Archimedes' principle states that an object wholly or partially immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object. In mathematical form, the buoyant force is F buoyant = rho fluid g V displaced for a uniform fluid under ordinary gravitational conditions. The principle is attributed to the ancient Greek mathematician and scientist Archimedes and is one of the foundations of hydrostatics.
It applies to liquids and gases and explains floating, sinking, apparent weight reduction in fluids, and the operation of instruments such as hydrometers. The principle does not state that the buoyant force equals the weight of the object. It equals the weight of displaced fluid. For a floating object, those quantities become equal because the object is in vertical equilibrium, but for a sinking object the displaced fluid weighs less than the object, so buoyancy is smaller than weight.
The physical basis of Archimedes' principle is the hydrostatic pressure gradient within a fluid. Under gravity, pressure increases with depth according to dp divided by dz = minus rho g when the vertical coordinate z increases upward and density is constant. A submerged object's lower surfaces therefore experience greater pressure than its upper surfaces. Integrating these pressure forces over the entire surface produces a net upward resultant.
The magnitude of that resultant is equal to rho g times the object's displaced volume, which is precisely the weight of the displaced fluid. This derivation shows that Archimedes' principle is not an independent mysterious rule but a consequence of the pressure distribution in a fluid under a body force such as gravity. If the fluid density varies significantly with position, the pressure field must be integrated using the local density rather than applying the simplest constant density formula without modification.
For a fully submerged object, the displaced fluid volume equals the object's external volume if the object is nonporous and the fluid does not enter it. Thus the buoyant force is F buoyant = rho fluid g V object. If the object is only partially submerged, as with a floating boat, only the submerged volume contributes to displacement. The equilibrium condition for floating is rho fluid g V displaced = mg.
Rearranging gives V displaced = m divided by rho fluid. If the object's average density is rho object = m divided by V object, the fraction of its volume submerged is V displaced divided by V object = rho object divided by rho fluid. This relation explains why a less dense object floats with only part of its volume submerged. It also shows why changing the object's mass, shape, enclosed volume, or the fluid density can alter its equilibrium immersion depth.
Archimedes' principle explains the apparent loss of weight experienced by an object when it is immersed in a fluid. If an object has true weight W = mg and buoyant force F buoyant, a supporting scale or spring balance measures a reduced apparent weight W apparent = mg minus F buoyant when the object is stationary and fully supported by the measuring device. This principle provides a practical method for measuring density.
If the object's mass is known and its apparent weight reduction in a fluid is measured, the buoyant force can be determined, allowing the volume of displaced fluid and therefore the object's density to be calculated under suitable conditions. Historically, such reasoning is associated with Archimedes' famous account involving the testing of a crown, although the popular story about Archimedes discovering the principle through a sudden bath time revelation is a later narrative and the exact historical details are uncertain.
The principle is closely connected to the distinction between mass, weight, density, and volume. The weight of displaced fluid is rho fluid V displaced g, so the object's own mass or density does not appear directly in the buoyant force equation. Object properties determine how much volume is displaced at a particular state and whether the object floats, sinks, or remains neutrally buoyant. Fluid properties determine the buoyant force for a specified displaced volume.
Gravity appears in both the object's weight and the displaced fluid's weight, which is why the simple submerged fraction for a floating body does not depend explicitly on g. This does not mean gravity is irrelevant. The pressure gradient responsible for hydrostatic buoyancy exists because of the gravitational field, and in an environment with negligible effective gravity the ordinary hydrostatic buoyancy mechanism would be greatly reduced.
Archimedes' principle applies to gases as well as liquids. A balloon or airship displaces a volume of surrounding air, and the buoyant force equals the weight of that displaced air. If the total weight of the balloon system is less than the weight of the displaced air, the net force is upward. Hot air balloons reduce the density of the air inside the envelope, lowering the total mass of the system without greatly changing the external displaced volume.
Helium and hydrogen balloons achieve buoyancy because the gases inside have lower densities than surrounding air, although the total system includes the envelope, payload, and other components. The same mathematical principle applies because air is a fluid. The magnitude of buoyancy can be smaller than in water for the same volume because atmospheric air has much lower density than water.
Archimedes' principle also underlies practical instruments and engineering systems. Hydrometers float at depths determined by the balance between their weight and the buoyant force from a liquid, allowing liquid density or related properties to be inferred. Ships are designed so that their displacement of water provides a buoyant force matching their weight under normal loading. Submarines change their effective density and displaced water relationship through ballast systems.
Density measurements can use immersion and apparent weight differences to determine the volume or density of irregular objects. Fluid displacement is also important in understanding why a vessel's maximum safe load is constrained by its hull geometry and freeboard. These applications demonstrate that Archimedes' principle is both a theoretical statement about pressure forces and a practical quantitative tool.
Archimedes' principle has limitations in how it should be interpreted. The standard formula assumes a well defined displaced volume and an appropriate hydrostatic pressure field. In moving fluids, additional forces such as drag, lift, added mass, and flow induced pressure variations may occur, so the total fluid force cannot necessarily be identified with hydrostatic buoyancy alone. Surface tension can contribute forces for small objects at liquid interfaces. Compressibility becomes important for gases and for fluids under sufficiently large pressure variations.
In nonuniform gravitational fields or strongly stratified fluids, the simple constant density formula may require modification. These qualifications do not invalidate the principle; they identify the conditions under which the familiar equation is a sufficient description. The central statement remains that hydrostatic pressure forces produce a buoyant resultant equivalent to the weight of the fluid that would occupy the displaced volume.
