Buoyancy (pronounced /ˈbɔɪənsi, ˈbuːjənsi/), or upthrust, is the net upward force that a fluid exerts on an immersed body or fluid parcel. It originates from the increase of hydrostatic pressure with depth: pressure acting over the submerged surface produces a larger resultant force on the bottom than on the top, yielding an upward resultant. Quantitatively, the buoyant force equals the weight of fluid displaced by the immersed volume (Archimedes’ principle), which can be expressed as the displaced fluid mass multiplied by the acceleration that produces the pressure gradient (typically gravity).
Whether an object floats, sinks, or remains neutrally buoyant depends on the comparison between its average density and that of the surrounding fluid. If the object’s weight exceeds the weight of displaced fluid it will sink; if lighter, buoyancy can balance weight and the object will float. The center of buoyancy—the centroid of the displaced volume—is central to stability analysis: relative positions of the center of buoyancy and the object’s center of gravity determine restoring moments and tendencies to overturn.
Buoyancy is best understood as an apparent force tied to the presence of an acceleration field (most commonly gravity) or a non-inertial reference frame that defines a downward direction; in accelerating frames buoyant effects can arise even without gravity, whereas absent any acceleration defining “down,” buoyancy does not manifest. In fluids and mixtures, buoyant forces drive convective motions and buoyancy-driven flows (e.g., separation of air and water, oil–water segregation, and thermal convection) whenever parcels of differing density experience an acceleration field.
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In continuum descriptions, buoyancy is incorporated into the conservation equations of mass, momentum, and energy and couples to constitutive relations and thermodynamic inequalities (for example, the Clausius–Duhem entropy inequality). Transport processes such as diffusion (often modeled by Fick’s laws, J = −D dφ/dx) interact with buoyancy-driven advection in many geophysical and engineering problems. Theoretical and applied treatments of buoyancy draw on a broad fluid-mechanical framework—statics and dynamics, Archimedes’ and Bernoulli’s principles, the Navier–Stokes equations, viscosity models, surface phenomena (capillarity, surface tension), and relevant gas laws—and rest on foundational contributions from scientists including Bernoulli, Boyle, Cauchy, Charles, Euler, Fick, Gay‑Lussac, Graham, Hooke, Newton, Navier, Noll, Pascal, Stokes, and Truesdell.
Practical evaluation of buoyancy combines body forces, geometry, and fluid properties: one locates the centroid of displaced volume, computes the displaced-fluid weight (density × volume × acceleration), and compares the buoyant force vector and its line of action with the object’s weight and center of gravity to predict flotation, sinking, or overturning.
Archimedes’ principle, formulated by Archimedes of Syracuse in 212 BC, states that any body wholly or partly immersed in a fluid experiences an upward (buoyant) force equal in magnitude to the weight of the fluid it displaces. In force terms the buoyant force has magnitude ρfluid Vdisplaced g and acts opposite to the gravitational force on the body; for a completely submerged body Vdisplaced equals the body volume, whereas for a floating body the displaced fluid weight equals the body weight. In vector/sign convention this can be written compactly as FB = −Fg = −ρfluid Vdisplaced g, the negative sign indicating that FB is directed opposite to the weight vector.
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Density (ρ = mass/volume) governs flotation: an object with ρobject < ρfluid will float, while one with ρobject > ρfluid will sink unless its effective displaced volume is increased (for example by enclosing air or by shape effects). For a fully submerged body in vertical equilibrium, net force zero gives 0 = FB − Fg = ρfluid V g − ρobject V g, which reduces to ρfluid = ρobject — the condition of neutral buoyancy.
Surface tension and capillary forces are not explicit in the classical statement of Archimedes’ principle; they modify the geometry and amount of displaced fluid and thus the effective buoyant support, but they do not change the fundamental equality when displacement is correctly accounted for. Practical manifestations include common wood floating on water (lower density than water), the change in buoyant support when a denser or lighter fluid is mixed (e.g., adding ethanol to water reduces fluid density and thus buoyant lift, as in Galileo’s ball demonstration), and the coin-on-mercury observation, where both mercury’s high density and its strong surface tension influence how the coin is supported. The principle applies equally to gases and liquids: any fluid of density ρ produces an upward force equal to ρ V g on an immersed body, with g the local gravitational acceleration.
