From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/archimedes-principle/index.md | 26 +++++++++++------------ 1 file changed, 13 insertions(+), 13 deletions(-) (limited to 'source/know/concept/archimedes-principle') diff --git a/source/know/concept/archimedes-principle/index.md b/source/know/concept/archimedes-principle/index.md index dc02d12..364461f 100644 --- a/source/know/concept/archimedes-principle/index.md +++ b/source/know/concept/archimedes-principle/index.md @@ -23,7 +23,7 @@ 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. +Let us consider a volume $$V$$ completely submerged in such a fluid. This volume will experience a downward force due to gravity, given by: $$\begin{aligned} @@ -31,10 +31,10 @@ $$\begin{aligned} = \int_V \va{g} \rho_\mathrm{b} \dd{V} \end{aligned}$$ -Where $\va{g}$ is the gravitational field, -and $\rho_\mathrm{b}$ is the density of the body. -Meanwhile, the pressure $p$ of the surrounding fluid exerts a force -on the entire surface $S$ of $V$: +Where $$\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 @@ -44,7 +44,7 @@ $$\begin{aligned} Where we have used the divergence theorem. Assuming [hydrostatic equilibrium](/know/concept/hydrostatic-pressure/), -we replace $\nabla p$, +we replace $$\nabla p$$, leading to the definition of the **buoyant force**: $$\begin{aligned} @@ -54,7 +54,7 @@ $$\begin{aligned} } \end{aligned}$$ -For the body to be at rest, we require $\va{F}_g + \va{F}_p = 0$. +For the body to be at rest, we require $$\va{F}_g + \va{F}_p = 0$$. Concretely, the equilibrium condition is: $$\begin{aligned} @@ -64,8 +64,8 @@ $$\begin{aligned} } \end{aligned}$$ -It is commonly assumed that $\va{g}$ is constant everywhere, with magnitude $\mathrm{g}$. -If we also assume that $\rho_\mathrm{f}$ is constant on the "submerged" side, +It is commonly assumed that $$\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} @@ -73,12 +73,12 @@ $$\begin{aligned} = \mathrm{g} (m_\mathrm{b} - m_\mathrm{f}) \end{aligned}$$ -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. +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 $m_\mathrm{f} < m_\mathrm{b}$ +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. -- cgit v1.2.3