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The density is by definition the mass in kilograms divided by the volume in liters. 1 liter of water weighs 1 kg, so the density of water is 1. The density of ethanol is .8 and the one of honey is 1.4.

Why do some items float where­as some others sink? How does a hydrometer work? This all comes from Archimedes' prin­ciple: "a body immersed in a fluid is subject to an upward force proportional to the mass of the fluid it displaces".

What does this mean? The fluid creates a force that opposes gravity. This force is propor­tion­al to the "mass of the fluid [the body] displaces" and it is upward. Gravity is propor­tional to the mass of the body and is downward.

The "mass of the fluid [the body] displaces" is the mass of the fluid if the fluid had the same volume as the immerged body. For example, an iron mass whose volume is 1 liter is immersed in water. As the density is the mass divided by the volume, the mass is the density multiplied by the volume. So the mass of the body is: 1 liter * 8 kg/liter = 8 kg. The "mass of the fluid [the body] displaces" is the mass of 1 liter of water, that is 1 kg. 8 kg being higher than 1 kg, gravity is stronger than Archimedes and the iron drowns.

More generally, if the density of the body is higher than the one of water (metal, stone) the force is downward, the body sinks. In the opposite case (wood, witch), the force is upward and the body goes up.

What if three quarters of the body are immerged? Archimedes' principle applies only to three quarters of the volume of the body: the volume of the "the fluid [the body] displaces" is no longer the volume of the body, it is only three quarters of it (the immerged part). If some wood whose volume is 1 liter and whose density is .9 is partly immersed in water (three quarters), the volume of the "the fluid it displaces" is equal to the volume of the immerged part of the wood: .75 liter. The density of water is 1 so .75 liter of water weighs .75 kg. Archimedes' prin­ciple applies to .75 kg. Gravity applies to the whole mass of the body, namely .9 kg. So gravity is stronger than Archimedes and the wood goes down. But its density (.9) is lower than the one of water (1), so it will not be completely immerged, it will remain partly immerged.

Is it possible to determine by how much it will be immerged?

Let us call x the proportion of the body that is immerged (for example, if three quarters of the body are immerged, x = .75). x can be found saying that at equilibrium, gravity and Archimedes' principle are equal. Gravity applies to the mass of the body, that is the volume of the body times its density: Vbody * dbody. Ar­chi­medes' principle applies to the "mass of the fluid [the body] displaces", that is the volume of the immerged part of the body times the density of water: Vbody * x * dwater. When the body is at rest, these two masses are equal (that is the two forces are equal):

Vbody * dbody = Vbody * x * dwater

so dwater = dbody / x.

This is precisely how the density (called "specific gravity" by wine makers and beer brewers) is measured using a hydrometer. The density of the hydrometer is known and x is measured to get the S. G. of the must. Thanks Archimedes.


May 29th 2002