

Archimedes' Principle was stated by the Greek mathematician Archimedes of Syracuse (287-212 B.C.). Tradition tells us that King Heiro of Syracuse was a close friend of Archimedes. Oftentimes Heiro advice the mathematician to solve technical problems. One day, Heiro summoned a goldsmith to make a golden crown, and he sent an exact amount of gold to the craftsman. However, upon returning the crown, Heiro became suspicious about the amount of gold that was used: perhaps the goldsmith kept apart some of the gold, and mixed the remainder with a cheaper metal to make the crown. King Heiro summoned Archimedes to determine whether the goldsmith had cheated him. The problem proved seemed to be impossible, because nothing was known about chemical analysis.

The solution came one night, when Archimedes took a bath. The tub was filled to the brim, and as he submerged himself into the water, he realized that the amount of water spilled was equal to his own volume. Archimedes could now measure the density of the metal by weighing the crown, and submerging it in water to obtain its volume. "Eureka"! Archimedes rushed out of the bathtub and rushed (supposedly naked) into the streets to announce that he had solved the problem. Indeed, the crown did not contain enough gold, and the goldsmith was beheaded.

Archimedes' Principle is the reason why boats remain buoyant (or sometimes sink), balloons rise and ice floats. A body will float in a given fluid depending on their relative densities: both the apparent density (mass per unit of volume) of the body, and that of the fluid determine the buoyant force. If the body is less dense than the fluid, it will rise (or float). If the body is denser than the fluid, it will drop (or sink).

The ratio of the two densities also determines how much of a floating body will be submerged. For instance, sea water has a density of 1024 kg/m3. Ice (-4 C) has a density of 917 kg/m3. Thus, an iceberg will be submerged for 917/1024 = 90%: only 10% is visible above the surface.

When calculating the buoyant force on an object, the shape and position of the object are also important. For instance, consider a steel ship. Steel has a larger density than water, so a solid block of steel would sink. However, a boat also has a large volume of air. The apparent density of the ship is equal to the mass of the steel and contained air divided by the entire volume of the ship. The apparent density is less than the density of water, and thus the ship will float.





Vitruvius: De Architectura, Book IX, ch. 9-12 of the preface
(translation by Joseph Gwilt, London, 1826)

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