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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.
2517:
2501:
1029:
1055:
2509:
2760:, 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.
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2614:. This situation is typically valid for a range of heel angles, beyond which the center 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.
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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
2609:
The stability of a buoyant object at the surface is more complex, and it may remain stable even if the center of gravity is above the center of buoyancy, provided that when disturbed from the equilibrium position, the center of buoyancy moves further to the same side that the center of gravity moves,
2556:
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
2193:
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
2746:
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
1307:
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
3101:
Note: In the absence of surface tension, the mass of fluid displaced is equal to the submerged volume multiplied by the fluid density. High repulsive surface tension will cause the body to float higher than expected, though the same total volume will be displaced, but at a greater distance from the
2586:
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
1123:
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
2733:
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
2545:
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
2541:
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
1308:
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.
2537:
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
2105:
Though the above derivation of
Archimedes principle is correct, a recent paper by the Brazilian 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
1135:
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 â 3 = 7 newtons. Buoyancy reduces the apparent
2477:
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
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962:
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
2491:
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
1948:
2055:
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
1229:
2700:
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.
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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.
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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.
1765:
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:
2560:
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.
3102:
object. Where there is doubt about the meaning of "volume of fluid displaced", this should be interpreted as the overflow from a full container when the object is floated in it, or as the volume of the object below the average level of the fluid.
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1008:
is an apparent force as a function of inertia. Buoyancy can exist without gravity in the presence of an inertial reference frame, but without an apparent "downward" direction of gravity or other source of acceleration, buoyancy does not exist.
2590:
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).
1124:
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.
1240:
2188:
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
1968:
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:
1676:
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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.
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2345:
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
1833:
2755:
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
1099:âwith 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.
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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.
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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 does not exert force on the part of the body which is outside of it.
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Buoyancy is a function of the force of gravity or other source of acceleration on objects of different densities, and for that reason is considered an apparent force, in the same way that
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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
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978:
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
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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.
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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
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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
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has difficulties to get under water due to its buoyancy. When no swimming forces are implied, the natural equilibrium of forces keeps about half of the duck off water.
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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.
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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.
2197:
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;
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Illustration of the stability of bottom-heavy (left) and top-heavy (right) ships with respect to the positions of their centres of buoyancy (CB) and gravity (CG)
873:
2602:. 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
1234:
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:
2734:
depth. Most military submarines operate with a slightly negative buoyancy and maintain depth by using the "lift" of the stabilizers with forward motion.
2534:
There are two pairs of opposing sides, therefore the resultant horizontal forces balance in both orthogonal directions, and the resultant force is zero.
1283:{\displaystyle {\frac {\text{density of object}}{\text{density of fluid}}}={\frac {\text{weight}}{{\text{weight}}-{\text{apparent immersed weight}}}}\,}
1067:
The
Galileo's Ball experiment, showing the different buoyancy of the same object, depending on its surrounding medium. The ball has certain buoyancy in
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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.
1616:
963:
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
3202:"Floater clustering in a standing wave: Capillarity effects drive hydrophilic or hydrophobic particles to congregate at specific points on a wave"
2194:
displaces water. For a sunken object, the entire volume displaces water, and there will be an additional force of reaction from the solid floor.
1943:{\displaystyle \mathbf {B} =\int \operatorname {div} \sigma \,dV=-\int \mathbf {f} \,dV=-\rho _{f}\mathbf {g} \int \,dV=-\rho _{f}\mathbf {g} V}
1075:
is added (which is less dense than water), it reduces the density of the medium, thus making the ball sink further down (reducing its buoyancy).
2564:
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.
1001:, but the principles remain the same. Examples of buoyancy driven flows include the spontaneous separation of air and water or oil and water.
1960:
The magnitude of buoyancy force may be appreciated a bit more from the following argument. Consider any object of arbitrary shape and volume
2902:
1681:
Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field. Let the
1224:{\displaystyle {\frac {\text{density of object}}{\text{density of fluid}}}={\frac {\text{weight}}{\text{weight of displaced fluid}}},\,}
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Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object
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Assuming the outer force field is conservative, that is it can be written as the negative gradient of some scalar valued function:
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2712:, and it rises and expands on the slightest upward perturbation, or falls and compresses on the slightest downward perturbation.
