A liquid of density ρ is rotating in a cylindrical vessel with constant angular velocity ω. The free surface of the liquid takes the shape of:

When a liquid rotates inside a cylindrical vessel with constant angular velocity ω, every fluid element experiences a centrifugal force directed radially outward due to rotation. At the same time, gravity acts vertically downward. To find the shape of the free surface, we use the condition that the pressure on the free surface is constant (equal to atmospheric pressure), and hence, the total potential energy per unit mass remains constant along the surface. Let us consider a point on the free surface at a radial distance r from the axis of rotation and vertical height z above some reference level.