From 71b9e1aa3050dd492761973ad4be73c6d65e7eb1 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 10 Apr 2021 19:58:09 +0200 Subject: Expand knowledge base --- .../know/concept/archimedes-principle/index.pdc | 94 ++++++++++++++++++++++ 1 file changed, 94 insertions(+) create mode 100644 content/know/concept/archimedes-principle/index.pdc (limited to 'content/know/concept/archimedes-principle') diff --git a/content/know/concept/archimedes-principle/index.pdc b/content/know/concept/archimedes-principle/index.pdc new file mode 100644 index 0000000..6335a77 --- /dev/null +++ b/content/know/concept/archimedes-principle/index.pdc @@ -0,0 +1,94 @@ +--- +title: "Archimedes' principle" +firstLetter: "A" +publishDate: 2021-04-10 +categories: +- Fluid statics +- Fluid mechanics +- Physics + +date: 2021-04-10T15:43:45+02:00 +draft: false +markup: pandoc +--- + +# 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 $V$: + +$$\begin{aligned} + \va{F}_p + = - \oint_S p \dd{\va{S}} +\end{aligned}$$ + +We rewrite this using Gauss' theorem, +and replace $\nabla p$ by demanding +[hydrostatic equilibrium](/know/concept/hydrostatic-pressure/): + +$$\begin{aligned} + \va{F}_p + = - \int_V \nabla p \dd{V} + = - \int_V \va{g} \rho_\mathrm{f} \dd{V} +\end{aligned}$$ + +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 $\va{g}$ has a constant direction +and magnitude $\mathrm{g}$ everywhere. +If we also assume that $\rho_\mathrm{b}$ and $\rho_\mathrm{f}$ are constant, +and only integrate over the "submerged" part, 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. -- cgit v1.2.3