= Buoyancy
{title2=$F_b$}
{wiki}
= Buoyant force
{synonym}
<Buoyancy> is the resultant upward <force> exerted by a resting <fluid> on an immersed body in a downward <gravitational field>. The <fluid> pushes inward, perpendicular to each part of the body's surface. Because <hydrostatic pressure> increases with depth, the upward push from below exceeds the downward push from above. Opposite horizontal <forces> cancel; <buoyancy> is the remaining vertical <force>. A sinking body still receives this upward support. https://openstax.org/books/university-physics-volume-1/pages/14-4-archimedes-principle-and-buoyancy[OpenStax, University Physics, §14.4: origin of buoyancy]{ref}
More precisely, <pressure> $p$ on a small surface of <area> $dA$ with outward unit <normal vector> $\mathbf n$ exerts
$$
d\mathbf F=-p\mathbf n\,dA.
$$
The minus sign says the <force> points into the body. On a <cube>, the top <force> points down, the bottom <force> points up, and the side <forces> cancel in pairs because opposite sides have the same <pressure> at each depth. This is a surface <traction>; <gravity> on the body itself is a separate <body force>. https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/dc299d2ff05553b3515dccc23fc0f7d2_2005reading1.pdf[MIT, A. Techet, Hydrodynamics reading 1, p. 5: pressure traction]{ref}
For a computed example, stipulate resting <water> with constant <mass density> $\rho_f=1000\ \mathrm{kg\,m^{-3}}$ and uniform <gravitational acceleration> $g=9.81\ \mathrm{m\,s^{-2}}$. A fully immersed <cube> has side $L=0.10\ \mathrm m$, top depth $h_t=0.20\ \mathrm m$ and bottom depth $h_b=0.30\ \mathrm m$. Each horizontal face has <area> $A=L^2=0.010\ \mathrm{m^2}$, and its <volume> is $V=L^3=0.001\ \mathrm{m^3}$. Its top and bottom <gauge pressures> are
$$
p_{g,t}=\rho_fgh_t=1962\ \mathrm{Pa},\qquad
p_{g,b}=\rho_fgh_b=2943\ \mathrm{Pa}.
$$
The corresponding <forces> relative to the common surface <pressure> are $p_{g,t}A=19.62\ \mathrm N$ downward and $p_{g,b}A=29.43\ \mathrm N$ upward. Therefore the net <buoyancy> is
$$
F_b=(p_{g,b}-p_{g,t})A=9.81\ \mathrm N.
$$
Using absolute <pressures> adds the same $p_{\mathrm{surface}}A$ to both face <forces>, leaving their difference unchanged. Zero <gauge pressure> at the surface does not mean zero absolute <pressure>. Increasing both depths by the same amount also leaves <buoyancy> unchanged in this constant-<mass density> model.
\Image[/hydrostatic-pressure-and-buoyancy.png]
{title=How a hydrostatic pressure difference supports an immersed cube}
{description=Left: a fully immersed <cube> of side $0.10\ \mathrm m$ has its top at depth $0.20\ \mathrm m$ and bottom at $0.30\ \mathrm m$. The inward <pressure> <forces> on its sides cancel; its horizontal faces have <area> $0.010\ \mathrm{m^2}$. The displayed face <forces> use <gauge pressure>, whose common surface offset cancels in the net $9.81\ \mathrm N$ upward <buoyancy>. Right: <gauge pressure> increases linearly with depth, marking $1962\ \mathrm{Pa}$ at the top and $2943\ \mathrm{Pa}$ at the bottom.}
<Archimedes' principle> turns this calculation into a rule for bodies of any shape. It explains <floating equilibrium> and lets us infer an immersed body's <volume> from the difference between its <force> from <gravity> and the <force> needed to support it. In particular, <buoyancy> depends on the displaced <volume> and the surrounding <mass density>, whereas the body's own <mass density> decides how its <force> from <gravity> compares with that support.
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