Question

A cylinder of radius $R$ full of liquid of density $\rho$ is rotated about its axis at $\omega rad/s$. The increase in pressure at the centre of the cylinder will be

• A. $\frac{\rho {\omega }^2{R}^2}{2}$
• B. $\frac{\rho {\omega }^{2}R}{2}$
• C. $\frac{{\rho }^2\omega R}{2}$
• D. $\frac{{\rho }^2{\omega }^2{R}^2}{2}$

Answer 1

The correct option is B $\frac{\rho {\omega }^{2}R}{2}$

Moment of inertia of a hollow cylinder $R$ about its axis of rotation

$I=M{R}^2$ .........$\left(1\right)$

Moment of inertia of cylinder fully filled with water about its axis of rotation.

$I^{\prime }=M{R}^2+\frac{1}{2}\times \pi R^{2}I\rho \times R^2$

$\therefore$increased torque on the cylinder $=(I^{\prime }-I)\alpha =\frac{1}{2}\times \pi R^{2}I\rho \times R^2\alpha$

Thus, torque$=\frac{1}{2}\times \pi R^{2}I\rho \times R^2\times \frac{d\Omega }{dt}$

or $F=\frac{\tau }{R}=\frac{\pi R^{4}l\rho {\omega }^2}{2}$

$P=\frac{F}{A}=\frac{\pi R^{3}l\rho {\omega }^2}{2\pi R^2}=\frac{Rl\rho {\omega }^2}{2}$

Pressure per unit length at the cetre$=\frac{R\rho {\omega }^2}{2}$