# 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 $S = \partial V$ 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

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Archimedes' principle** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**buoyancy** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Archimedes' principle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**buoyancy**