A solid sphere of mass m is attached to a light spring of force constant k so that it can roll without slipping along a horizontal surface. Calculate the period of oscillation made by sphere.

Explanation

Let at instant, extension of spring is x, v is speed of centre of mass and `omega` is angular speed of cylinder

Energy of system
`E=1/2mv^2+1/2lomega^2+1/2kx^2`
`=1/2mv^2+1/2*2/5mR^2(v/R)^2+1/2kx^2`
`=7/10mv^2+1/2kx^2`
As total energy is constant
`(dE)/(dt)=0`
`0=7/10m*2v(dv)/(dt)+1/2k2x(dx)/(dx)`
`=7/5mva+(kx)/(2)v`
`a=-5/7k/mx=-omega^2x`
`T=2pisqrt((7m)/(5k))`