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The forces at work in buoyancy. The object floats at rest because the upward force of buoyancy is equal to the downward force of
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3074:
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showing that the depth to which a floating object will sink, and the volume of fluid it will displace, is independent of the
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in contact is the sum of the buoyancies of each cube. This analogy can be extended to an arbitrary number of cubes.
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967:) is equivalent to the weight of the fluid that would otherwise occupy the submerged volume of the object, i.e. the
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A simplified explanation for the integration of the pressure over the contact area may be stated as follows:
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1112:(capillarity) acting on the body, but this additional force modifies only the amount of fluid displaced and
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2492:(typically less than 0.1% except for objects of very low average density such as a balloon or light foam).
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restraint force T in order to remain fully submerged. An object which tends to sink will eventually have a
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thus providing a positive righting moment. If this occurs, the floating object is said to have a positive
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134:
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is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is
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1174:{\displaystyle {\text{apparent immersed weight}}={\text{weight}}-{\text{weight of displaced fluid}}\,}
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2926: â Classic science experiment demonstrating the Archimedes' principle and the ideal gas law
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2997: â Mixture of sand, silt or clay with water, which creates a liquefied soil when agitated
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3015: â Gas-filled organ that contributes to the ability of a fish to control its buoyancy
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then inserted into the quotient of weights, which has been expanded by the mutual volume
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Buoyancy force = weight of object in empty space â weight of object immersed in fluid
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If the object would otherwise float, the tension to restrain it fully submerged is:
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is the mass density of the fluid. Taking the pressure as zero at the surface, where
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of the displaced volume of fluid. The weight force on the object acts through its
17:
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1685:-axis point downward. In this case the field is gravity, so ÎŚ = −
993:
Buoyancy also applies to fluid mixtures, and is the most common driving force of
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2627:
1382:
1321:
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732:
83:
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1671:{\displaystyle \nabla (p+\Phi )=0\Longrightarrow p+\Phi ={\text{constant}}.\,}
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1293:(This formula is used for example in describing the measuring principle of a
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due to the buoyancy force upon it and appears to float higher because of the
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currents. In these cases, the mathematical modelling is altered to apply to
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1450:. In this case the stress tensor is proportional to the identity tensor:
1394:
The equation to calculate the pressure inside a fluid in equilibrium is:
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3048:
2949: â Thermometer containing several glass vessels of varying density
2743:
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2527:
Consider a cube immersed in a fluid with the upper surface horizontal.
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2034:
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256:
3003: â Mixing process of warm, salty water with colder, fresher water
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2967: â Technique for measuring the density of a living person's body
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955:
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1304:
Example: If you drop wood into water, buoyancy will keep it afloat.
251:
2973: â Property necessary for a gas to be considered a lifting gas
2415:
When a sinking object settles on the solid floor, it experiences a
1757:
So pressure increases with depth below the surface of a liquid, as
1442:
is the force density exerted by some outer field on the fluid, and
27:
Upward force that opposes the weight of an object immersed in fluid
3243:
3227:"Using surface integrals for checking Archimedes' law of buoyancy"
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2331:) at every location, since the density depends on temperature and
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2905: â Measure of fluid stability against vertical displacement
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1091:), Archimedes' principle may be stated thus in terms of forces:
2178:{\displaystyle F_{\text{net}}=0=mg-\rho _{f}V_{\text{disp}}g\,}
958:
of a partially or fully immersed object. In a column of fluid,
3009: â Watercraft capable of independent underwater operation
2855:
2767:
2621:
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This analogy is valid for variations in the size of the cube.
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Assuming
Archimedes' principle to be reformulated as follows,
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2943: â Branch of fluid mechanics that studies fluids at rest
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1428:{\displaystyle \mathbf {f} +\operatorname {div} \,\sigma =0}
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910:
2985: â Legal limit to which a merchant ship may be loaded
1805:{\displaystyle \mathbf {B} =\oint \sigma \,d\mathbf {A} .}
2911: â Equipment for controlling the buoyancy of a diver
925:
2890: â Variation in pressure as a function of elevation
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Approximation of an arbitrary volume as a group of cubes
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Pages displaying short descriptions of redirect targets
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3114:"The diving "Law-ers": A brief resume of their lives"
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Pages displaying wikidata descriptions as a fallback
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2044:is the volume of the displaced body of liquid, and
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108:. Unsourced material may be challenged and removed.
2917: â equipment to regulate buoyancy of airships
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3119:South Pacific Underwater Medicine Society Journal
2961: â Device used to measure density of liquids
2542:depth multiplied by the area of the top surface.
