Two identical simple pendulums each of length 'l' are connected by a weightless spring as shown. In equilibrium, the pendulums are vertical and the spring is horizontal and undformed. The time period of small oscillations of the linked pendulums, when they are deflected form their equilibrium positive through equal displacements in the same vertical plane in the opposite directions and released, is

Options

A. `2pi sqrt((l)/(g))`
B. `2pi sqrt((l)/((g+(2kl)/(m))))`
C. `2pi sqrt((l)/((g+(kl)/(m))))`
D. `2pi sqrt((2l)/(g)+(m)/(k))`

Answer

B

Explanation

At the disturbed position:

Restoring torque on particle is
`tau =- [(mg xx l sin theta)+(2kl thetaxx l cos theta)]`
for small, `theta, sin theta rarr theta: cos theta rarr 1` and
`tau = I alpha, alpha =- omega^(2) theta : T = (2pi)/(omega)`