A solid cylinder of mass M is attached to the spring by means of a frintionless pivot. The cylinder is placed on floor. Find the time period of small oscillation of the system, if the floor is sufficiently rought to prevent any slipping of cylinder. Assume stiffness of spring to be K.

Options

A. `2pisqrt((7M)/(5K))`
B. `2pisqrt((5M)/(3K))`
C. `2pisqrt((3M)/(2K))`
D. `2pisqrt((M)/(K))`

Answer

C

Explanation

Let the cylinder is slightly displaced from mean position
the total energy of the system when elongation in spring is x.
`E=(1)/(2)kx^(2)+(1)/(2)Mv^(2)+(1)/(2)xx((1)/(2)MR^(2))omega^(2)`
Differentiating the above expression we get
`2kx(dx)/(dt)+M2v(dv)/(dt)+(1)/(2)MR^(2)(2v)/(R^(2))(dv)/(dt)=0`
`2kx+(2M+M)(dv)/(dt)=0`
`(dv)/(dt)=-((2K)/(3M))x`
`T=2pisqrt((3M)/(2k))`