2979: â Engineering discipline of marine vessels
2016:{\displaystyle B=\rho _{f}V_{\text{disp}}\,g,\,}
2955: â Watertight buoyant body of a watercraft
2488:The final result would be measured in Newtons.
2246:{\displaystyle mg=\rho _{f}V_{\text{disp}}g,\,}
1093:
2937: â Ballast carried to counteract buoyancy
2538:multiplied by the area of the bottom surface.
2478:particular information, this formula applies:
1502:{\displaystyle \sigma _{ij}=-p\delta _{ij}.\,}
988:accelerating due to a force other than gravity
2299:{\displaystyle m=\rho _{f}V_{\text{disp}}.\,}
1600:{\displaystyle \mathbf {f} =-\nabla \Phi .\,}
867:
8:
3142:. Archived from the original on 2 April 2011
1114:the spatial distribution of the displacement
1108:Archimedes' principle does not consider the
2802:. Unsourced material may be challenged and
2656:. Unsourced material may be challenged and
1350:. Unsourced material may be challenged and
319:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
71:Learn how and when to remove these messages
3326:. Oxford University Press US. p. 42.
3180:. Oxford University Press US. p. 41.
1127:Suppose a rock's weight is measured as 10
1104:buoyant force = weight of displaced fluid.
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2676:Learn how and when to remove this message
2504:Pressure distribution on an immersed cube
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1525:. Using this the above equation becomes:
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974:For this reason, an object whose average
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2722:Submarine § Submersion and trimming
2335:. For this reason, a ship may display a
1556:{\displaystyle \mathbf {f} =\nabla p.\,}
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3295:Revista Brasileira de Ensino de Fisica
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2838:Density column of liquids and solids:
3291:"A downward buoyant force experiment"
3225:Lima, FĂĄbio M. S. (22 January 2012).
2587:force, will push the object back up.
1079:Archimedes' principle is named after
7:
2991: â Float used to support a boat
2899: â Floating structure or device
2800:adding citations to reliable sources
2654:adding citations to reliable sources
1348:adding citations to reliable sources
1118:buoyancy = weight of displaced fluid
106:adding citations to reliable sources
2546:volume of the cube in its absence.
2467:{\displaystyle N=mg-\rho _{f}Vg.\,}
2405:{\displaystyle T=\rho _{f}Vg-mg.\,}
2313:regardless of geographic location.
1697:is the gravitational acceleration,
3214:from the original on 21 July 2011.
2704:If an object at equilibrium has a
2319:Note: If the fluid in question is
1653:
1632:
1620:
1590:
1587:
1543:
1020:of the displaced volume of fluid.
25:
3289:Lima, FĂĄbio M. S. (11 May 2014).
990:defining a "downward" direction.
52:This article has multiple issues.
3045:Longman Pronunciation Dictionary
2772:
2626:
2095:{\displaystyle B=\rho _{f}Vg.\,}
1933:
1902:
1874:
1838:
1795:
1777:
1747:{\displaystyle p=\rho _{f}gz.\,}
1577:
1536:
1405:
1320:
1131:when suspended by a string in a
1032:A metallic coin (an old British
891:
184:
82:
41:
3308:10.1590/S1806-11172014000200009
2915:Buoyancy compensator (aviation)
2862:(with blue food colouring) and
2729:rise and dive by filling large
93:needs additional citations for
60:or discuss these issues on the
3322:Pickover, Clifford A. (2008).
3176:Pickover, Clifford A. (2008).
3075:English Pronouncing Dictionary
1644:
1635:
1623:
1:
3372:NASA's definition of buoyancy
2909:Buoyancy compensator (diving)
2052:at the location in question.
210:in tone and meet Knowledge's
3078:(18th ed.), Cambridge:
2323:, it will not have the same
1964:surrounded by a liquid. The
980:non-inertial reference frame
3231:European Journal of Physics
3413:
3261:10.1088/0143-0807/33/1/009
3080:Cambridge University Press
2719:
2571:
2512:Forces on an immersed cube
2050:gravitational acceleration
1819:can be transformed into a
1047:
29:
3160:: CS1 maint: unfit URL (
1213:weight of displaced fluid
1167:weight of displaced fluid
1273:apparent immersed weight
1151:apparent immersed weight
1116:, so the principle that
1023:
378:ClausiusâDuhem (entropy)
328:Fick's laws of diffusion
3361:Elementary Hydrostatics
2935:Diving weighting system
2903:BruntâVäisälä frequency
536:NavierâStokes equations
474:Material failure theory
30:For the 2019 film, see
3021: â Reaction force
2867:
2742:The height to which a
2583:
2521:
2513:
2505:
2468:
2406:
2300:
2247:
2179:
2096:
2017:
1944:
1806:
1748:
1672:
1601:
1557:
1503:
1429:
1391:
1312:Forces and equilibrium
1284:
1225:
1175:
1097:
1076:
1045:
320:
260:
3324:Archimedes to Hawking
3178:Archimedes to Hawking
3112:Acott, Chris (1999).
