Archimedes' Law: the history of the discovery and the essence of the phenomenon for dummies. Buoyancy force

ARCHIMEDES' LAW– the law of statics of liquids and gases, according to which a body immersed in a liquid (or gas) is acted upon by a buoyant force equal to the weight of the liquid in the volume of the body.

The fact that a certain force acts on a body immersed in water is well known to everyone: heavy bodies seem to become lighter - for example, our own body when immersed in a bath. When swimming in a river or in the sea, you can easily lift and move very heavy stones along the bottom - ones that we cannot lift on land; the same phenomenon is observed when, for some reason, a whale is washed up on the shore - the animal cannot move outside the aquatic environment - its weight exceeds the capabilities of its muscular system. At the same time, lightweight bodies resist immersion in water: sinking a ball the size of a small watermelon requires both strength and dexterity; It will most likely not be possible to immerse a ball with a diameter of half a meter. It is intuitively clear that the answer to the question - why a body floats (and another sinks) is closely related to the effect of the liquid on the body immersed in it; one cannot be satisfied with the answer that light bodies float and heavy ones sink: a steel plate, of course, will sink in water, but if you make a box out of it, then it can float; however, her weight did not change. To understand the nature of the force acting on a submerged body from the side of a liquid, it is enough to consider a simple example (Fig. 1).

Cube with an edge a immersed in water, and both the water and the cube are motionless. It is known that the pressure in a heavy liquid increases in proportion to depth - it is obvious that a higher column of liquid presses more strongly on the base. It is much less obvious (or not at all obvious) that this pressure acts not only downwards, but also sideways and upwards with the same intensity - this is Pascal's law.

If we consider the forces acting on the cube (Fig. 1), then due to the obvious symmetry, the forces acting on the opposite side faces are equal and oppositely directed - they try to compress the cube, but cannot affect its balance or movement. There remain forces acting on the upper and lower faces. Let h– depth of immersion of the upper face, r– fluid density, g– acceleration of gravity; then the pressure on the upper face is equal to

r· g · h = p 1

and on the bottom

r· g(h+a)= p 2

The pressure force is equal to the pressure multiplied by the area, i.e.

F 1 = p 1 · a\up122, F 2 = p 2 · a\up122 , where a- cube edge,

and strength F 1 is directed downwards and the force F 2 – up. Thus, the action of the liquid on the cube is reduced to two forces - F 1 and F 2 and is determined by their difference, which is the buoyancy force:

F 2 – F 1 =r· g· ( h+a)a\up122 – r gha· a 2 = pga 2

The force is buoyant, since the lower edge is naturally located below the upper one and the force acting upward is greater than the force acting downward. Magnitude F 2 – F 1 = pga 3 is equal to the volume of the body (cube) a 3 multiplied by the weight of one cubic centimeter of liquid (if we take 1 cm as a unit of length). In other words, the buoyant force, often called Archimedean force, is equal to the weight of the liquid in the volume of the body and is directed upward. This law was established by the ancient Greek scientist Archimedes, one of the greatest scientists on Earth.

If a body of arbitrary shape (Fig. 2) occupies a volume inside the liquid V, then the effect of a liquid on a body is completely determined by the pressure distributed over the surface of the body, and we note that this pressure is completely independent of the material of the body - (“the liquid doesn’t care what to press on”).

To determine the resulting pressure force on the surface of the body, you need to mentally remove from the volume V given body and fill (mentally) this volume with the same liquid. On the one hand, there is a vessel with liquid at rest, on the other hand, inside the volume V– a body consisting of a given liquid, and this body is in equilibrium under the influence of its own weight (the liquid is heavy) and the pressure of the liquid on the surface of the volume V. Since the weight of liquid in the volume of a body is equal to pgV and is balanced by the resultant pressure forces, then its value is equal to the weight of the liquid in the volume V, i.e. pgV.

