Question

Liquid surface in a rotating fluid about its axis is of the shape of

• A. Ellipsoid
• B. Sphere
• C. Parabola
• D. Paraboloid

Answer 1

The correct option is D Paraboloid

For a rotating fluid about an axis, pressure difference for a point at a distance $x$ from the axis is given by
$\Delta p=\frac{1}{2}\rho {\omega }^2{x}^2$
[ Pressure will increase by amount $\Delta p$ on moving away from the axis of rotation]

And height of liquid $y$ above the centre level also increases with distance $x$ from axis of rotation.
$\Delta P=\rho gy=\frac{1}{2}\rho {\omega }^2{x}^2$
$\Rightarrow y=\frac{{\omega }^2}{2g}{x}^2$
Above equation is similar to equation of parabola $y=a{x}^2$.


When this parabola is traced in three dimensions by rotating it by ${360}^{\circ }$ about $Y-$ axis (axis of rotation of fluid), it will form a paraboloid.
$\Rightarrow$Shape of liquid surface is a paraboloid.