3071:Roach, Peter (2011),
2837:
2581:
2519:
2511:
2503:
2469:
2407:
2301:
2248:
2180:
2097:
2018:
1945:
1823:with the help of the
1807:
1749:
1673:
1602:
1558:
1504:
1430:
1385:
1285:
1226:
1176:
1066:
1050:Archimedes' principle
1031:
1024:Archimedes' principle
982:, which either has a
965:Archimedes' principle
531:Bernoulli's principle
524:Archimedes' principle
321:
254:
2965:Hydrostatic weighing
2796:improve this section
2758:buoyancy compensator
2696:Compressible objects
2650:improve this section
2426:
2364:
2263:
2204:
2120:
2063:
1976:
1834:
1773:
1715:
1617:
1573:
1532:
1457:
1448:Cauchy stress tensor
1401:
1344:improve this section
1299:hydrostatic weighing
1241:
1191:
1146:
1016:of an object is the
623:Cohesion (chemistry)
445:Infinitesimal strain
281:
204:improve this article
102:improve this article
3253:2012EJPh...33..101L
2947:Galileo thermometer
2884:, also known as Air
2882:Atmosphere of Earth
2311:gravitational field
984:gravitational field
541:Poiseuille equation
272:Continuum mechanics
266:Part of a series on
2977:Naval architecture
2888:Archimedes paradox
2868:
2618:Fluids and objects
2612:metacentric height
2584:
2522:
2514:
2506:
2464:
2402:
2296:
2243:
2175:
2092:
2013:
1940:
1802:
1744:
1668:
1597:
1553:
1499:
1425:
1392:
1280:
1221:
1171:
1077:
1046:
1014:center of buoyancy
747:Magnetorheological
742:Electrorheological
479:Fracture mechanics
316:
261:
18:Center of buoyancy
2832:
2831:
2824:
2686:
2685:
2678:
2600:center of gravity
2289:
2233:
2168:
2130:
2002:
1662:
1380:
1379:
1372:
1277:
1274:
1266:
1261:
1252:
1251:
1248:
1247:density of object
1215:
1214:
1211:
1202:
1201:
1198:
1197:density of object
1168:
1160:
1152:
1064:
1018:center of gravity
1006:centrifugal force
954:that opposes the
884:
883:
759:
758:
693:
692:
462:Contact mechanics
385:
384:
314:
247:
246:
239:
229:
228:
212:quality standards
178:
177:
170:
152:
75:
16:(Redirected from
3404:
3351:Falling in Water
3338:
3337:
3319:
3313:
3312:
3310:
3286:
3280:
3279:
3277:
3275:
3246:
3222:
3216:
3215:
3213:
3207:. 23 June 2005.
