Question

A cylinder of water, is rotating about its own axis with uniform angular velocity $\omega$. The shape of free surface of water will be

• A. Parabola
• B. Elliptical
• C. Circular
• D. Spherical

Answer 1

The correct option is A Parabola

Let the density of water be $\rho$ and the cylinder is rotating with angular velocity $w.$

Also assume an elimentary cylinder of length $x$ and cross-section $A$

Mass of water contained in an elimentary cylinder , $M=\rho V=\rho \left(Ax\right)$

Centripetal force will act at the centre of mass i.e at $\frac{x}{2}$

$F_C=Mr{w}^2=\left(\rho Ax\right)\frac{x}{2}{w}^2$

$F_C=\frac{\rho A{x}^2{w}^2}{2}$

Now for x direction: $F_B-F_O=F_C$

$(P_B-P_O)A=\frac{\rho A{x}^2{w}^2}{2}$

$(P_o+\rho gy)-P_o=\frac{\rho x^2{w}^2}{2}$ where $P_o$ is the atmospheric pressure

$⟹y-\frac{w^2{x}^2}{2g}=0$ which is an equation of parabola.