In hydrostatic weighing the submerged object is in static equilibrium, so the vector sum of forces vanishes: F_net = 0 = F_B + F_T − F_g, where the buoyant force F_B equals the weight of displaced fluid (ρ_fluid V g), F_T is the tension (apparent weight) measured by a force probe, and F_g = m g is the object’s true weight. Rearranging this balance yields the displaced volume directly as V = (m g − F_T) / (ρ_fluid g); thus displacement can be determined from force measurements without geometric volume measurement.
Substituting V into the definition of object density ρ_obj = m / V gives ρ_obj = (m g ρ_fluid) / (m g − F_T). Equivalently, writing densities as a ratio removes explicit g: ρ_obj / ρ_fluid = F_g / (F_g − F_app), with F_app ≡ F_T the apparent submerged weight. These relations show that relative density is obtained solely from the true weight and the submerged force reading.
In practice one measures the object’s mass (or true weight F_g = m g) and the apparent weight while fully submerged (F_app) using a force probe; knowing the fluid density ρ_fluid then permits computation of V or ρ_obj from the formulas above. Careful unit consistency is required so that m g is treated as the weight force.
Underlying assumptions are that the object is entirely submerged and stationary (no acceleration), the fluid has uniform density, buoyancy equals the weight of displaced fluid (Archimedes’ principle), and other effects such as surface tension or fluid motion are negligible. Under these conditions the force-balance method (hydrostatic weighing) provides a robust, instrument-friendly means to determine material density and forms the operating principle of devices such as the dasymeter, which infer density from measured buoyant and apparent-weight forces.
Forces and equilibrium
Static equilibrium of a fluid is described by the local balance between applied body forces and internal stresses, f + ∇·σ = 0, where f is the external force density and σ the Cauchy stress tensor. For a simple (isotropic) fluid σ = −pI, and the equilibrium reduces to f = ∇p. When the external field is conservative, f = −∇Φ, pressure and potential combine to a spatially uniform quantity so that p + Φ = const; consequently the free surface of an open fluid coincides with an equipotential of the applied field.
Under Earth gravity (taking z positive downward) this gives the familiar hydrostatic law p = ρf g z when the pressure at the free surface is set to zero: pressure grows linearly with depth. Because pressure on a submerged body varies with vertical position, lower surfaces experience larger normal stresses than upper surfaces, and the resulting net force is directed upward. Formally the buoyant force is the surface integral of the fluid stress over the wetted area, B = ∮ σ·n dA, which by the divergence theorem equals the volume integral −∫ f dV. For uniform gravitational body force this yields the vector relation B = ρf V g acting opposite the gravitational pull on the displaced fluid, and in scalar form the buoyant magnitude equals the weight of displaced fluid, B = ρf Vdisp g — the standard statement of Archimedes’ principle for homogeneous, static fluids.
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More general theoretical treatments extend this evaluation to non‑uniform fluids and arbitrary body shapes; such analyses (notably work by F. M. S. Lima) predict and experiments confirm non‑intuitive outcomes, for example a net downward buoyant force on a block in contact with the container bottom under certain stratifications and geometries. For an unconstrained floating body in static equilibrium the net vertical force vanishes, mg = ρf Vdisp g, so the submerged volume satisfies m = ρf Vdisp. This relation is independent of the magnitude of g (and thus of geographic location) but depends on local fluid density; variations of seawater density with temperature and salinity therefore alter ship draft and motivate load‑mark systems such as the Plimsoll line.
When additional contact or restraint forces are present, those constraint forces account for the difference between weight and buoyancy: a tether needed to hold a buoyant object fully submerged has tension T = ρf V g − m g, whereas the normal reaction on an object resting on the bottom is N = m g − ρf V g. Buoyant force can also be measured experimentally as the loss of apparent weight on immersion (weight_in_air − weight_in_fluid); because air density is small relative to most solids and liquids, neglecting air buoyancy typically introduces errors ≲0.1% except for very low‑density specimens.
