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author | Prefetch | 2022-10-14 23:25:28 +0200 |
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committer | Prefetch | 2022-10-14 23:25:28 +0200 |
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diff --git a/source/know/concept/archimedes-principle/index.md b/source/know/concept/archimedes-principle/index.md new file mode 100644 index 0000000..4bab87b --- /dev/null +++ b/source/know/concept/archimedes-principle/index.md @@ -0,0 +1,89 @@ +--- +title: "Archimedes' principle" +date: 2021-04-10 +categories: +- Fluid statics +- Fluid mechanics +- Physics +layout: "concept" +--- + +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 entire surface $S$ of $V$: + +$$\begin{aligned} + \va{F}_p + = - \oint_S p \dd{\va{S}} + = - \int_V \nabla p \dd{V} +\end{aligned}$$ + +Where we have used the divergence theorem. +Assuming [hydrostatic equilibrium](/know/concept/hydrostatic-pressure/), +we replace $\nabla p$, +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: + +$$\begin{aligned} + 0 + = \mathrm{g} (m_\mathrm{b} - m_\mathrm{f}) +\end{aligned}$$ + +In other words, the mass $m_\mathrm{b}$ of the entire 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 the displaced mass $m_\mathrm{f} < m_\mathrm{b}$ +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. |