Archimedes principle
   HOME

TheInfoList



OR:

Archimedes' principle (also spelled Archimedes's principle) states that the upward
buoyant force Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight 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 pr ...
that is exerted on a body immersed in a
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
, whether fully or partially, is equal to the
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
of the fluid that the body displaces. Archimedes' principle is a
law of physics Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) a ...
fundamental to
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
. It was formulated by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
of
Syracuse Syracuse may refer to: Places Italy * Syracuse, Sicily, or spelled as ''Siracusa'' * Province of Syracuse United States *Syracuse, New York **East Syracuse, New York ** North Syracuse, New York * Syracuse, Indiana *Syracuse, Kansas *Syracuse, M ...
.


Explanation

In ''
On Floating Bodies ''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language work consisting of two books written by Archimedes of Syracuse (287 – c. 212 BC), one of the most important mathematicians, physicis ...
'', Archimedes suggested that (c. 246 BC): Archimedes' principle allows the buoyancy of any floating object partially or fully immersed in a fluid to be calculated. The downward force on the object is simply its weight. The upward, or buoyant, force on the object is that stated by Archimedes' principle above. Thus, the net force on the object is the difference between the magnitudes of the buoyant force and its weight. If this net force is positive, the object rises; if negative, the object sinks; and if zero, the object is neutrally buoyant—that is, it remains in place without either rising or sinking. In simple words, Archimedes' principle states that, when a body is partially or completely immersed in a fluid, it experiences an apparent loss in weight that is equal to the weight of the fluid displaced by the immersed part of the body(s).


Formula

Consider a cuboid immersed in a fluid, its top and bottom faces orthogonal to the direction of gravity (assumed constant across the cube's stretch). The fluid will exert a
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance '' normal'' is used in the geometric sense and means perpendicular, as oppos ...
on each face, but only the normal forces on top and bottom will contribute to buoyancy. The
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
difference between the bottom and the top face is directly proportional to the height (difference in depth of submersion). Multiplying the pressure difference by the area of a face gives a net force on the cuboid ⁠ ⁠—  the buoyancy ⁠ ⁠—  equaling in size the weight of the fluid displaced by the cuboid. By summing up sufficiently many arbitrarily small cuboids this reasoning may be extended to irregular shapes, and so, whatever the shape of the submerged body, the buoyant force is equal to the weight of the displaced fluid. :\text = \text - \text\, The
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). The weight of the object in the fluid is reduced, because of the force acting on it, which is called upthrust. In simple terms, the principle states that the buoyant force (Fb) on an object is equal to the weight of the fluid displaced by the object, or the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
( ρ) of the fluid multiplied by the submerged volume (V) times the
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
(g) We can express this relation in the equation: : F_ = \rho g V where F_ denotes the buoyant force applied onto the submerged object, \rho denotes the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the fluid, V represents the volume of the displaced fluid and g is the acceleration due to
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy. Suppose a rock's weight is measured as 10 newtons when suspended by a string in a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
with gravity acting on it. Suppose that, when the rock is lowered into the 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 buoyant 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 through the water than it is to pull it out of the water. For a fully submerged object, Archimedes' principle can 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 volume is : \frac = \frac .\, (This formula is used for example in describing the measuring principle of a
dasymeter A dasymeter was meant initially as a device to demonstrate the buoyant effect of gases like air (as shown in the adjacent pictures). A dasymeter which allows weighing acts as a densimeter used to measure the density of gases. Principle The ...
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 of a living person's body. It is a direct application of Archimedes' principle, t ...
.) Example: If you drop wood into water, buoyancy will keep it afloat. Example: A helium balloon in a moving car. When increasing speed or driving in a curve, the air moves in the opposite direction to the car's acceleration. However, due to buoyancy, the balloon is pushed "out of the way" by the air and will drift in the same direction as the car's acceleration. When an object is immersed in a liquid, the liquid exerts an upward force, which is known as the buoyant force, that is proportional to the weight of the displaced liquid. The sum force acting on the object, then, is equal to the difference between the weight of the object ('down' force) and the weight of displaced liquid ('up' force). Equilibrium, or neutral buoyancy, is achieved when these two weights (and thus forces) are equal.


