Buoyant Force (Archimedes' Principle)

The reason buoyancy exists is that pressure in a fluid increases with depth. The bottom of a submerged body sits deeper than its top, so it is pushed up harder than the top is pushed down, and the net of that imbalance — added over the whole surface — comes out exactly equal to the weight of the fluid the body displaced. That is Archimedes' principle, and the derivation makes clear why the material of the body is irrelevant. The fluid only ever touches the outside; it has no way of knowing what is inside the shape.

Displace 1 m³ of fresh water and you get $1000 \times 1 \times 9.80665 = 9807$ N of lift — about a tonne-force — whether the displacer is steel, styrofoam or a sealed void. This is why a ship's size is quoted as its displacement: a 10 000 tonne vessel is one that settles until it has pushed aside 10 000 tonnes of water, roughly 10 000 m³ of it. A submarine hovers by adjusting ballast until the two figures match, and a hot-air balloon does the identical arithmetic in air, where a 2500 m³ envelope displaces about 3 tonnes of atmosphere.

The story of Archimedes leaping from the bath comes from Vitruvius, writing two centuries after the fact, and the crown test as popularly told — measuring the overflow — would have been far too crude to catch the adulteration. The method that actually works, and that Archimedes' own writing supports, is to weigh the crown in air and again suspended in water: the difference is the buoyant force, which gives the volume, which gives the density. That comparison is still the standard way to measure the density of an irregular solid, and it is the principle inside every hydrometer.

Two substitutions account for most wrong answers, and they are easy to state. First, $\rho$ is the density of the fluid, never of the object — the object's density decides whether it floats, but it plays no part in this equation. Second, $V$ is the displaced volume, which equals the object's volume only when the object is fully submerged. A floating body displaces just the part below the waterline, and it settles until the fluid it has pushed aside weighs exactly what the body weighs. Fully submerged: displaced volume equals object volume. Floating: displaced weight equals object weight. Reaching for the wrong one of those two is the classic slip.

Then the details that matter in practice. Fluid density is not one number: sea water at 1025 kg/m³ gives 2.5% more lift than fresh at 1000, which sounds trivial until a loaded ship moves from ocean into a river and sinks noticeably lower — the reason Plimsoll load lines carry separate marks for fresh and salt, summer and winter. What a scale reads for a submerged object is the apparent weight, true weight minus $F_b$, not the buoyant force itself. And air is a fluid too: everything weighed on a bench is buoyed by about 1.2 kg per cubic metre of its volume, which is negligible for a steel block and a genuine correction when calibrating precision masses.