can freely slide on a smooth vertical rod. The ring is symmetrically attached with two springs, as shown, each of stiffness `k`. Ring is displaced such that each spring makes an angle `theta` with the horizontal. If the ring is slightly displaced vertically, then time period is.......

Options

A. `2pi sqrt((m)/(k))`
B. `2pi sqrt((m sin theta)/(2k))`
C. `(2pi)/(sin theta) sqrt((m)/(2k))`
D. `(pi)/(sin theta) sqrt((m)/(2k))`

Answer

C

Explanation

Let `x` be the displacement of the ring from equilibrium position and `x'` be the extension in the spring:

`Sin theta = (x')/(x) :. F_(rest) = 2kx' cos (90 - theta)`

`rArr x' = x sin theta = ma & a = omega^(2)x, T = (2pi)/(omega) = ?`