TheInfoList

Buoyancy (), or upthrust, is an upward
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

exerted by a
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
that opposes the
weight In science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as ...

of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object. The pressure difference results in a net upward force on the object. The magnitude of the force is proportional to the pressure difference, and (as explained by
Archimedes' principle Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external f ...
) is equivalent to the weight of the fluid that would otherwise occupy the submerged volume of the object, i.e. the displaced fluid. For this reason, an object whose average
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

is greater than that of the fluid in which it is submerged tends to sink. If the object is less dense than the liquid, the force can keep the object afloat. This can occur only in a
non-inertial reference frame A non-inertial reference frame is a frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...
, which either has a
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

or is defining a "downward" direction. The center of buoyancy of an object is the center of gravity of the displaced volume of fluid.

# Archimedes' principle

Archimedes' principle is named after
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

of
Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily Syracuse ( ; it, Siracusa , or scn, Seragusa, label=none ; lat, Syrācūsae ; grc-att, wikt:Συράκουσαι, Συράκουσαι, Syrákousai ; grc-dor, wikt:Συράκοσ ...

, who first discovered this law in 212 BC.. For objects, floating and sunken, and in gases as well as liquids (i.e. a
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
), Archimedes' principle may be stated thus in terms of forces: —with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object. More tersely: buoyant force = weight of displaced fluid. Archimedes' principle does not consider the
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Gerridae, water strid ...

(capillarity) acting on the body, but this additional force modifies only the amount of fluid displaced and the spatial distribution of the displacement, so the principle that ''buoyancy = weight of displaced fluid'' remains valid. The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In simple terms, the principle states that the buoyancy force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational acceleration, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy. This is also known as upthrust. Suppose a rock's weight is measured as 10
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
s when suspended by a string in a
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gree ...

with gravity acting upon it. Suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyancy force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water. Assuming Archimedes' principle to be reformulated as follows, :$\text = \text - \text\,$ then inserted into the quotient of weights, which has been expanded by the mutual volume :$\frac = \frac , \,$ yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes.: :$\frac = \frac \,$ (This formula is used for example in describing the measuring principle of a
dasymeter Historical drawing of a dasymeter (in German: "Fig. 109. Baroscope. (Apparatus for proving the buoyancy of air)") A dasymeter was meant initially as a device to demonstrate the buoyant effect of gases like air (as shown in the adjacent pictures). ...
and of
hydrostatic weighing Hydrostatic weighing, also referred to as "underwater weighing", "hydrostatic body composition analysis", and "hydrodensitometry" is a technique for measuring the density The density (more precisely, the volumetric mass density; also known as ...
.) Example: If you drop wood into water, buoyancy will keep it afloat. Example: A helium balloon in a moving car. During a period of increasing speed, the air mass inside the car moves in the direction opposite to the car's acceleration (i.e., towards the rear). The balloon is also pulled this way. However, because the balloon is buoyant relative to the air, it ends up being pushed "out of the way", and will actually drift in the same direction as the car's acceleration (i.e., forward). If the car slows down, the same balloon will begin to drift backward. For the same reason, as the car goes round a curve, the balloon will drift towards the inside of the curve.

# Forces and equilibrium

The equation to calculate the pressure inside a fluid in equilibrium is: :$\mathbf+\operatorname\,\sigma=0$ where f is the force density exerted by some outer field on the fluid, and ''σ'' is the
Cauchy stress tensor In continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis C ...
. In this case the stress tensor is proportional to the identity tensor: : $\sigma_=-p\delta_.\,$ Here ''δ''''ij'' is the
Kronecker delta In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. Using this the above equation becomes: :$\mathbf=\nabla p.\,$ Assuming the outer force field is conservative, that is it can be written as the negative gradient of some scalar valued function: :$\mathbf=-\nabla\Phi.\,$ Then: :$\nabla\left(p+\Phi\right)=0 \Longrightarrow p+\Phi = \text.\,$ Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field. Let the ''z''-axis point downward. In this case the field is gravity, so Φ = −''ρfgz'' where ''g'' is the gravitational acceleration, ''ρf'' is the mass density of the fluid. Taking the pressure as zero at the surface, where ''z'' is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is :$p=\rho_f g z.\,$ So pressure increases with depth below the surface of a liquid, as ''z'' denotes the distance from the surface of the liquid into it. Any object with a non-zero vertical depth will have different pressures on its top and bottom, with the pressure on the bottom being greater. This difference in pressure causes the upward buoyancy force. The buoyancy force exerted on a body can now be calculated easily, since the internal pressure of the fluid is known. The force exerted on the body can be calculated by integrating the stress tensor over the surface of the body which is in contact with the fluid: :$\mathbf=\oint \sigma \, d\mathbf.$ The
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

