Fluid mechanics

Archimedes’ principle

An object immersed in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid.

An object feels lighter underwater because the surrounding fluid exerts a net upward force on it. The force equals the weight of the fluid it displaces.

F_B=\rho_{\!f}gV_{\mathrm{déplacé}}=m_{\!f}g

Definition: Buoyant force

$F_B$ acts upward and equals the weight of displaced fluid; $\rho_f$ is fluid density and $V_{\mathrm{disp}}$ is the object’s submerged volume. It arises because fluid pressure is greater beneath the object.

Vary fluid density and submerged volume to compare displaced-fluid weight with buoyancy.

Floating, sinking, and suspension

Compare weight $P=mg$ with buoyancy. A floating object at rest has buoyancy equal to its weight and is only partly submerged. An object sinks if its weight exceeds the maximum buoyancy; an object whose average density is below that of the fluid can float.

Example: An object in water

An object displaces $2.0\times10^{-3}\,\mathrm{m^3}$ of water. Find the buoyant force using $\rho=1000\,\mathrm{kg/m^3}$ and $g=9.8\,\mathrm{m/s^2}$ .

Solution

$F_B=\rho gV=1000\times9.8\times0.002=19.6\,\mathrm N$ .

Quick check

Buoyancy depends on the displaced volume, not absolute depth: a fully submerged object of fixed volume in nearly incompressible water experiences nearly the same buoyant force at different depths. For a floating object, balance gives $ho_{ m object}V_{ m total}= ho_{ m fluid}V_{ m submerged}$ . Thus a less dense block floats with a smaller fraction submerged, while an object whose average density exceeds the fluid density cannot float freely.

Example: Submerged fraction of a floating object

A wooden block of density $600,mathrm{kg/m^3}$ floats in water of density $1000,mathrm{kg/m^3}$ . Force balance gives $V_{ m submerged}/V_{ m total}= ho_{ m wood}/ ho_{ m water}=0.60$ , so about $60%$ of its volume lies below the surface. Adding a load makes the block sink farther, displacing more water and increasing buoyancy until forces balance again.

An object is fully immersed in a liquid. Its buoyant force equals:

A block floats at rest on water. Then:

References

David Halliday, Robert Resnick, Jearl Walker (2018). Fundamentals of Physics