Finally, Archimedes’ relations apply strictly to static or quasi‑static situations. Bodies undergoing acceleration or rapid motion require a full dynamical treatment of the surrounding fluid (added mass, unsteady pressures and viscous forces). Note: as of January 2016 the underlying material was indicated to lack cited references and would benefit from reliable sourcing.
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A simple, rigorous way to derive buoyancy uses a cube with its top face horizontal. The four vertical faces are congruent and sit at identical depth profiles, so each vertical face supports the same hydrostatic pressure distribution; the total forces on opposite vertical faces are equal and opposite, and therefore all horizontal components cancel, leaving no net horizontal resultant on the cube.
Hydrostatic pressure on a horizontal face is uniform because that face lies at a single depth. The upward force on the bottom face is therefore the pressure at the bottom depth multiplied by the bottom area; the downward force on the top face is the pressure at the top depth multiplied by the top area. For a cube the top and bottom areas are equal, so the net vertical (buoyant) force is the pressure difference times that area. Using p = ρgh, this difference equals ρg(h_bottom − h_top)A, which is ρg times the cube’s volume. Thus the resultant upward force equals the weight of the fluid that would occupy the cube’s volume in the absence of the cube. In the absence of other supports, the downward force on the cube is its own weight, so equilibrium depends on the comparison between buoyant force and weight.
The argument is scale-invariant: changing the cube’s linear dimensions does not alter the qualitative pressure distribution or the cancellation of horizontal forces, and the buoyant force continues to equal the weight of displaced fluid. When two cubes touch, pressures on the contacting faces are mutual and cancel, so the external force balance for the pair is the sum of the individual balances; buoyancies are additive. By tessellating an arbitrary solid with sufficiently small cubes, one obtains an approximation of any shape whose accuracy improves as cube size decreases; in the limit of infinitesimal cubes the discrete cube-sum reproduces the continuous body exactly, so the cube-based derivation generalizes to arbitrary bodies.
Inclined or otherwise non-horizontal faces do not invalidate this reasoning: the resultant hydrostatic force on any inclined face acts normal to that face and may be decomposed into vertical and horizontal components. Horizontal components continue to cancel pairwise, while vertical components sum to the same pressure-difference result obtained for horizontal layers. Consequently, the buoyant force on any immersed body equals the weight of the fluid displaced, independent of shape or orientation.
Static stability
A floating body is acted on by two forces whose lines of action determine its attitude: an upward buoyant force passing through the center of buoyancy (CB), the centroid of the displaced fluid volume, and the downward gravitational force through the center of gravity (CG). A small vertical displacement alters the displaced volume and therefore buoyancy; if the increased buoyant force exceeds the change in weight support the net result is an unbalanced upward force that returns the body toward its original immersion, a form of vertical stability.
Rotational (transverse) stability concerns the response to a small angular displacement (heel). Depending on the turning moments produced by the relative positions and movements of CB and CG the body may produce a restoring moment (stable), an amplifying moment (unstable), or no net moment (neutral). An immediately restoring torque is guaranteed when CG lies below CB, since any small tilt then produces opposing moments from weight and buoyancy. However, a vessel can still be rotationally stable with CG above CB if, upon heeling, the CB shifts laterally more than the CG so that the resulting moment tends to right the body. This behaviour is formalized by the metacentric concept: for small angles the instantaneous center about which CB moves is the metacenter (M), and a positive metacentric height (GM = distance from CG to M) implies a righting moment.
Metacentric stability is angle-dependent. As heel increases the geometry of the submerged volume changes and the CB trajectory may no longer produce a restoring lever; beyond a certain angle GM can fall to zero or become negative, creating capsize risk. During a single heeling event the sign and magnitude of the righting moment can change more than once as the submerged shape evolves, and some hull forms exhibit multiple distinct equilibrium attitudes. In practical terms, a bottom‑heavy vessel with a low CG achieves intrinsic stability by geometry (CG beneath CB), whereas a top‑heavy vessel relies on favorable lateral shifts of CB and a positive GM at small angles and is therefore more vulnerable to loss of stability as heel grows.
