Archimedes’ Principle

Eureka

The story — possibly apocryphal, first recorded by Vitruvius two centuries after the fact — is that King Hiero II of Syracuse asked Archimedes to determine whether a golden crown had been adulterated with silver, without melting it down. Archimedes supposedly noticed, while stepping into a bath, that the water level rose by an amount equal to the volume of his submerged body. He realized that by measuring the volume of water displaced by the crown and comparing it to an equal weight of pure gold, he could detect the fraud. He leapt from the bath shouting “Eureka!” (“I have found it!”). Whether or not the bath story is true, the principle Archimedes formalized in On Floating Bodies (c. 250 BCE) is exact and general: a body immersed in fluid is buoyed up by a force equal to the weight of the fluid displaced.

The principle

The buoyant force arises from the pressure gradient in the fluid. Pressure increases with depth (P = ρgh), so the bottom of a submerged object experiences greater upward pressure than the top experiences downward pressure. The net upward force is F_b = ρ_fluid · g · V_displaced, where V_displaced is the volume of fluid pushed aside by the object. At equilibrium, this buoyant force exactly balances the component of the object’s weight that would otherwise make it sink or rise. For a floating object, the submerged fraction equals the ratio of object density to fluid density: f = ρ_obj / ρ_fluid. If ρ_obj < ρ_fluid, the object floats with part above the surface. If ρ_obj > ρ_fluid, no amount of submersion generates enough buoyant force, and the object sinks.

Why ships float

Steel has a density about eight times that of water. A solid block of steel sinks. Yet aircraft carriers made of steel float easily. The key is that the relevant density is the average density of the entire hull, including all the air inside. A ship’s hull encloses a vast volume of air, making the overall density of the ship-plus-air system much less than the density of water. As long as the average density stays below 1.0 g/cm³, the ship floats. Puncture the hull and let water flood the interior — replacing air with water — and the average density rises above 1.0. The ship sinks because the buoyant force can no longer match the weight.

Hot air balloons

Archimedes’ principle applies to any fluid, including air. A hot air balloon rises because heating the air inside the envelope reduces its density below that of the surrounding cooler air. The buoyant force on the balloon equals the weight of the displaced ambient air, and since the balloon (heated air plus envelope plus basket) weighs less than an equal volume of ambient air, there is a net upward force. The same principle governs helium balloons: helium has a density of about 0.164 kg/m³ at sea level, versus 1.225 kg/m³ for air, providing a substantial buoyant lift.

Icebergs and the 90% rule

Ice has a density of about 0.917 g/cm³, and seawater is about 1.025 g/cm³. The submerged fraction is 0.917/1.025 ≈ 0.895, meaning roughly 89.5% of an iceberg sits below the waterline. The familiar saying that “90% of an iceberg is underwater” is very nearly correct. This is also why ice cubes float with most of their volume submerged, and why the Titanic’s lookouts could not see the vast bulk of the iceberg that lay below the surface.

The simulation

This simulation models a rectangular block in a tank of fluid. The object experiences two forces: its weight (downward), equal to ρ_obj · g · V_total, and the buoyant force (upward), equal to ρ_fluid · g · V_submerged. A damped spring-like differential equation governs the vertical position, so the object bobs realistically toward equilibrium with decaying oscillations. You can drag the object up or down and release it to watch it settle. The “Drop” button releases the object from above the surface. Try setting the fluid density to 13.6 to simulate mercury, where even iron floats.