3206:
3198:
3192:
3191:
3173:
3167:
3165:
3159:
3151:
3149:
3147:
3109:
3103:
3099:
3093:
3092:
3068:
3062:
3061:
3047:(3rd ed.),
3037:
2971:Lighter than air
2920:
2893:
2827:
2820:
2816:
2813:
2807:
2776:
2768:
2681:
2674:
2670:
2667:
2661:
2630:
2622:
2568:Static stability
2496:Simplified model
2473:
2471:
2470:
2465:
2453:
2452:
2411:
2409:
2408:
2403:
2382:
2381:
2305:
2303:
2302:
2297:
2291:
2290:
2287:
2281:
2280:
2252:
2250:
2249:
2244:
2235:
2234:
2231:
2225:
2224:
2184:
2182:
2181:
2176:
2170:
2169:
2166:
2160:
2159:
2132:
2131:
2128:
2101:
2099:
2098:
2093:
2081:
2080:
2022:
2020:
2019:
2014:
2004:
2003:
2000:
1994:
1993:
1949:
1947:
1946:
1941:
1936:
1931:
1930:
1905:
1900:
1899:
1877:
1841:
1817:surface integral
1811:
1809:
1808:
1803:
1798:
1780:
1753:
1751:
1750:
1745:
1733:
1732:
1677:
1675:
1674:
1669:
1663:
1660:
1606:
1604:
1603:
1598:
1580:
1562:
1560:
1559:
1554:
1539:
1508:
1506:
1505:
1500:
1494:
1493:
1472:
1471:
1434:
1432:
1431:
1426:
1408:
1375:
1368:
1364:
1361:
1355:
1324:
1316:
1289:
1287:
1286:
1281:
1278:
1276:
1275:
1272:
1267:
1264:
1259:
1258:
1253:
1250:density of fluid
1249:
1246:
1245:
1230:
1228:
1227:
1222:
1216:
1212:
1209:
1208:
1203:
1200:density of fluid
1199:
1196:
1195:
1180:
1178:
1177:
1172:
1169:
1166:
1161:
1158:
1153:
1150:
1065:
946:is a net upward
941:
940:
937:
936:
933:
930:
927:
924:
921:
918:
913:
912:
909:
906:
903:
900:
897:
876:
869:
862:
708:
673:Gay-Lussac's law
663:Combined gas law
613:Capillary action
498:
341:
325:
323:
322:
317:
315:
313:
305:
297:
263:
242:
235:
224:
221:
215:
188:
187:
180:
173:
166:
162:
159:
153:
151:
110:
86:
78:
67:
45:
44:
37:
21:
3412:
3411:
3407:
3406:
3405:
3403:
3402:
3401:
3392:Fluid mechanics
3377:
3376:
3347:
3342:
3341:
3334:
3321:
3320:
3316:
3288:
3287:
3283:
3273:
3271:
3224:
3223:
3219:
3211:
3204:
3200:
3199:
3195:
3188:
3175:
3174:
3170:
3152:
3145:
3143:
3111:
3110:
3106:
3100:
3096:
3090:
3070:
3069:
3065:
3059:
3039:
3038:
3034:
3029:
3024:
2924:Cartesian diver
2918:
2891:
2877:
2844:rubbing alcohol
2828:
2817:
2811:
2808:
2793:
2777:
2766:
2753:
2740:
2724:
2718:
2706:compressibility
2698:
2682:
2671:
2665:
2662:
2647:
2631:
2620:
2576:
2570:
2498:
2444:
2424:
2423:
2373:
2362:
2361:
2282:
2272:
2261:
2260:
2226:
2216:
2202:
2201:
2161:
2151:
2123:
2118:
2117:
2072:
2061:
2060:
2042:
2031:
1995:
1985:
1974:
1973:
1922:
1891:
1832:
1831:
1821:volume integral
1771:
1770:
1724:
1713:
1712:
1702:
1690:
1615:
1614:
1571:
1570:
1530:
1529:
1523:Kronecker delta
1520:
1482:
1460:
1455:
1454:
1399:
1398:
1376:
1365:
1359:
1356:
1341:
1325:
1314:
1262:
1239:
1238:
1189:
1188:
1144:
1143:
1120:remains valid.
1110:surface tension
1054:
1052:
1044:of the mercury.
1042:surface tension
1026:
915:
894:
890:
880:
851:
850:
849:
769:
761:
760:
714:Viscoelasticity
705:
695:
694:
682:
632:
628:Surface tension
592:
495:
493:Fluid mechanics
485:
484:
483:
397:
395:Solid mechanics
387:
386:
338:
330:
306:
298:
279:
278:
243:
232:
231:
230:
225:
219:
216:
201:
189:
185:
174:
163:
157:
154:
111:
109:
99:
87:
46:
42:
35:
32:Buoyancy (film)
28:
23:
22:
15:
12:
11:
5:
3410:
3408:
3400:
3399:
3394:
3389:
3379:
3378:
3375:
3374:
3369:
3353:
3346:
3345:External links
3343:
3340:
3339:
3332:
3314:
3281:
3237:(1): 101â113.
3217:
3193:
3186:
3168:
3104:
3094:
3088:
3063:
3057:
3041:Wells, John C.
