Question

Two plates 1 and 2 move with velocities $v$ and $2v$ as shown. If the sphere does not slide relative to the plates and assuming the mass of each body as $m$, find the kinetic energy of the system (plates + sphere).

• A. $\frac{123}{40}m{v}^2$
• B. $\frac{230}{40}m{v}^2$
• C. $\frac{123}{40}m{v}^3$
• D. $\frac{120}{50}m{v}^2$

Answer 1

The correct option is A $\frac{123}{40}m{v}^2$

Total energy= $K{E}_{P1}+K{E}_{P2}+K{E}_{sphere}$ ....(1)
calculation of kinetic energy of plates,

$=\frac{1}{2}m{v}^2+\frac{1}{2}m(2v{)}^2=\frac{5}{2}m{v}^2$

Calculation for sphere,
$KE=\frac{1}{2}I{\omega }^2+\frac{1}{2}m{v}_m^2$
where, $I=\frac{2}{5}m{R}^2$ for sphere.

let sphere moving with speed $v_{c}m$
and $\omega =\frac{3v}{2R}$
$v=R\omega -v_{cm}v=R\times \frac{3v}{2R}-v_{cm}{v}_{cm}=\frac{v}{2}$
so the kinetic energy of sphere,

$KE=\frac{1}{2}\times \frac{2}{5}m{R}^2(\frac{3v}{2R}{)}^2+\frac{1}{2}m(\frac{v}{2}{)}^2$
$KE=\frac{9}{20}m{v}^2+\frac{m{v}^2}{8}$

So,finall kinnetic energy= $\frac{5}{2}m{v}^2+KE=\frac{9}{20}m{v}^2+\frac{m{v}^2}{8}=\frac{123}{40}m{v}^2$