A solid cylinder of mass 'm' is attached to a horizontal spring with a force constant K. The cylinder can roll without slipping along the horizontal plane as shown in the figure. Find the time period of centre of mass of the cylinder

A solid cylinder of mass 'm' is attached to a horizontal spring with a

1.	A solid cylinder of mass 'm' is attached to a horizontal spring with a force constant K. The cylinder can roll without slipping along the horizontal plane as shown in the figure. Find the time period of centre of mass of the cylinder
Answer» Solution :Let x be the displacement of the centre of mass of disc towards left for no slipping. `a_(cm) = R alpha , V_(cm) = R omega and x = R theta` Total energy of the disc is `E= 1/2 MV_c^2 + 1/2 I omega^2 + 1/2 Kx^2 ` constant `(dE)/(DT) =0 ` as E is constant `impliesMV_c a_c + I omega alpha + Kx V_c = 0 impliesMR^2 alpha omega + (MR^2)/(2) omega alpha + KR^2 omega theta = 0 ` `(3MR^2)/(2) alpha + K theta = 0 , alpha =-((2K)/(3M)) theta impliesT = 2pi sqrt((alpha)/(theta)) = 2pi sqrt((3M)/(2K))`