can be transformed into a
volume integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with the help of the
Gauss theorem In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...
: :$\mathbf=\int \operatorname\sigma \, dV = -\int \mathbf\, dV = -\rho_f \mathbf \int\,dV=-\rho_f \mathbf V$ where ''V'' is the measure of the volume in contact with the fluid, that is the volume of the submerged part of the body, since the fluid doesn't exert force on the part of the body which is outside of it. The magnitude of buoyancy force may be appreciated a bit more from the following argument. Consider any object of arbitrary shape and volume ''V'' surrounded by a liquid. The
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

the liquid exerts on an object within the liquid is equal to the weight of the liquid with a volume equal to that of the object. This force is applied in a direction opposite to gravitational force, that is of magnitude: :$B = \rho_f V_\text\, g, \,$ where ''ρf'' is the
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

of the fluid, ''Vdisp'' is the volume of the displaced body of liquid, and ''g'' is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing air drag, drag). This is the steady gain in speed caused exclusively by the force of ''gravitational attraction' ...
at the location in question. If this volume of liquid is replaced by a solid body of exactly the same shape, the force the liquid exerts on it must be exactly the same as above. In other words, the "buoyancy force" on a submerged body is directed in the opposite direction to gravity and is equal in magnitude to :$B = \rho_f V g. \,$ Though the above derivation of Archimedes principle is correct, a recent paper by the Brazillian physicist Fabio M. S. Lima brings a more general approach for the evaluation of the buoyant force exerted by any fluid (even non-homogeneous) on a body with arbitrary shape. Interestingly, this method leads to the prediction that the buoyant force exerted on a rectangular block touching the bottom of a container points downward! Indeed, this downward buoyant force has been confirmed experimentally. The
net force In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...

on the object must be zero if it is to be a situation of fluid statics such that Archimedes principle is applicable, and is thus the sum of the buoyancy force and the object's weight :$F_\text = 0 = m g - \rho_f V_\text g \,$ If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it tends to rise. An object whose weight exceeds its buoyancy tends to sink. Calculation of the upwards force on a submerged object during its
accelerating In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ( ...

period cannot be done by the Archimedes principle alone; it is necessary to consider dynamics of an object involving buoyancy. Once it fully sinks to the floor of the fluid or rises to the surface and settles, Archimedes principle can be applied alone. For a floating object, only the submerged volume displaces water. For a sunken object, the entire volume displaces water, and there will be an additional force of reaction from the solid floor. In order for Archimedes' principle to be used alone, the object in question must be in equilibrium (the sum of the forces on the object must be zero), therefore; :$mg = \rho_f V_\text g, \,$ and therefore :$m = \rho_f V_\text. \,$ showing that the depth to which a floating object will sink, and the volume of fluid it will displace, is independent of the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

regardless of geographic location. :(''Note: If the fluid in question is
seawater Seawater, or salt water, is water Water is an Inorganic compound, inorganic, Transparency and translucency, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of E ...

, it will not have the same
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

(''ρ'') at every location, since the density depends on temperature and
salinity Salinity () is the saltiness or amount of dissolved in a body of , called (see also ). It is usually measured in g/L or g/kg (grams of salt per liter/kilogram of water; the latter is dimensionless and equal to ‰). Salinity is an important ...

