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Buoyancy Equation: Prove Archimedes’ Principle from Pressure

Derive the buoyancy equation from the pressure difference across a submerged cylinder and identify the weight of displaced fluid.

Question

Referring to Figure 14.20, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is F2−F1F_2-F_1 and that the ends of the cylinder have equal areas AA. Note that the volume of the cylinder (and that of the fluid it displaces) equals (h2−h1)A(h_2-h_1)A.

OpenStax Figure 14.20: a submerged cylinder with equal end areas A; top at depth h1 has downward force F1, bottom at depth h2 has upward force F2.

Figure 14.20. Source: OpenStax, University Physics Volume 1, Section 14.4. Depths are measured downward from the same fluid surface.

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Evidence boundary

This proof uses OpenStax University Physics Volume 1, Chapter 14, Problem 73 and its Figure 14.20: equal cylinder end areas, depths h1 and h2, and buoyancy F2 − F1. It assumes a static constant-density fluid and uniform gravity. It proves the pressure force, without inventing measurements or assuming the cylinder is floating in equilibrium.