Categories: Fluid mechanics, Fluid statics, Physics.

# Archimedes’ principle

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 $$V$$ completely submerged in such a fluid. This volume will experience a downward force due to gravity, given by:

\begin{aligned} \va{F}_g = \int_V \va{g} \rho_\mathrm{b} \dd{V} \end{aligned}

Where $$\va{g}$$ is the gravitational field, and $$\rho_\mathrm{b}$$ is the density of the body. Meanwhile, the pressure $$p$$ of the surrounding fluid exerts a force on the surface $$S$$ of $$V$$:

\begin{aligned} \va{F}_p = - \oint_S p \dd{\va{S}} = - \int_V \nabla p \dd{V} \end{aligned}

The last step follows from Gauss’ theorem. We replace $$\nabla p$$ by assuming hydrostatic equilibrium, leading to the definition of the buoyant force:

\begin{aligned} \boxed{ \va{F}_p = - \int_V \va{g} \rho_\mathrm{f} \dd{V} } \end{aligned}

For the body to be at rest, we require $$\va{F}_g + \va{F}_p = 0$$. Concretely, the equilibrium condition is:

\begin{aligned} \boxed{ \int_V \va{g} (\rho_\mathrm{b} - \rho_\mathrm{f}) \dd{V} = 0 } \end{aligned}

It is commonly assumed that $$\va{g}$$ is constant everywhere, with magnitude $$\mathrm{g}$$. If we also assume that $$\rho_\mathrm{f}$$ is constant on the “submerged” side, and zero on the “non-submerged” side, we find:

\begin{aligned} 0 = \mathrm{g} (\rho_\mathrm{b} - \rho_\mathrm{f}) V = \mathrm{g} (m_\mathrm{b} - m_\mathrm{f}) \end{aligned}

In other words, the mass $$m_\mathrm{b}$$ of the submerged portion $$V$$ of the body, is equal to the mass $$m_\mathrm{f}$$ of the fluid it displaces. This is the best-known version of Archimedes’ principle.

Note that if $$\rho_\mathrm{b} > \rho_\mathrm{f}$$, then, even if the entire body is submerged, the displaced mass $$m_\mathrm{f} < m_\mathrm{b}$$, and the object will continue to sink.

## References

1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.