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, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
. In this case the stress tensor is proportional to the identity tensor: : \sigma_=-p\delta_.\, Here ''δij'' is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. 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(p+\Phi)=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, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
can be transformed into a
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
with the help of the
Gauss theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vo ...
: : \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 of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
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 drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
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. \, The
net force Net Force may refer to: * Net force, the overall force acting on an object * ''NetForce'' (film), a 1999 American television film * Tom Clancy's Net Force, a novel series * Tom Clancy's Net Force Explorers, a young adult novel series {{disam ...
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 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, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
regardless of geographic location. : (''Note: If the fluid in question is
seawater Seawater, or salt water, is water from a sea or ocean. On average, seawater in the world's oceans has a salinity of about 3.5% (35 g/L, 35 ppt, 600 mM). This means that every kilogram (roughly one liter by volume) of seawater has appro ...
, it will not have the same
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
(''ρ'') at every location. For this reason, a ship may display a
Plimsoll line The waterline is the line where the hull of a ship meets the surface of the water. Specifically, it is also the name of a special marking, also known as an international load line, Plimsoll line and water line (positioned amidships), that indi ...
.)'' 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 restraint force T in order to remain fully submerged. An object which tends to sink will eventually have a
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance '' normal'' is used in the geometric sense and means perpendicular, as oppos ...
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, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance '' normal'' is used in the geometric sense and means perpendicular, as oppos ...
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.


Refinements

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. water striders) t ...
(capillarity) acting on the body. Moreover, Archimedes' principle has been found to break down in complex fluids. There is an exception to Archimedes' principle known as the bottom (or side) case. This occurs when a side of the object is touching the bottom (or side) of the vessel it is submerged in, and no liquid seeps in along that side. In this case, the net force has been found to be different from Archimedes' principle, owing to the fact that since no fluid seeps in on that side, the symmetry of pressure is broken.


Principle of floatation

Archimedes' principle shows the buoyant force and displacement of fluid. However, the concept of Archimedes' principle can be applied when considering why objects float. Proposition 5 of Archimedes' treatise ''
On Floating Bodies ''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language work consisting of two books written by Archimedes of Syracuse (287 – c. 212 BC), one of the most important mathematicians, physicis ...
'' states that In other words, for an object floating on a liquid surface (like a boat) or floating submerged in a fluid (like 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. The term is also sometimes used historically or colloquially to refer to remotely op ...
in water or
dirigible An airship or dirigible balloon is a type of aerostat or lighter-than-air aircraft that can navigate through the air under its own power. Aerostats gain their lift from a lifting gas that is less dense than the surrounding air. In early ...
in air) the weight of the displaced liquid equals the weight of the object. Thus, only in the special case of floating does the buoyant force acting on an object equal the objects weight. Consider a 1-ton block of solid iron. As iron is nearly eight times as dense as water, it displaces only 1/8 ton of water when submerged, which is not enough to keep it afloat. Suppose the same iron block is reshaped into a bowl. It still weighs 1 ton, but when it is put in water, it displaces a greater volume of water than when it was a block. The deeper the iron bowl is immersed, the more water it displaces, and the greater the buoyant force acting on it. When the buoyant force equals 1 ton, it will sink no farther. When any boat displaces a weight of water equal to its own weight, it floats. This is often called the "principle of flotation": A floating object displaces a weight of fluid equal to its own weight. Every ship, submarine, and dirigible must be designed to displace a weight of fluid at least equal to its own weight. A 10,000-ton ship's hull must be built wide enough, long enough and deep enough to displace 10,000 tons of water and still have some hull above the water to prevent it from sinking. It needs extra hull to fight waves that would otherwise fill it and, by increasing its mass, cause it to submerge. The same is true for vessels in air: a dirigible that weighs 100 tons needs to displace 100 tons of air. If it displaces more, it rises; if it displaces less, it falls. If the dirigible displaces exactly its weight, it hovers at a constant altitude. While they are related to it, the principle of flotation and the concept that a submerged object displaces a volume of fluid equal to its own volume are ''not'' Archimedes' principle. Archimedes' principle, as stated above, equates the ''buoyant force'' to the weight of the fluid displaced. One common point of confusion regarding Archimedes' principle is the meaning of displaced volume. Common demonstrations involve measuring the rise in water level when an object floats on the surface in order to calculate the displaced water. This measurement approach fails with a buoyant submerged object because the rise in the water level is directly related to the volume of the object and not the mass (except if the effective density of the object equals exactly the fluid density).


Eureka

Archimedes reportedly exclaimed "Eureka" after he realized how to detect whether a crown is made of impure gold. While he did not use Archimedes' principle in the widespread tale and used displaced water only for measuring the volume of the crown, there is an alternative approach using the principle: Balance the crown and pure gold on a scale in the air and then put the scale into water. According to Archimedes' principle, if the density of the crown differs from the density of pure gold, the scale will get out of balance under water.


References


External links

* {{Authority control Fluid dynamics
Principle A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, such as the l ...
Force Buoyancy Scientific laws