Archimedes' Principle

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Physical Law

Let $V$ be a compact body with a piecewise smooth boundary, immersed in an incompressible fluid $F$.

Then the net buoyancy $\mathbf F$ effected upon $V$ by $F$ is equal to the weight of $F$ displaced.


The line of action of $\mathbf F$ passes through the center of gravity of the displaced volume of $F$.


This is often quoted (and probably better considered) as the informal statement:

A body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.


Proof

Let $V$ be the immersed body.

Let $S = \partial V$ be its boundary.

Recall Gauss's Theorem for a smooth vector field $\mathbf F$ defined over $V$:

$(1): \quad \ds \oint_S \mathbf F \cdot \rd \mathbf S = \int_V \nabla \cdot \mathbf F \rd \mathbf V$

provided that $\partial V$ is piecewise smooth and compact.


The pressure on $S$ depends only on the depth of $V$ within $F$.

Accounting for atmospheric pressure $p_0$, $\mathbf F$ is given by:

$p = -\rho g z + p_0$

where:

$\rho$ is the density of the fluid
$g = 9.81 \ldots$ is the gravitational acceleration
$z$ is the vertical displacement.


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Letting $\mathbf F = -p \cdot \mathbf k$ (with $\mathbf k$ a unit vector in the $z$ direction) we see that the left hand side of $(1)$ becomes the buoyancy force acting on the object, for it is the sum over the surface of the $z$ component of the pressure.

Clearly $\nabla \cdot \mathbf F = \rho g$, so we have:

$\ds \int_V \nabla \cdot \mathbf F \rd V = \rho g \int_V \rd V = \rho g V$

where we have let $V$ denote the scalar volume of $V$.

Note that we have assumed incompressibility and thus constant density of the fluid.

This is precisely the weight of the fluid in the volume $V$ Ref?.

The proof is complete.


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$\blacksquare$


Also known as

Archimedes' Principle is also known as the Basic Law of Hydrostatics.


Also see

Not to be confused with the Archimedean Principle.


Source of Name

This entry was named for Archimedes of Syracuse.


Historical Note

Archimedes' Principle was discovered by Archimedes of Syracuse.

Hence he created the science of hydrostatics.


Sources

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