Many objects float when placed on a liquid, but some float higher than others, and some do not float at all, sinking instead. Archimedes’ principle balances the forces, and predicts how much of a body is submerged, and how much is non-submerged.
In truth, there is no real distinction between the submerged and non-submerged parts, since the latter is surrounded by another fluid (air), which has a pressure and thus affects it. The right thing to do is treat the entire body as being submerged in a fluid with varying properties.
Let us consider a volume completely submerged in such a fluid. This volume will experience a downward force due to gravity, given by:
Where is the gravitational field, and is the density of the body. Meanwhile, the pressure of the surrounding fluid exerts a force on the entire surface of :
Where we have used the divergence theorem. Assuming hydrostatic equilibrium, we replace , leading to the definition of the buoyant force:
For the body to be at rest, we require . Concretely, the equilibrium condition is:
It is commonly assumed that is constant everywhere, with magnitude . If we also assume that is constant on the “submerged” side, and zero on the “non-submerged” side, we find:
In other words, the mass of the entire body is equal to the mass of the fluid it displaces. This is the best-known version of Archimedes’ principle.
Note that if , then the displaced mass even if the entire body is submerged, and the object will therefore continue to sink.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.