A solid cylinder of mass `m` is attached to a horizontal spring with force constant `k`. The cylinder can roll without slipping along the horizontal plane. (See the accompanying figure.) Show that the center of mass of the cylinder executes simple harmonic motion with a period `T = 2pisqrt((3m)/(2k))`, if displaced from mean position.

Answer

A

Explanation

In displaced position,
`E = (1)/(2)kx^(2) + (1)/(2)mv^(2) + (1)/(2)I omega^(2)`
putting `I = (1)/(2)mR^(2)`
and `omega = (v)/(R)`
we get `E = (1)/(2)kx^(2) + (3)/(4)mv^(2)`
Since, E = constant
`:. (dE)/(dt) = 0`
or `0 = (1)/(2)k ((dx)/(dt))(2x) + (3)/(4)m(2v)(dv)/(dt)`
putting, `(dx)/(dt) = v` and `(dv)/(dt) = a`
we get, `F = (ma) = - ((2k)/(3))x`
Since, `F prop - x` motion is simple harmonic
`k_(e) = -(2k)/(3)`
`T = 2pi sqrt((m)/(k_(e))) = 2pi sqrt((3m)/(2k))` Hence proved.