3031:
3030:
3028:
3025:
3023:
3022:
3016:
3010:
3004:
3001:Salt fingering
2998:
2992:
2986:
2980:
2974:
2968:
2962:
2956:
2950:
2944:
2938:
2932:
2927:
2921:
2912:
2906:
2900:
2894:
2885:
2878:
2876:
2873:
2848:food colouring
2830:
2829:
2780:
2778:
2771:
2765:
2762:
2752:
2749:
2739:
2736:
2717:
2714:
2697:
2694:
2684:
2683:
2634:
2632:
2625:
2619:
2616:
2574:Ship stability
2572:Main article:
2569:
2566:
2497:
2494:
2486:
2485:
2475:
2474:
2462:
2459:
2456:
2451:
2447:
2443:
2440:
2437:
2434:
2431:
2413:
2412:
2400:
2397:
2394:
2391:
2388:
2385:
2380:
2376:
2372:
2369:
2343:
2342:
2307:
2306:
2294:
2285:
2279:
2275:
2271:
2268:
2256:and therefore
2254:
2253:
2241:
2238:
2229:
2223:
2219:
2215:
2212:
2209:
2186:
2185:
2173:
2164:
2158:
2154:
2150:
2147:
2144:
2141:
2138:
2135:
2126:
2103:
2102:
2090:
2087:
2084:
2079:
2075:
2071:
2068:
2040:
2037:of the fluid,
2029:
2024:
2023:
2011:
2008:
1998:
1992:
1988:
1984:
1981:
1951:
1950:
1939:
1935:
1929:
1925:
1921:
1918:
1915:
1912:
1908:
1904:
1898:
1894:
1890:
1887:
1884:
1881:
1876:
1872:
1869:
1866:
1863:
1860:
1856:
1853:
1850:
1847:
1844:
1840:
1813:
1812:
1801:
1797:
1793:
1789:
1786:
1783:
1779:
1755:
1754:
1742:
1739:
1736:
1731:
1727:
1723:
1720:
1700:
1688:
1679:
1678:
1666:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1608:
1607:
1595:
1592:
1589:
1586:
1583:
1579:
1564:
1563:
1551:
1548:
1545:
1542:
1538:
1516:
1510:
1509:
1497:
1492:
1489:
1485:
1481:
1478:
1475:
1470:
1467:
1463:
1436:
1435:
1424:
1421:
1418:
1414:
1411:
1407:
1378:
1377:
1328:
1326:
1319:
1313:
1310:
1291:
1290:
1270:
1256:
1232:
1231:
1219:
1206:
1182:
1181:
1164:
1156:
1102:More tersely:
1048:Main article:
1025:
1022:
882:
881:
879:
878:
871:
864:
856:
853:
852:
848:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
771:
770:
767:
766:
763:
762:
757:
756:
755:
754:
749:
744:
736:
735:
729:
728:
727:
726:
721:
716:
706:
701:
700:
697:
696:
691:
690:
684:
683:
681:
680:
675:
670:
665:
660:
655:
650:
644:
641:
640:
634:
633:
631:
630:
625:
620:
618:Chromatography
615:
610:
604:
601:
600:
594:
593:
591:
590:
571:
570:
569:
550:
538:
533:
521:
508:
505:
504:
496:
491:
490:
487:
486:
482:
481:
476:
471:
470:
469:
459:
454:
449:
448:
447:
442:
432:
427:
422:
417:
416:
415:
405:
399:
398:
393:
392:
389:
388:
383:
382:
381:
380:
372:
371:
367:
366:
365:
364:
359:
354:
346:
345:
339:
336:
335:
332:
331:
326:
312:
309:
304:
301:
295:
292:
289:
286:
275:
274:
268:
267:
245:
244:
227:
226:
192:
190:
183:
176:
175:
90:
88:
81:
76:
50:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3409:
3398:
3395:
3393:
3390:
3388:
3385:
3384:
3382:
3373:
3370:
3367:
3363:
3362:
3357:
3354:
3352:
3349:
3348:
3344:
3335:
3333:9780195336115
3329:
3325:
3318:
3315:
3309:
3304:
3300:
3296:
3292:
3285:
3282:
3270:
3266:
3262:
3258:
3254:
3250:
3245:
3240:
3236:
3232:
3228:
3221:
3218:
3210:
3203:
3197:
3194:
3189:
3187:9780195336115
3183:
3179:
3172:
3169:
3163:
3157:
3141:
3137:
3133:
3129:
3125:
3121:
3120:
3115:
3108:
3105:
3098:
3095:
3091:
3089:9780521152532
3085:
3081:
3077:
3076:
3067:
3064:
3060:
3058:9781405881180
3054:
3050:
3046:
3042:
3036:
3033:
3026:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2983:Plimsoll line
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2931:
2928:
2925:
2922:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2889:
2886:
2883:
2880:
2879:
2874:
2872:
2865:
2861:
2857:
2853:
2852:vegetable oil
2849:
2845:
2841:
2836:
2826:
2823:
2815:
2805:
2801:
2797:
2791:
2790:
2786:
2781:This section
2779:
2775:
2770:
2769:
2763:
2761:
2759:
2750:
2748:
2745:
2737:
2735:
2732:
2728:
2723:
2715:
2713:
2711:
2707:
2702:
2695:
2693:
2691:
2680:
2677:
2669:
2659:
2655:
2651:
2645:
2644:
2640:
2635:This section
2633:
2629:
2624:
2623:
2617:
2615:
2613:
2607:
2605:
2601:
2597:
2592:
2588:
2580:
2575:
2567:
2565:
2562:
2558:
2554:
2551:
2547:
2543:
2539:
