Question

A solid cylinder of mass m and radius R is kept in equilibrium on a horizontal rough surface. Three unstretched springs of spring constants k, 2k, 3k are attached to cylinder as shown in the figure. Find period of small oscillations. (Given that surface is rough enough to prevent slipping of cylinder.)

Answer 1

Let there is sufficient friction to support pure rolling

Let the cylinder rotates by an angle $\theta$ clockwise

Compression in the upper spring$=\left(2R\right)\times \left(\theta \right)$

Compression in the middle spring$=(R+\frac{R}{2})\times \left(\theta \right)=\frac{3R\theta }{2}$

Compression in the lower spring$=\left(R\right)\times \left(\theta \right)$

Torque applied by upper spring$=F\times Leverarm=k\times 2R\theta \times 2R=4k{R}^2\theta$

Torque applied by middle spring$=F\times Leverarm=2k\times \frac{3}{2}R\theta \times \frac{3}{2}R=\frac{9}{2}k{R}^2\theta$

Torque applied by lower spring$=F\times Leverarm=3k\times R\theta \times R=3k{R}^2\theta$

Net anticlockwise torque$=4k{R}^2\theta +\frac{9}{2}k{R}^2\theta +3k{R}^2\theta =\frac{23}{2}k{R}^2\theta$

Taking the torque about the lowest point of the cylinder$=I\alpha =\frac{23}{2}k{R}^2\theta$

$\Rightarrow (\frac{m{R}^2}{2}+m{R}^2)\alpha =\frac{23}{2}k{R}^2\theta$

$\Rightarrow \left(\frac{3}{2}m{R}^2\right)\alpha =\frac{23}{2}k{R}^2\theta$

$\Rightarrow 3m\times \alpha =23k\theta$

$\Rightarrow \alpha =\frac{23k\theta }{3m}$

$\alpha ={\omega }^2\theta =\frac{23k\theta }{3m}\Rightarrow \omega =\sqrt{\frac{23k}{3m}}$

$\omega =\frac{2\pi }{T}=\sqrt{\frac{23k}{3m}}$

$T=2\pi \sqrt{\frac{3m}{23k}}$