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

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

Where g⃗\va{g} is the gravitational field, and ρb\rho_\mathrm{b} is the density of the body. Meanwhile, the pressure pp of the surrounding fluid exerts a force on the entire surface SS of VV:

F⃗p=−∮SpdS⃗=−∫V∇pdV\begin{aligned} \va{F}_p = - \oint_S p \dd{\va{S}} = - \int_V \nabla p \dd{V} \end{aligned}

Where we have used the divergence theorem. Assuming hydrostatic equilibrium, we replace ∇p\nabla p, leading to the definition of the buoyant force:

F⃗p=−∫Vg⃗ρfdV\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 F⃗g+F⃗p=0\va{F}_g + \va{F}_p = 0. Concretely, the equilibrium condition is:

∫Vg⃗(ρb−ρf)dV=0\begin{aligned} \boxed{ \int_V \va{g} (\rho_\mathrm{b} - \rho_\mathrm{f}) \dd{V} = 0 } \end{aligned}

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

0=g(mb−mf)\begin{aligned} 0 = \mathrm{g} (m_\mathrm{b} - m_\mathrm{f}) \end{aligned}

In other words, the mass mbm_\mathrm{b} of the entire body is equal to the mass mfm_\mathrm{f} of the fluid it displaces. This is the best-known version of Archimedes’ principle.

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

References

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