# Rotation - Rotating Vessel

When at rest, the surface of mass of liquid is horizontal at *PQ* as shown in the figure. When this mass of liquid is rotated about a vertical axis at constant angular velocity *ω* radian per second, it will assume the surface *ABC* which is parabolic. Every particle is subjected to centripetal force or centrifugal force *CF = mω*^{2}*x* which produces centripetal acceleration towards the center of rotation. Other forces that acts are gravity force *W = mg* and normal force *N*.

$\tan \theta = \dfrac{CF}{W}$

$\tan \theta = \dfrac{m\omega^2x}{mg}$

Where tan θ is the slope at the surface of paraboloid at any distance *x* from the axis of rotation.

From Calculus, *y’* = slope, thus

$\dfrac{dy}{dx} = \tan \theta$

$\dfrac{dy}{dx} = \dfrac{\omega^2x}{g}$

$dy = \dfrac{\omega^2}{g}x ~ dx$

$\displaystyle \int dy = \dfrac{\omega^2}{g} \int x ~ dx$

For cylindrical vessel of radius *r* revolved about its vertical axis, the height *h* of paraboloid is

**Other Formulas**

By squared-property of parabola, the relationship of *y*, *x*, *h* and *r* is defined by

Volume of paraboloid of revolution

Important conversion factor