Why things float

An immersed object is pushed up by the weight of the fluid it shoves aside — Archimedes figured it out in the bath.

Archimedes' principle

A submerged object feels an upward buoyant force equal to the weight of the fluid it displaces: Fb = rho_fluid * V_displaced * g. Fully submerged means V_displaced is the object's whole volume; partially submerged means only the underwater part counts.

F_b = \rho_{fluid} V_{disp}\, g

Float or sink

Compare the buoyant force at full submersion with the object's weight. If the object's density is less than the fluid's, it floats, riding just deep enough that the displaced fluid's weight matches its own — so the submerged fraction equals rho_object/rho_fluid. Denser than the fluid and it sinks; exactly equal and it hangs neutrally.

\frac{V_{sub}}{V_{total}} = \frac{\rho_{obj}}{\rho_{fluid}}

Worked example

A 1.5 kg block of volume 2.0e-3 m^3 (density 750 kg/m^3) floats in water. Fully wet it would feel Fb = (1000)(2.0e-3)(9.8) = 19.6 N, but it weighs only (1.5)(9.8) = 14.7 N, so it floats with 14.7/19.6 = 75 percent of its volume submerged — exactly the ratio 750/1000.

\frac{V_{sub}}{V} = \frac{750}{1000} = 0.75

Flashcard

F_b = \rho_f V g

Buoyant force equals displaced fluid weight

The fluid pushes up with the weight of whatever volume the object shoved aside. A 2-liter bottle fully underwater in water feels about 19.6 N of upward push, whatever is inside it.

Check your understanding

A solid of density 600 kg/m^3 floats in water (density 1000 kg/m^3). What fraction of its volume sits below the surface?

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