Having mentally made the reverse replacement - placing it in volume V given body and noting that this replacement will not affect the distribution of pressure forces on the surface of the volume V, we can conclude: a body immersed in a heavy liquid at rest is acted upon by an upward force (Archimedean force), equal to the weight of the liquid in the volume of the given body.

Similarly, it can be shown that if a body is partially immersed in a liquid, then the Archimedean force is equal to the weight of the liquid in the volume of the immersed part of the body. If in this case the Archimedean force is equal to the weight, then the body floats on the surface of the liquid. Obviously, if, during complete immersion, the Archimedean force is less than the weight of the body, then it will drown. Archimedes introduced the concept of "specific gravity" g, i.e. weight per unit volume of a substance: g = pg; if we assume that for water g= 1, then a solid body of matter for which g> 1 will drown, and when g < 1 будет плавать на поверхности; при g= 1 a body can float (hover) inside a liquid. In conclusion, we note that Archimedes' law describes the behavior of balloons in the air (at rest at low speeds).

Vladimir Kuznetsov

Archimedes' law is the law of statics of liquids and gases, according to which a body immersed in a liquid (or gas) is acted upon by a buoyant force equal to the weight of the liquid in the volume of the body.

Background

"Eureka!" (“Found!”) - this is the exclamation, according to legend, made by the ancient Greek scientist and philosopher Archimedes, who discovered the principle of repression. Legend has it that the Syracusan king Heron II asked the thinker to determine whether his crown was made of pure gold without harming the royal crown itself. It was not difficult to weigh the crown of Archimedes, but this was not enough - it was necessary to determine the volume of the crown in order to calculate the density of the metal from which it was cast and determine whether it was pure gold. Then, according to legend, Archimedes, preoccupied with thoughts about how to determine the volume of the crown, plunged into the bath - and suddenly noticed that the water level in the bath had risen. And then the scientist realized that the volume of his body displaced an equal volume of water, therefore, the crown, if lowered into a basin filled to the brim, would displace a volume of water equal to its volume. A solution to the problem was found and, according to the most common version of the legend, the scientist ran to report his victory to the royal palace, without even bothering to get dressed.

However, what is true is true: it was Archimedes who discovered the principle of buoyancy. If a solid body is immersed in a liquid, it will displace a volume of liquid equal to the volume of the part of the body immersed in the liquid. The pressure that previously acted on the displaced liquid will now act on the solid body that displaced it. And, if the buoyant force acting vertically upward turns out to be greater than the force of gravity pulling the body vertically downward, the body will float; otherwise it will sink (drown). Speaking modern language, a body floats if its average density is less than the density of the liquid in which it is immersed.

Archimedes' Law and Molecular Kinetic Theory

In a fluid at rest, pressure is produced by the impacts of moving molecules. When a certain volume of liquid is displaced solid body, the upward impulse of the collisions of the molecules will fall not on the liquid molecules displaced by the body, but on the body itself, which explains the pressure exerted on it from below and pushing it towards the surface of the liquid. If the body is completely immersed in the liquid, the buoyant force will continue to act on it, since the pressure increases with increasing depth, and the lower part of the body is subjected to more pressure than the upper, which is where the buoyant force arises. This is the explanation of buoyant force at the molecular level.

This pushing pattern explains why a ship made of steel, which is much denser than water, remains afloat. The fact is that the volume of water displaced by a ship is equal to the volume of steel submerged in water plus the volume of air contained inside the ship's hull below the waterline. If we average the density of the shell of the hull and the air inside it, it turns out that the density of the ship (as a physical body) is less than the density of water, therefore the buoyancy force acting on it as a result of upward impulses of impact of water molecules turns out to be higher than the gravitational force of attraction of the Earth, pulling the ship towards to the bottom - and the ship floats.