. For this reason, a ship may display a .)'' It can be the case that forces other than just buoyancy and gravity come into play. This is the case if the object is restrained or if the object sinks to the solid floor. An object which tends to float requires a
tension Tension may refer to: Science * Psychological stress * Tension (physics), a force related to the stretching of an object (the opposite of compression) * Tension (geology), a stress which stretches rocks in two opposite directions * Voltage or elect ...
restraint force T in order to remain fully submerged. An object which tends to sink will eventually have a
normal force In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (ve ...

of constraint N exerted upon it by the solid floor. The constraint force can be tension in a spring scale measuring its weight in the fluid, and is how apparent weight is defined. If the object would otherwise float, the tension to restrain it fully submerged is: :$T = \rho_f V g - m g . \,$ When a sinking object settles on the solid floor, it experiences a
normal force In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (ve ...

of: :$N = m g - \rho_f V g . \,$ Another possible formula for calculating buoyancy of an object is by finding the apparent weight of that particular object in the air (calculated in Newtons), and apparent weight of that object in the water (in Newtons). To find the force of buoyancy acting on the object when in air, using this particular information, this formula applies: : Buoyancy force = weight of object in empty space − weight of object immersed in fluid The final result would be measured in Newtons. Air's density is very small compared to most solids and liquids. For this reason, the weight of an object in air is approximately the same as its true weight in a vacuum. The buoyancy of air is neglected for most objects during a measurement in air because the error is usually insignificant (typically less than 0.1% except for objects of very low average density such as a balloon or light foam).

## Simplified model

A simplified explanation for the integration of the pressure over the contact area may be stated as follows: Consider a cube immersed in a fluid with the upper surface horizontal. The sides are identical in area, and have the same depth distribution, therefore they also have the same pressure distribution, and consequently the same total force resulting from hydrostatic pressure, exerted perpendicular to the plane of the surface of each side. There are two pairs of opposing sides, therefore the resultant horizontal forces balance in both orthogonal directions, and the resultant force is zero. The upward force on the cube is the pressure on the bottom surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal bottom surface of the cube is the hydrostatic pressure at that depth multiplied by the area of the bottom surface. Similarly, the downward force on the cube is the pressure on the top surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal top surface of the cube is the hydrostatic pressure at that depth multiplied by the area of the top surface. As this is a cube, the top and bottom surfaces are identical in shape and area, and the pressure difference between the top and bottom of the cube is directly proportional to the depth difference, and the resultant force difference is exactly equal to the weight of the fluid that would occupy the volume of the cube in its absence. This means that the resultant upward force on the cube is equal to the weight of the fluid that would fit into the volume of the cube, and the downward force on the cube is its weight, in the absence of external forces. This analogy is valid for variations in the size of the cube. If two cubes are placed alongside each other with a face of each in contact, the pressures and resultant forces on the sides or parts thereof in contact are balanced and may be disregarded, as the contact surfaces are equal in shape, size and pressure distribution, therefore the buoyancy of two cubes in contact is the sum of the buoyancies of each cube. This analogy can be extended to an arbitrary number of cubes. An object of any shape can be approximated as a group of cubes in contact with each other, and as the size of the cube is decreased, the precision of the approximation increases. The limiting case for infinitely small cubes is the exact equivalence. Angled surfaces do not nullify the analogy as the resultant force can be split into orthogonal components and each dealt with in the same way.

## Static stability

A floating object is stable if it tends to restore itself to an equilibrium position after a small displacement. For example, floating objects will generally have vertical stability, as if the object is pushed down slightly, this will create a greater buoyancy force, which, unbalanced by the weight force, will push the object back up. Rotational stability is of great importance to floating vessels. Given a small angular displacement, the vessel may return to its original position (stable), move away from its original position (unstable), or remain where it is (neutral). Rotational stability depends on the relative lines of action of forces on an object. The upward buoyancy force on an object acts through the center of buoyancy, being the
centroid In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of the displaced volume of fluid. The weight force on the object acts through its
center of gravity In physics, the center of mass of a distribution of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
. A buoyant object will be stable if the center of gravity is beneath the center of buoyancy because any angular displacement will then produce a 'righting moment'. The stability of a buoyant object at the surface is more complex, and it may remain stable even if the centre of gravity is above the centre of buoyancy, provided that when disturbed from the equilibrium position, the centre of buoyancy moves further to the same side that the centre of gravity moves, thus providing a positive righting moment. If this occurs, the floating object is said to have a positive
metacentric height The metacentric height (GM) is a measurement of the initial static stability of a floating body. It is calculated as the distance between the centre of gravity of a ship and its metacentre. A larger metacentric height implies greater initial stab ...
. This situation is typically valid for a range of heel angles, beyond which the centre of buoyancy does not move enough to provide a positive righting moment, and the object becomes unstable. It is possible to shift from positive to negative or vice versa more than once during a heeling disturbance, and many shapes are stable in more than one position.

