The Archimedes paradox, named after Archimedes of Syracuse, or the hydrostatic paradox,[1] states that an object can float in a quantity of water that has less volume than the object itself, if its average density is less than that of water. A more general formulation of the paradox is that "that any quantity of water, or other fluid, how small soever, may be made to balance and support any quantity, or any weight, how great soever".[2]

The implication of this is that a large, massive object can float in a relatively small volume of liquid, provided that it is surrounded by it. One extreme application of the paradox is that a battleship can float in a few buckets of water, provided that the water surrounds the hull completely and that the ship would have floated had it been in open water.

Origin

Archimedes' principle (also referred to as the Law of Buoyancy), as stated in Book 1 of Archimedes' work On Floating Bodies, states that "the buoyant force is equal to the weight of the displaced fluid." Simon Stevin developed this idea into the hydrostatic paradox in his De Beghinselen des Waterwichts (Elements of Hydrostatics) (1586). His proof rested partly on the principle of solidification: if a large proportion of the water in a vessel were to be frozen without affecting its weight or volume, leaving just a narrow region of liquid between it and the container, then the forces everywhere would be the same as before solidification, and therefore the frozen water would remain floating in the same position as its liquid precursor. The frozen water could then be replaced by any object of the same Volume, shape, weight and centre of gravity, and the object would continue to float.[3]

In the case of a ship, the upward force exerted on it is equal to the weight of water of equal volume to the part of the ship that is submerged. If this upward force is greater than the weight of the ship, then the ship will float.

The Archimedes Paradox implies that if a mould of the hull of ship is made and a relatively small amount of water is placed in the mould, then the ship would float on the thin layer of water between itself and the mould, even though the total volume of water is much less than the volume of the ship.

Explanation

The paradox originates from the fact that the volume of the immersed part of the object is important, not the actual volume of water that is displaced by it. In other words, no fluid needs to be actually displaced for Archimedes' principle to take effect. The object merely needs to be surrounded by the fluid.

One method offered to visualize the solution to the paradox is to conduct a simple thought experiment. Instead of a ship suspended in the water, imagine a lightweight bucket filled with water. Since the density of the bucket of water is the same as the water in the dock, the bucket would remain suspended, or floating. Nothing changes hydrostatically by replacing the bucket with a ship of equal or lower density than water (which it would have to be or else it would sink in open water anyway), therefore the ship would also float.

Another way of looking at it is that the buoyant force is the integral of the ambient pressure over the surface. In this model the volume is not directly relevant. When the object is immersed at the interface between two fluids, there is a different pressure distribution with depth on the parts exposed to the different fluids.

References

1. ^ Koehl, George M., "Archimedes' Principle and the Hydrostatic Paradox—Simple Demonstrations", The American Journal of Physics, Volume 17 (9), American Association of Physics Teachers, Dec 1, 1949, p.579, accessed 2012-01-09
2. ^ Hutton, Charles, Mathematical and Philosophical Dictionary, 1795, accessed 2012-01-09
3. ^ Stephen Gaukroger, John Schuster, The hydrostatic paradox and the origins of Cartesian dynamics, Stud. Hist. Phil. Sci. 33 (2002) 535–572, accessed 2012-01-09; also published in John Schuster, Descartes-Agonistes: Physico-mathematics, Method & Corpuscular-Mechanism 1618-33, ISBN 9400747454