Formulation and explanations

The fact that a certain force acts on a body immersed in water is well known to everyone: heavy bodies seem to become lighter - for example, our own body when immersed in a bath. When swimming in a river or sea, you can easily lift and move very heavy stones along the bottom - ones that cannot be lifted on land. At the same time, lightweight bodies resist immersion in water: sinking a ball the size of a small watermelon requires both strength and dexterity; It will most likely not be possible to immerse a ball with a diameter of half a meter. It is intuitively clear that the answer to the question - why a body floats (and another sinks) is closely related to the effect of the liquid on the body immersed in it; one cannot be satisfied with the answer that light bodies float and heavy ones sink: a steel plate, of course, will sink in water, but if you make a box out of it, then it can float; however, her weight did not change.

The existence of hydrostatic pressure results in a buoyant force acting on any body in a liquid or gas. Archimedes was the first to determine the value of this force in liquids experimentally. Archimedes' law is formulated as follows: a body immersed in a liquid or gas is subject to a buoyancy force equal to the weight of the amount of liquid or gas that is displaced by the immersed part of the body.

Formula

The Archimedes force acting on a body immersed in a liquid can be calculated by the formula: F A = ρ f gV Fri,

where ρl is the density of the liquid,

g – free fall acceleration,

Vpt is the volume of the body part immersed in the liquid.

The behavior of a body located in a liquid or gas depends on the relationship between the modules of gravity Ft and the Archimedean force FA, which act on this body. The following three cases are possible:

1) Ft > FA – the body sinks;

2) Ft = FA – the body floats in liquid or gas;

3) Ft< FA – тело всплывает до тех пор, пока не начнет плавать.

And static gases.

Encyclopedic YouTube

• 1 / 5

  Archimedes' law is formulated as follows: a body immersed in a liquid (or gas) is acted upon by a buoyant force equal to the weight of the liquid (or gas) in the volume of the immersed part of the body. The force is called by the power of Archimedes:

  F A = ​​ρ g V , (\displaystyle (F)_(A)=\rho (g)V,)

  Where ρ (\displaystyle \rho )- density of liquid (gas), g (\displaystyle (g)) is the acceleration of free fall, and V (\displaystyle V)- the volume of the submerged part of the body (or the part of the volume of the body located below the surface). If a body floats on the surface (uniformly moves up or down), then the buoyancy force (also called the Archimedean force) is equal in magnitude (and opposite in direction) to the force of gravity acting on the volume of liquid (gas) displaced by the body, and is applied to the center of gravity of this volume.

  It should be noted that the body must be completely surrounded by liquid (or intersect with the surface of the liquid). So, for example, Archimedes' law cannot be applied to a cube that lies at the bottom of a tank, hermetically touching the bottom.

  As for a body that is in a gas, for example in air, to find the lifting force it is necessary to replace the density of the liquid with the density of the gas. For example, a helium balloon flies upward due to the fact that the density of helium is less than the density of air.

  Archimedes' law can be explained using the difference in hydrostatic pressure using the example of a rectangular body.

  P B − P A = ρ g h (\displaystyle P_(B)-P_(A)=\rho gh) F B − F A = ​​ρ g h S = ρ g V , (\displaystyle F_(B)-F_(A)=\rho ghS=\rho gV,)

  Where P A, P B- pressure at points A And B, ρ - fluid density, h- level difference between points A And B, S- horizontal cross-sectional area of ​​the body, V- volume of the immersed part of the body.

  In theoretical physics, Archimedes' law is also used in integral form:

  F A = ​​∬ S p d S (\displaystyle (F)_(A)=\iint \limits _(S)(p(dS))),

  Where S (\displaystyle S) - surface area, p (\displaystyle p)- pressure at an arbitrary point, integration is carried out over the entire surface of the body.

  In the absence of a gravitational field, that is, in a state of weightlessness, Archimedes' law does not work. Astronauts are quite familiar with this phenomenon. In particular, in zero gravity there is no phenomenon of (natural) convection, therefore, for example, air cooling and ventilation of living compartments spacecraft produced forcibly by fans.