2535:
2532:
2528:
2525:
2518:
2510:
2502:
2495:
2493:
2489:
2484:
2481:
2480:
2479:
2460:
2457:
2454:
2449:
2445:
2441:
2438:
2435:
2432:
2429:
2422:
2421:
2420:
2418:
2398:
2395:
2392:
2389:
2386:
2383:
2378:
2374:
2370:
2367:
2360:
2359:
2358:
2355:
2353:
2349:
2340:
2338:
2337:Plimsoll line
2334:
2328:
2326:
2322:
2316:
2315:
2314:
2312:
2292:
2283:
2277:
2273:
2269:
2266:
2259:
2258:
2257:
2239:
2236:
2227:
2221:
2217:
2213:
2210:
2207:
2200:
2199:
2198:
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2112:
2107:
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2077:
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2059:
2058:
2057:
2053:
2051:
2047:
2043:
2036:
2032:
2009:
2006:
1996:
1990:
1986:
1982:
1979:
1972:
1971:
1970:
1967:
1963:
1958:
1956:
1937:
1927:
1923:
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1916:
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1861:
1858:
1854:
1851:
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1845:
1842:
1830:
1829:
1828:
1826:
1825:Gauss theorem
1822:
1818:
1799:
1791:
1787:
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1329:This section
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1007:
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996:
991:
989:
985:
981:
977:
972:
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966:
961:
957:
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950:exerted by a
949:
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704:
699:
698:
689:
685:
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674:
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669:
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664:
661:
659:
658:Charles's law
656:
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566:non-Newtonian
564:
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452:Compatibility
450:
446:
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440:Finite strain
438:
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344:Conservations
342:
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195:reads like a
193:This article
191:
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118:
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113:Find sources:
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103:
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91:This article
89:
85:
80:
79:
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65:
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59:
58:
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39:
38:
33:
19:
3366:Google Books
3360:
3356:W. H. Besant
3323:
3317:
3298:
3294:
3284:
3272:. Retrieved
3234:
3230:
3220:
3196:
3177:
3171:
3156:cite journal
3144:. Retrieved
3123:
3117:
3107:
3097:
3072:
3066:
3044:
3035:
3013:Swim bladder
2941:Hydrostatics
2869:
2818:
2812:January 2016
2809:
2794:Please help
2782:
2754:
2741:
2725:
2703:
2699:
2687:
2672:
2666:January 2016
2663:
2648:Please help
2636:
2608:
2593:
2589:
2585:
2563:
2559:
2555:
2552:
2548:
2544:
2540:
2536:
2533:
2529:
2526:
2523:
2490:
2487:
2482:
2476:
2417:normal force
2414:
2356:
2352:normal force
2344:
2330:
2318:
2308:
2255:
2196:
2191:accelerating
2187:
2108:
2104:
2054:
2045:
2038:
2027:
2025:
1961:
1959:
1954:
1952:
1814:
1764:
1758:
1756:
1705:
1698:
1694:
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1682:
1680:
1609:
1565:
1517:
1513:
1511:
1443:
1439:
1437:
1393:
1366:
1360:January 2016
1357:
1342:Please help
1330:
1306:
1303:
1292:
1233:
1183:
1138:
1126:
1122:
1117:
1107:
1103:
1101:
1098:
1094:
1078:
1036:) floats in
1013:
1011:
1003:
992:
973:
943:
886:
885:
733:Smart fluids
678:Graham's law
584:
577:
573:
562:
548:Pascal's law
544:
527:
515:
370:Inequalities
248:
233:
217:
194:
164:
155:
145:
138:
131:
124:
112:
100:Please help
95:verification
92:
68:
61:
55:
54:Please help
51:
3301:(2): 2309.
