Categories: Fluid mechanics, Fluid statics, Physics.

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.

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

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