  Generalizations

  A certain analogue of Archimedes' law is also valid in any field of forces that act differently on a body and on a liquid (gas), or in a non-uniform field. For example, this refers to the field of inertia forces (for example, centrifugal force) - centrifugation is based on this. An example for a field of a non-mechanical nature: a diamagnetic material in a vacuum is displaced from a region of a magnetic field of higher intensity to a region of lower intensity.

  Derivation of Archimedes' law for a body of arbitrary shape

  Hydrostatic pressure of fluid at depth h (\displaystyle h) There is p = ρ g h (\displaystyle p=\rho gh). At the same time we consider ρ (\displaystyle \rho ) fluids and gravitational field strength constant values, A h (\displaystyle h)- parameter. Let's take a body of arbitrary shape that has a non-zero volume. Let us introduce a right orthonormal coordinate system O x y z (\displaystyle Oxyz), and choose the direction of the z axis to coincide with the direction of the vector g → (\displaystyle (\vec (g))). We set zero along the z axis on the surface of the liquid. Let us select an elementary area on the surface of the body d S (\displaystyle dS). It will be acted upon by the fluid pressure force directed into the body, d F → A = − p d S → (\displaystyle d(\vec (F))_(A)=-pd(\vec (S))). To get the force that will act on the body, take the integral over the surface:

  F → A = − ∫ S p d S → = − ∫ S ρ g h d S → = − ρ g ∫ S h d S → = ∗ − ρ g ∫ V g r a d (h) d V = ∗ ∗ − ρ g ∫ V e → z d V = − ρ g e → z ∫ V d V = (ρ g V) (− e → z) (\displaystyle (\vec (F))_(A)=-\int \limits _(S)(p \,d(\vec (S)))=-\int \limits _(S)(\rho gh\,d(\vec (S)))=-\rho g\int \limits _(S)( h\,d(\vec (S)))=^(*)-\rho g\int \limits _(V)(grad(h)\,dV)=^(**)-\rho g\int \limits _(V)((\vec (e))_(z)dV)=-\rho g(\vec (e))_(z)\int \limits _(V)(dV)=(\ rho gV)(-(\vec (e))_(z)))

  When moving from the surface integral to the volume integral, we use the generalized Ostrogradsky-Gauss theorem.

  ∗ h (x, y, z) = z; ∗ ∗ g r a d (h) = ∇ h = e → z (\displaystyle ()^(*)h(x,y,z)=z;\quad ^(**)grad(h)=\nabla h=( \vec (e))_(z))

  We find that the modulus of the Archimedes force is equal to ρ g V (\displaystyle \rho gV), and it is directed to the side, opposite direction vector of gravitational field strength.

  Another wording (where ρ t (\displaystyle \rho _(t))- body density, ρ s (\displaystyle \rho _(s))- the density of the medium in which it is immersed).

  The dependence of pressure in a liquid or gas on the depth of immersion of a body leads to the appearance of a buoyant force (or otherwise the Archimedes force), acting on any body immersed in a liquid or gas.

  The Archimedean force is always directed opposite to the force of gravity, therefore the weight of a body in a liquid or gas is always less than the weight of this body in a vacuum.

  The magnitude of the Archimedean force is determined by Archimedes' law.

  

  The law is named after the ancient Greek scientist Archimedes, who lived in the 3rd century BC.

  

  The discovery of the fundamental law of hydrostatics is the greatest achievement of ancient science. Most likely, you already know the legend about how Archimedes discovered his law: “One day the Syracusan king Hiero called him and said.... And what happened next? ...

  Archimedes' law was first mentioned by him in his treatise "On Floating Bodies." Archimedes wrote: “bodies heavier than the liquid, immersed in this liquid, will sink until they reach the very bottom, and in the liquid they will become lighter by the weight of the liquid in a volume equal to the volume of the immersed body.”

