Question

A solid cylinder attached to a horizontal massless spring, can roll without slipping along a horizontal surface. If the cylinder is displaced and released, the corresponding time period is $2\pi \sqrt{\frac{pM}{qk}}.$ The value of $p+q=$ ______.

Answer 1

Answer: 5.00

At any instant of rolling, the cylinder has rotational and translational kinetic energies.
$K_{\text{(rotational) }}=\frac{1}{2}I{\omega }^2=\frac{1}{2}\left[\frac{1}{2}M{R}^2\right]\left[\frac{v^2}{R^2}\right]=\frac{1}{4}M{v}^2$
$K_{\text{(translational) }}=\frac{1}{2}M{v}^2$
$U$ (potential energy of the system) $=\frac{1}{2}k{x}^2$
As, $K_{\text{total }}+U=$ constant
$E=\frac{3}{4}M{v}^2+\frac{1}{2}k{x}^2=$ constant
$\Rightarrow \frac{dE}{dt}=\frac{3}{4}M(2v)\frac{dv}{dt}+\frac{1}{2}k(2x)\frac{dx}{dt}=0$
$\Rightarrow \frac{dv}{dt}=-\left(\frac{2k}{3M}\right)x$
$\Rightarrow \omega =\sqrt{\frac{2k}{3M}}$
or $T=2\pi \sqrt{\frac{3M}{2k}}$
As $p=3,q=2$
$\Rightarrow p+q=5$