2953:Hull (ship)
1071:, but once
752:Ferrofluids
653:Boyle's law
425:Hooke's law
403:Deformation
206:to make it
3381:Categories
3073:Cambridge
3027:References
2959:Hydrometer
2846:(with red
2727:Submarines
2720:See also:
2716:Submarines
1081:Archimedes
1034:pound coin
995:convection
805:Gay-Lussac
768:Scientists
668:Fick's law
648:Atmosphere
467:frictional
420:Plasticity
408:Elasticity
128:newspapers
117:"Buoyancy"
57:improve it
3244:1110.5264
3132:0813-1988
3007:Submarine
2995:Quicksand
2930:Dasymeter
2864:aluminium
2783:does not
2690:submarine
2637:does not
2446:ρ
2442:−
2390:−
2375:ρ
2274:ρ
2218:ρ
2153:ρ
2149:−
2111:net force
2074:ρ
1987:ρ
1924:ρ
1920:−
1907:∫
1893:ρ
1889:−
1871:∫
1868:−
1855:σ
1852:
1846:∫
1788:σ
1785:∮
1726:ρ
1654:Φ
1645:⟹
1633:Φ
1621:∇
1591:Φ
1588:∇
1585:−
1544:∇
1484:δ
1477:−
1462:σ
1417:σ
1331:does not
1295:dasymeter
1269:−
1163:−
969:displaced
845:Truesdell
775:Bernoulli
724:Rheometer
719:Rheometry
559:Newtonian
553:Viscosity
303:φ
291:−
220:July 2023
158:July 2014
63:talk page
3387:Buoyancy
3269:54556860
3209:Archived
3140:16986801
3043:(2008),
2875:See also
2840:baby oil
2738:Balloons
2710:unstable
2596:centroid
2333:salinity
2321:seawater
1661:constant
1085:Syracuse
999:continua
960:pressure
944:upthrust
887:Buoyancy
703:Rheology
608:Adhesion
588:Pressure
574:Buoyancy
519:Dynamics
357:Momentum
197:textbook
3358:(1889)
3274:8 April
3249:Bibcode
3146:13 June
3049:Longman
2989:Pontoon
2804:removed
2789:sources
2764:Density
2744:balloon
2731:ballast
2658:removed
2643:sources
2348:tension
2325:density
2048:is the
2035:density
2033:is the
1521:is the
1446:is the
1352:removed
1337:sources
1297:and of
1129:newtons
1073:ethanol
1038:mercury
976:density
971:fluid.
790:Charles
598:Liquids
512:Statics
457:Bending
257:gravity
208:neutral
202:Please
142:scholar
3330:
3267:
3184:
3138:
3130:
3086:
3055:
3019:Thrust
2751:Divers
2604:moment
2329:ρ
2026:where
1953:where
1693:where
1610:Then:
1438:where
1265:weight
1260:weight
1210:weight
1159:weight
1133:vacuum
986:or is
956:weight
942:), or
840:Stokes
835:Pascal
825:Navier
820:Newton
810:Graham
785:Cauchy
688:Plasma
583:
581:Mixing
576:
561:
543:
526:
514:
502:Fluids
435:Strain
430:Stress
413:linear
362:Energy
144:
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130:
123:
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3397:Force
3364:from
3265:S2CID
3239:arXiv
3212:(PDF)
3205:(PDF)
3126:(1).
2860:water
2688:As a
1966:force
1512:Here
1089:fluid
1069:water
952:fluid
948:force
815:Hooke
795:Euler
780:Boyle
638:Gases
149:JSTOR
135:books
3328:ISBN
3276:2021
3182:ISBN
3162:link
3148:2009
3136:OCLC
3128:ISSN
3084:ISBN
3053:ISBN
2897:Buoy
2787:any
2785:cite
2641:any
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2288:disp
2232:disp
2167:disp
2109:The
2041:disp
2001:disp
1815:The
1388:duck
1335:any
1333:cite
1012:The
830:Noll
800:Fick
352:Mass
337:Laws
121:news
3303:doi
3257:doi
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