  Another formula for determining Archimedean force:

  

  It is interesting that the Archimedes force is zero when a body immersed in a liquid is tightly pressed to the bottom with its entire base.

  WEIGHT OF A BODY IMMEDIED ​​IN A LIQUID (OR GAS)

  Body weight in vacuum Po=mg.
  If a body is immersed in a liquid or gas,
  That P = Po - Fa = Po - Pzh

  The weight of a body immersed in a liquid or gas is reduced by the amount of buoyant force acting on the body.

  Or else:

  A body immersed in a liquid or gas loses as much weight as the liquid it displaced weighs.

  BOOKSHELF

  TURNS OUT

  The density of organisms living in water is almost no different from the density of water, so they don’t need strong skeletons!

  

  Fish regulate their diving depth by changing the average density of their body. To do this, they only need to change the volume of the swim bladder by contracting or relaxing the muscles.

  Off the coast of Egypt, there is an amazing fagak fish. The approach of danger forces the fagak to quickly swallow water. At the same time, rapid decomposition of food products occurs in the fish esophagus with the release of a significant amount of gases. Gases fill not only the active cavity of the esophagus, but also the blind outgrowth attached to it. As a result, the phagak's body swells greatly, and, in accordance with Archimedes' law, it quickly floats to the surface of the reservoir. Here he swims, hanging upside down, until the gases released in his body disappear. After this, gravity lowers it to the bottom of the reservoir, where it takes refuge among the bottom algae.

  

  Chilim (water chestnut) produces heavy fruits under water after flowering. These fruits are so heavy that they can easily drag the entire plant to the bottom. However, at this time, in the chilim growing in deep water, swellings appear on the petioles of the leaves, giving it the necessary lifting force, and it does not sink.

  One of the first physical laws studied by students high school. Any adult remembers at least approximately this law, no matter how far he is from physics. But sometimes it is useful to return to the exact definitions and formulations - and understand the details of this law that may have been forgotten.

  What does Archimedes' law say?

  There is a legend that the ancient Greek scientist discovered his famous law while taking a bath. Having plunged into a container filled to the brim with water, Archimedes noticed that the water splashed out - and experienced an epiphany, instantly formulating the essence of the discovery.

  Most likely, in reality the situation was different, and the discovery was preceded by long observations. But this is not so important, because in any case, Archimedes managed to discover the following pattern:

  • plunging into any liquid, bodies and objects experience several multidirectional forces at once, but directed perpendicular to their surface;
  • the final vector of these forces is directed upward, so any object or body, finding itself in a liquid at rest, experiences pushing;
  • in this case, the buoyancy force is exactly equal to the coefficient that is obtained if the product of the volume of the object and the density of the liquid is multiplied by the acceleration of free fall.

  So, Archimedes established that a body immersed in a liquid displaces a volume of liquid that is equal to the volume of the body itself. If only part of a body is immersed in a liquid, then it will displace the liquid, the volume of which will be equal to the volume of only the part that is immersed.

  The same principle applies to gases - only here the volume of the body must be correlated with the density of the gas.

  You can formulate a physical law a little more simply - the force that pushes an object out of a liquid or gas is exactly equal to the weight of the liquid or gas displaced by this object during immersion.

  The law is written in the form of the following formula:

  
  

  What is the significance of Archimedes' law?

  The pattern discovered by the ancient Greek scientist is simple and completely obvious. But at the same time its significance for everyday life cannot be overstated.

  It is thanks to the knowledge of the pushing of bodies by liquids and gases that we can build river and sea vessels, as well as airships and balloons for aeronautics. Heavy metal ships do not sink due to the fact that their design takes into account Archimedes' law and numerous consequences from it - they are built so that they can float on the surface of the water, and do not sink. Aeronautics operate on a similar principle - they use the buoyancy of air, becoming, as it were, lighter in the process of flight.