Question

A solid cylinder of mass m is attached to a horizontal spring with force constant k. The cylinder can roll without slipping along the horizontal plane. (See the accompanying figure.) Show that the center of mass of the cylinder executes simple harmonic motion with a period $T=2\pi \sqrt{\frac{3m}{2k}}$, if displaced from mean position.

Answer 1

In displaced position,
$E=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$
Putting $I=\frac{1}{2}m{R}^2$
and $\omega =\frac{v}{R}$
we get $E=\frac{1}{2}k{x}^2+\frac{3}{4}m{v}^2$
Since E=constant
$\therefore$ $\frac{dE}{dt}=0$
or $0=\frac{1}{2}k\left(\frac{dx}{dt}\right)\left(2x\right)+\frac{3}{4}m\left(2v\right)\frac{dv}{dt}$
Putting $\frac{dx}{dt}=v$ and $\frac{dv}{dt}=a$
we get,
$F=\left(ma\right)=-\left(\frac{2k}{3}\right)x$
Since $F\propto -x$, motion is simple harmonic
$k_e=-\frac{2k}{3}$
$T=2\pi \sqrt{\frac{m}{k_e}}=2\pi \sqrt{\frac{3m}{2k}}$
Hence Proved.