# Fluids and objects

The atmosphere's density depends upon altitude. As an
airship An airship, dirigible balloon or blimp is a type of aerostat An aerostat (From greek language, Greek ἀήρ ''aer'' (air) + στατός ''statos'' (standing) through French) is a lighter than air aircraft that gains its lift through the ...

rises in the atmosphere, its buoyancy decreases as the density of the surrounding air decreases. In contrast, as a
submarine A submarine (or sub) is a watercraft capable of independent operation underwater. It differs from a submersible, which has more limited underwater capability. It is also sometimes used historically or colloquially to refer to remotely operated ...

expels water from its buoyancy tanks, it rises because its volume is constant (the volume of water it displaces if it is fully submerged) while its mass is decreased.

## Compressible objects

As a floating object rises or falls, the forces external to it change and, as all objects are compressible to some extent or another, so does the object's volume. Buoyancy depends on volume and so an object's buoyancy reduces if it is compressed and increases if it expands. If an object at equilibrium has a
compressibility In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
less than that of the surrounding fluid, the object's equilibrium is stable and it remains at rest. If, however, its compressibility is greater, its equilibrium is then , and it rises and expands on the slightest upward perturbation, or falls and compresses on the slightest downward perturbation.

### Submarines

Submarines A submarine (or sub) is a watercraft Watercraft, also known as water vessels or waterborne vessels, are vehicles A vehicle (from la, vehiculum) is a machine A machine is any physical system with ordered structural and functional p ...

rise and dive by filling large
ballast Ballast is material that is used to provide stability to a vehicle or structure. Ballast, other than cargo In economics, the word cargo refers in particular to goods or produce being conveyed—generally for Commerce, commercial gain—by ...

tanks with seawater. To dive, the tanks are opened to allow air to exhaust out the top of the tanks, while the water flows in from the bottom. Once the weight has been balanced so the overall density of the submarine is equal to the water around it, it has neutral buoyancy and will remain at that depth. Most military submarines operate with a slightly negative buoyancy and maintain depth by using the "lift" of the stabilizers with forward motion.

### Balloons

The height to which a
balloon A balloon is a flexible bag that can be inflated with a gas, such as , , , , and . For special tasks, balloons can be filled with smoke, liquid , granular media (e.g. sand, flour or rice), or light sources. Modern day balloons are made from ...

rises tends to be stable. As a balloon rises it tends to increase in volume with reducing atmospheric pressure, but the balloon itself does not expand as much as the air on which it rides. The average density of the balloon decreases less than that of the surrounding air. The weight of the displaced air is reduced. A rising balloon stops rising when it and the displaced air are equal in weight. Similarly, a sinking balloon tends to stop sinking.

### Divers

Underwater divers are a common example of the problem of unstable buoyancy due to compressibility. The diver typically wears an exposure suit which relies on gas-filled spaces for insulation, and may also wear a buoyancy compensator, which is a variable volume buoyancy bag which is inflated to increase buoyancy and deflated to decrease buoyancy. The desired condition is usually neutral buoyancy when the diver is swimming in mid-water, and this condition is unstable, so the diver is constantly making fine adjustments by control of lung volume, and has to adjust the contents of the buoyancy compensator if the depth varies.

# Density

If the weight of an object is less than the weight of the displaced fluid when fully submerged, then the object has an average density that is less than the fluid and when fully submerged will experience a buoyancy force greater than its own weight. If the fluid has a surface, such as water in a lake or the sea, the object will float and settle at a level where it displaces the same weight of fluid as the weight of the object. If the object is immersed in the fluid, such as a submerged submarine or air in a balloon, it will tend to rise. If the object has exactly the same density as the fluid, then its buoyancy equals its weight. It will remain submerged in the fluid, but it will neither sink nor float, although a disturbance in either direction will cause it to drift away from its position. An object with a higher average density than the fluid will never experience more buoyancy than weight and it will sink. A ship will float even though it may be made of steel (which is much denser than water), because it encloses a volume of air (which is much less dense than water), and the resulting shape has an average density less than that of the water.

* * * * * * * * * * * * * * * * * * * * * * * *