States of buoyancy
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Buoyancy is the mechanical response of any body that occupies volume within a fluid and is subject to gravity or other acceleration. The magnitude of the buoyant effect depends jointly on the volume of fluid displaced, the fluid’s density, and the external acceleration field acting on the system.
Hydrostatic pressure gradients produce the upward (buoyant) force; according to Archimedes’ principle this force equals the weight of the fluid displaced by the immersed portion of the body. The net vertical force on the body is therefore the buoyant force minus the body’s weight (or its effective weight in a non‑inertial frame).
Three limiting states follow from this balance. If the buoyant force exceeds weight (positive buoyancy), the body experiences a net upward acceleration and will tend to rise toward the free surface. If the forces balance (neutral buoyancy), the body remains at a fixed depth in the absence of other influences; however, currents, drag, added‑mass effects and external accelerations can displace this equilibrium. If buoyancy is less than weight (negative buoyancy), a net downward force causes the body to sink toward regions of higher pressure or the bed.
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In geographical and environmental systems these states govern vertical distribution and transport: for example, the floating or sinking of icebergs and ships, depth control of submersibles, deposition and resuspension of sediments, and ascent of warm air or pollutant plumes. Changes in fluid density, submerged volume, or the acceleration environment modify stratification, residence times and vertical mixing in natural waters and the atmosphere.
The following summary consolidates material on compressible objects in a hydrostatic buoyancy context; previous versions of this treatment were noted (January 2016) to lack cited sources.
In a hydrostatic environment the external pressure on an immersed body varies with depth, and because most materials are at least slightly compressible this pressure variation induces changes in the body’s volume. Compressibility κ quantifies this sensitivity of volume V to pressure P (κ = −(1/V)·(∂V/∂P)); an increase in external pressure generally reduces V, while a decrease allows expansion. Because buoyant force = ρ_fluid · V_displaced · g, any pressure-induced volume change produces a directly proportional change in buoyancy. The magnitude of the buoyant response also depends on the ambient fluid density ρ_fluid (which itself may vary with salinity, temperature or phase); identical volumetric changes yield different buoyant-force changes in fluids of different density.
An immersed body may occupy a depth at which gravitational and buoyant forces balance (a neutral depth or equilibrium). The dynamical stability of this equilibrium under small vertical perturbations is governed by the relative compressibilities of the object and its surrounding fluid. If the object is less compressible than the fluid (object κ < fluid κ), a small displacement produces a restoring change in buoyancy and the equilibrium is stable. Conversely, if the object is more compressible than the fluid (object κ > fluid κ), perturbations are amplified: an upward displacement reduces pressure, the object expands, buoyancy increases and it rises further; a downward displacement increases pressure, the object compresses, buoyancy decreases and it sinks further (positive feedback and instability).
This compressibility-driven criterion has practical consequences for the design and analysis of systems whose buoyancy depends on depth-varying pressure and stratified fluid properties. Examples include buoyancy control in submersibles and balloons, the contrasting stability of gas-filled versus solid bodies, and the vertical response of objects in fluids with nonuniform density profiles. Accounting simultaneously for pressure-dependent volume changes and the local fluid-density field is therefore essential for predicting vertical motion and stability.
Submarines
Submergence is effected by flooding large external ballast tanks: vents at the tank tops let trapped air escape upward while seawater enters through lower openings, raising the vessel’s mass until the desired buoyant condition is reached. When the submarine’s mean density matches that of the surrounding seawater it is neutrally buoyant and, in the absence of other forces, will neither ascend nor descend. Precise control of buoyancy and longitudinal trim is achieved with internal sealed trim tanks of low compressibility; small transfers of water between these tanks and the external ballast, or discharge of water overboard, permit fine adjustments to overall mass distribution and attitude. Operational practice in many military submarines is to maintain a slight negative buoyancy so the hull is statically heavier than the water; in that state depth and attitude are actively sustained by hydrodynamic lift produced by control surfaces while the vessel is underway. Accordingly, buoyancy control operates in two complementary regimes: a hydrostatic mode using ballast and trim tanks for coarse depth setting and static balance, and a hydrodynamic mode using forward motion and control-surface forces for fine depth-keeping and maneuvering.
