The Buoyancy Approach to U-tube Problems

In this note we unify two physical situations treatable with hydrostatics: an object floating on a denser fluid and an open U-shaped tube with two immiscible fluids. We begin by reviewing the problem of a partially floating uniform, rectangular prism of horizontal area A immersed in a denser fluid, with respective densities ρ1 < ρh for the prism and fluid (the subscripts stand for light and heavy): see Fig. 1. We define three horizontal levels within the solid, y0, y1, and y2, corresponding to the bottom, flotation line, and top of the prism. The buoyant force is ρh (y1 − y0)gA upwards, and the weight of the prism is ρ1 (y2 − y0)gA downwards. By Newton's second law, these two forces balance at equilibrium. After dividing by the common horizontal area, one obtains y1−y0y2−y0=ρ1ρh.(1) A detailed derivation can be found for example in Ref. 1, Section 13-7.