A uniform stick of mass `M` and length `L` is pivoted its come its ands are fast to two spring each of the constant `K`. In the position shown in figure the same through a small length when spring is display through a small angle of and released. The stick :

Options

A. executes non- periodic motion
B. executes periodic motion which in its not simple harmonic
C. executes`S.H.M.` of frequency `(1)/(2pi)sqrt((6K)/(M))`
D. executes`S.H.M.` of frequency `(1)/(2pi)sqrt((K)/(2M))`

Answer

C

Explanation

if stick is rotated through x small angle `theta `the spring as streached by a distance `(L)/(2) g` Therefore resulting force is spring `= K((k)/(2) g) ` each spring causes a temper `tau( = Fd) = ((kL theta)/(2)) (L)/(2)` an the same direction Therefore equation of rotation motion is
`tau = 1a`
`- 2((KL theta)/(2)) (k)/(2) = 1 a with 1 = (ML^(2))/(12)`
` a = - ((68)/(M)) gg`
syanderd equation of angle `S.H.M` is `a = - omega^(2) theta`
`omega^(2) = (6k)/(M)`
`frequency f = (omega)/(2pi) = (1)/(2pi) sqrt((6k)/(M))`