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Why Things Float: Buoyancy and Archimedes

The upward push every fluid gives, the 2,200-year-old principle that predicts it exactly, and the density rule that tells you in advance whether something floats or sinks.

The mysterious upward push

Lift a heavy rock while it is underwater and it feels startlingly light; let go of a beach ball at the bottom of a pool and it rockets to the surface. Fluids push up on whatever is submerged in them. We call that upward push the buoyant force, and it is not a new law of nature — it follows directly from the depth-pressure rule you already know.

Here is the mechanism. A submerged block has its bottom face deeper than its top face. Since pressure grows with depth, the fluid pushes up on the bottom harder than it pushes down on the top. The sideways pushes cancel, but that top–bottom mismatch leaves a net upward force. Buoyancy is nothing more than the sum of the pressure forces over the object's whole surface.

Archimedes' principle

When you work out that top–bottom pressure difference, something elegant drops out. The net upward force exactly equals the weight of the fluid that the object pushes out of the way. This is Archimedes' principle, named for the Greek thinker who — legend says — leapt from his bath shouting *Eureka!* The bathtub story may be embroidered, but the principle is rock solid.

F_B = \rho_{\text{fluid}}\, V_{\text{disp}}\, g

The buoyant force equals the density of the fluid times the displaced volume times g — that is, the weight of the fluid pushed aside.

A floating object sinks just deep enough that the weight of displaced fluid equals its own weight; a denser object cannot displace enough and sinks.

Read the formula carefully: the buoyant force depends on the density of the fluid and the displaced volume — never on the object's own density or on how deep it sits (as long as it stays fully submerged). A steel cube and a wooden cube of the same size feel the same buoyant force in water. What differs is their weight, and that is what decides the contest.

Float or sink — the density rule

An object settles wherever buoyant force and weight balance. Fully submerge it and compare: if the fluid it displaces weighs more than the object, the upward push wins and it rises until part of it pokes out; if the displaced fluid weighs less, it sinks. Because both weights share the same volume and the same g, the whole contest reduces to a comparison of densities — object versus fluid.

\dfrac{V_{\text{submerged}}}{V_{\text{object}}} = \dfrac{\rho_{\text{object}}}{\rho_{\text{fluid}}}

For a floating object, the fraction of its volume below the surface equals the ratio of its density to the fluid's density.

This one ratio explains a classic image. Ice has density 917\text{ kg/m}^3 and seawater about 1025\text{ kg/m}^3, so a floating iceberg sits with 917/1025 \approx 0.89 — nearly ninety percent — of its bulk hidden underwater. The visible peak really is just “the tip of the iceberg.” Play with the interactive below: change the object and fluid densities and watch the floating line move.

Float-or-sink lab: raise the object's density toward the fluid's and it rides lower; exceed the fluid's density and it sinks. The submerged fraction always equals the density ratio.

Apparent weight, and catching a forger

When an object hangs from a scale underwater, the scale reads its apparent weight: the true weight minus the buoyant force. That single fact is a precision densitometer. Weigh something in air, then weigh it submerged; the difference is the buoyant force, which hands you the displaced volume, which hands you the object's density — no need to melt or cut it. This is exactly the trick attributed to Archimedes for testing a king's crown.