A uniform rod of length (L) and mass (M) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants (k). The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. 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. ?
(##JMA_CHMO_C10_009_Q01##).

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))`