From 8f883023e6354648727479aec029f418b30ef2dc Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 15 Apr 2021 20:53:17 +0200
Subject: Expand knowledge base

---
 .../know/concept/archimedes-principle/index.pdc    | 22 ++++++++++++----------
 1 file changed, 12 insertions(+), 10 deletions(-)

(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
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
-- 
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