A cylindrical container of an incompressible liquid rotates with $\omega$  as shown. The density of liquid is $\rho .$ The liquid acquires parabolic shape. Find the height of the liquid. Radius of container is $r.$ 

 

The force experienced by element $dx$ at distance $x$ from the centre $dF=Adx\rho {\omega }^{2}x$ and $dP=\frac{dF}{A}=\rho {\omega }^{2\text{'}}xdx$

$P={\displaystyle {\int }_0^r\rho {\omega }^{2}xdx=h\rho g}$ or $h=\frac{r^2{\omega }^2}{2g}$

Note: This technique is employed to make parabolic telescopic mirrors. Liquid glass is rotated and allowed to solidify while rotating.