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

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