A free balloon ascending through the atmosphere experiences expansion of both the external air and the gas within its envelope because of the vertical pressure gradient. The envelope and enclosed gas cannot expand as freely as an equivalent parcel of ambient air, so the balloon’s bulk volume increases less than that of the surrounding parcel; consequently the balloon’s mean (bulk) density declines with altitude but more slowly than the ambient air density.
According to Archimedes’ principle, the buoyant force on the balloon equals the weight of the displaced air. As ambient pressure and density fall with height, the mass of displaced air decreases and the upward buoyant force diminishes. The balloon attains neutral buoyancy at the altitude where its total weight equals the weight of the displaced air; because the balloon’s density varies less rapidly with altitude than the ambient density, small vertical displacements produce restoring buoyant imbalances that tend to return the balloon to that equilibrium level.
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During descent the reverse occurs: increasing ambient density raises the weight of displaced air and therefore the buoyant force, which decelerates and ultimately halts downward motion when buoyancy again balances weight. Thus the elastic properties of the envelope, the compressibility and thermal behavior of the internal gas, and the vertical stratification of atmospheric pressure and density together determine the equilibrium altitude and the sensitivity of buoyant force to altitude changes; these factors govern where and how stably an atmospheric balloon will settle.
Divers
Underwater buoyancy is inherently unstable because any gas-filled components of the diver or equipment change volume as ambient pressure changes with depth; compressed gas occupies less space at greater depth, reducing the displaced water and therefore buoyant force. Exposure suits rely on trapped gas layers for thermal insulation, so compression of these spaces both diminishes insulating volume and reduces the suit’s contribution to overall buoyancy. To manage these depth-dependent variations, divers use a buoyancy compensator—a controllable, variable-volume bladder that can be inflated to increase buoyancy or vented to decrease it. The operational aim while swimming mid-water is neutral buoyancy, yet this condition is precarious: small alterations in depth or in any gas volume rapidly shift equilibrium and demand corrective action. Divers typically achieve fine, immediate control by varying lung volume through breathing, and they use adjustments to the buoyancy compensator for larger or more sustained changes in buoyancy. Additionally, consumption of breathing gas during a dive lowers the diver’s mass over time, producing a progressive increase in net buoyancy and therefore reducing the quantity of compensating gas required in the buoyancy compensator to remain neutrally buoyant at a given depth.
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Density
Density — mass per unit volume — determines how fluids and objects arrange vertically and whether bodies float, sink, or remain suspended. A simple density column, constructed from stratified liquids such as baby oil, coloured rubbing alcohol, vegetable oil, wax, and coloured water, with a denser solid (e.g. aluminium) introduced, visually demonstrates layered fluids and provides comparative density measurements.
Buoyancy arises from the interaction between an object’s average density and the surrounding fluid’s density. If an object displaces a mass of fluid whose weight exceeds the object’s weight when fully submerged, the net upward (buoyant) force exceeds its weight; the object therefore has a lower mean density than the fluid and is positively buoyant. At a free surface this condition produces flotation: the object垂 comes to equilibrium with a submerged volume that displaces a mass of fluid equal to the object’s mass. When fully immersed, a positively buoyant body will accelerate toward regions of lower hydrostatic pressure or toward the free surface because the buoyant force exceeds its weight.
Neutral buoyancy occurs when an object’s average density exactly matches the fluid density, so buoyant force and weight balance and the object remains suspended; however, this balance is marginal and small disturbances can displace the body. Negative buoyancy arises when the object’s mean density exceeds that of the fluid, so the buoyant force is insufficient and the object sinks.
Engineering exploits average-density control to make dense materials float: enclosing large volumes of low-density material (commonly air) within a heavier shell reduces the system’s overall density. Thus a steel hull displaces enough water, via its enclosed air-filled volume, to balance the total weight of ship plus contents and remain afloat.