A cylindrical vessel filled with liquid is rotated about its vertical axis of symmetry with a constant angular velocity omega. The shape of the free…

The height $y$ of the liquid surface as a function of the radial distance $r$ from the axis of rotation is given by the balance of centrifugal force and gravity: $\frac{dy}{dr} = \frac{\omega^2 r}{g}$. Integrating this yields $y = \frac{\omega^2 r^2}{2g}$, which is the equation of a parabola. Therefore, the free surface forms a paraboloid of revolution.