diff options
Diffstat (limited to 'content/know/concept/archimedes-principle/index.pdc')
| -rw-r--r-- | content/know/concept/archimedes-principle/index.pdc | 22 |
1 files changed, 12 insertions, 10 deletions
diff --git a/content/know/concept/archimedes-principle/index.pdc b/content/know/concept/archimedes-principle/index.pdc index 6335a77..0837cc9 100644 --- a/content/know/concept/archimedes-principle/index.pdc +++ b/content/know/concept/archimedes-principle/index.pdc @@ -44,16 +44,19 @@ on the surface $S$ of $V$: $$\begin{aligned} \va{F}_p = - \oint_S p \dd{\va{S}} + = - \int_V \nabla p \dd{V} \end{aligned}$$ -We rewrite this using Gauss' theorem, -and replace $\nabla p$ by demanding -[hydrostatic equilibrium](/know/concept/hydrostatic-pressure/): +The last step follows from Gauss' theorem. +We replace $\nabla p$ by assuming +[hydrostatic equilibrium](/know/concept/hydrostatic-pressure/), +leading to the definition of the **buoyant force**: $$\begin{aligned} - \va{F}_p - = - \int_V \nabla p \dd{V} - = - \int_V \va{g} \rho_\mathrm{f} \dd{V} + \boxed{ + \va{F}_p + = - \int_V \va{g} \rho_\mathrm{f} \dd{V} + } \end{aligned}$$ For the body to be at rest, we require $\va{F}_g + \va{F}_p = 0$. @@ -66,10 +69,9 @@ $$\begin{aligned} } \end{aligned}$$ -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: +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 |