A solid uniform cylinder of mass `m` performs small oscillations due to the action of two springs of stiffness `k` each (figure). Find the period of these oscillation in the absendce of sliding.
`x`

Explanation

Let `k_(1)` and `k_(2)` be the spring constant of left and right sides springs. As the rolling of the solide cylinder is pure its lowest point becomes the instantaneous centre of rotation. If `theta` be the smaller angular displacement of its upper most point relavtive to its equilibrium position, the deformation of each spring becomes `(2Rtheta)`. Since the mechanical energy of oscillation of the solide cylinder is conserved, `E=T+U=` constant
i.e. `(1)/(2) I_(p)(dot(theta))^(2)+(1)/(2) k_(1)(2Rtheta)^(2)+(1)/(2) k_(2)(2Rtheta)^(2)=` constant
Differentiating w.r.t. time
`(1)/(2) I_(p)2 dot (theta) ddot (theta)+(1)/(2) (k_(1)+k_(2))4R^(2)2 thetadot (theta)=0`
or, `((mR^(2))/(2)+mR^(2))ddot(theta)+4 R^(2)k theta=0`
`(` Because `I_(p)=I_(C)+mR^(2)=(mR^(2))/(2)+mR^(2))`
Hence `ddot (theta)=-(8)/(3)(k)/(m)theta`
Thus `omega_(0)=(8k)/(3m)` and sourght time period
`T=(2pi)/(omega_(0))=2pisqrt((3m)/(8k))=pisqrt((3m)/(2k))`