Question

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

• A. $\frac{125}{40}m{v}^2$
• B. $\frac{123}{40}m{v}^2$
• C. $\frac{23}{90}m{v}^2$
• D. $\frac{23}{40}m{v}^2$

Answer 1

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



In the above diagram, C is the point about which the whole body is rotating.
From the similar triangle we get,
$\frac{y}{v}=\frac{2R-y}{2v}$
$\Rightarrow y=\frac{2}{3}R$

$KE=\frac{1}{2}{I}_c{\omega }_c^2$
$\omega =\frac{v}{y}=\frac{3v}{2R}$
$I_c=\frac{2}{5}m{R}^2+m(OC{)}^2$
$\Rightarrow OC=\frac{R}{3}$, as $OC=(R-y)$
$I_c=\frac{23}{45}m{R}^2$

$KE=\frac{1}{2}\left(\frac{23}{45}m{R}^2\right)\left(\right(\frac{3}{2}\frac{v}{R}{)}^2$
$K{E}_T=K{E}_1+K{E}_2+K{E}_{ball}$
$\Rightarrow K{E}_T=\frac{123}{40}m{v}^2$

For detailed solution watch next video.