A uniform stick of mass `M` and length `L` is pivoted its come its ends are attached 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 periodec motion
B. executes periodic motion which is not simple harmonic .
C. executes SHM of frequency `(1)/(2pi)sqrt((6K)/(M))`.
D. ececutes SHM of frequency `(1)/(2pi)sqrt((K)/(2M))`.

Answer

C

Explanation

If the stick is rotated through a small angle `theta`, the spring is stretched by a distance `(Ltheta)/(2)`. Therefore resoring force in spring `=((K"g"theta)/(2))`. Each spring causes a torque
`tau(=Fd)=((KLtheta)/(2))(L)/(2)`
In the same direction. Therefore equation of rotational motions is `tau=Ialpha`.
`-2((KLtheta)/(2))(L)/(2)=Ialpha` with `I=(ML^2)/(12)`
`alpha=-((6K)/(M))theta`
Standard equation of angular SHM is `alpha=-omega^2theta`
`omega^2=(6K)/(M)impliesf=(omega)/(2pi)=(1)/(2pi)sqrt((6K)/(M))`