Submerged volume of floating body via density ratio

If a body is fully or partially submerged in a fluid, an Archimedes force starts acting on it from the fluid pushing it in the opposite direction of the gravity force.

The ratio of the body’s volume submerged in a fluid to its total volume depends on the ratio of the densities of the body and the fluid.

Conditions:

1. \(\rho \le \rho_\text{fl}\), so the body must be floating. See below for the description of the symbols.

2. The body must be in static equilibrium.

Links:

1. Physics LibreTexts, formula 10.3.17.

submerged_volume

volume submerged in the fluid, which is equal to the volume of the displaced fluid.

Symbol:

V_fl

Latex:

\(V_\text{fl}\)

Dimension:

volume

body_volume

Total volume of the body.

Symbol:

V

Latex:

\(V\)

Dimension:

volume

body_density

density of the body.

Symbol:

rho

Latex:

\(\rho\)

Dimension:

mass/volume

fluid_density

density of the fluid.

Symbol:

rho_fl

Latex:

\(\rho_\text{fl}\)

Dimension:

mass/volume

law

V_fl / V = rho / rho_fl

Latex:

\[\frac{V_\text{fl}}{V} = \frac{\rho}{\rho_\text{fl}}\]