A uniform rod of length `L` and mass `M` is pivotedat the centre. Its two ends are attached to two springs of equal spring constants. `k`. The springs as shown in the figure, and the rod is free to oscillate in hte horizontal plane. the rod is gently pushed through a small angle `theta` in one direction and released. the frequency of oscilllation 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))`