Question

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 oscillations in the absence of sliding.

Answer 1

Let's consider the cylinder is rotating at its place, so that its center of mass has no linear displacement. we are going to discuss here the oscillatory motion about the point of contact.

We can write $AO=2R$ and the angular displacement $\theta$ about $o$ is $\theta$ , So,

$A^{\prime }A=2R\theta$

So, extension in one spring= Contraction in other spring

$=R\theta$

Here we consider that $\theta$ is very small so that extension and compression remain linear.

Now the restoring force acting on spring as well as the cylinder is:

$F=$ force due to both springs

$=(k+k)R\theta$

$\Rightarrow F=2kR\theta$

Moment of this force is$=F.2R=4R^2\theta .k$ (about point $o$)

Now the moment of inertia of the cylinder about point of contact is:

$I_0=I_{C.M}+m{R}^2=\frac{1}{2}m{R}^2+m{R}^2$

$\Rightarrow I_0=\frac{3}{2}m{R}^2$

So the angular torque of the body is:

$\Rightarrow I_0\frac{d^2\theta }{d{t}^2}=\left(\frac{3}{2}m{R}^2\right)\frac{d^2\theta }{d{t}^2}$

So, the equation motion of cylinder:

$\left(\frac{3}{2}m{R}^2\right)\frac{d^2\theta }{d{t}^2}=-4k{R}^2\theta$

$\Rightarrow \frac{d^2\theta }{d{t}^2}+\left(\frac{8k}{3m}\right)\theta =0$

Angular frequency $=\omega =\sqrt{\frac{8k}{m}}$

So, Time Period $T=2\pi \sqrt{\frac{m}{8k}}=\pi \sqrt{\frac{m}{2k}}$