A U-tube filled with liquid is being rotated about an axis with angular speed ω as shown in the figure. The difference in heights of liquid in two arms is

Let us take a small element of length dr at a distance r from axis of rotation.Mass of the element dm = (Adr.ρ) where A = cross-sectional area of the tube. p = pressure at a radial distance r.p + dp = pressure at a radial distance (r + dr)Net radially inward thrust on the element = (p + dp)A - p.A (p + dp)A - p.A = dm. ω2.r(p + dp) - p = ρω2rdr