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A solid cylinder attached to horizontal ...

A solid cylinder attached to horizontal massless spring can roll without slipping along a horizontal surface. Find time period of oscillation.

A

`2pisqrt(M/(2k))`

B

`pisqrt((3M)/(2k))`

C

`pisqrt((2M)/(3k))`

D

`2pisqrt((3M)/(2k))`

Text Solution

Verified by Experts

The correct Answer is:
D
At any instant of rolling the cylinder has rotational and translation kinetic energy
`K_("rot") = (1)/(2) l omega^(2) = (1)/(2) [(1)/(2) MR^(2)] [(v^(2))/(R^(2))] = (1)/(4) Mv^(2)`
`K_("trans") = (1)/(2) Mv^(2)`
`U_("(potential)") = (1)/(2) kx^(2)`
`rArr` Total energy `E = K_("rot") + K_("trans") + U`

`rArr E = (3)/(4) Mv^(2) +(1)/(2) kx^(2)`
Here `((dE)/(dt)) - (3)/(4) M (2v) (dV)/(dt) + (1)/(2) k2x ((dx)/(dt)) = 0`
or, `((dv)/(dt)) = - ((2k)/(3M)) x`
Hence `alpha = - omega^(2)x`
Therefore the motion is simple harmonic
`omega = sqrt((2k)/(3M))`
`rArr T = 2pi sqrt((3M)/(2k))`
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