A uniform rod of length l and mass M is pivoted at the centre. Its two ends are attached to two ends are attached to two springs of equal spring constant k. the springs are fixed to rigid support as shown in figure and the rod is free to oscillate in the horizontal plane. the rod is gently pushed through a small angle `theta` in one direction and released. the frequency of oscillation is

Options

A. `1/2pisqrt((2k)/(M))`
B. `1/2pisqrt((k)/(M))`
C. `1/2pisqrt((6k)/(M))`
D. `1/2pisqrt((24k)/(M))`

Answer

C

Explanation

Torque about `P = kx (L)/(2) + (kx) (L)/(2) = kxL = (L^(2))/(2) theta (':. x = (L)/(2)sinthta = (L)/(2)theta)`
`t = Ialpha`
`rArr -(KL^(2))/(2)theta = (ML^(2))/(12)alpha rArr (-6Ktheta)/(M) = alpha`
`rArr alpha = -(6K)/(M) theta = -omega^(2)theta`
`rArr omega = sqrt((6K)/(M))` and `f = (omega)/(2pi) = (1)/(2pi)sqrt((6K)/(M))`