Archimedes force acting on a body. Buoyancy force

Often scientific discoveries are the result of simple chance. But only people with a trained mind can appreciate the importance of a simple coincidence and draw far-reaching conclusions from it. It is thanks to the chain random events Archimedes' law appeared in physics, explaining the behavior of bodies in water.

Tradition

In Syracuse, legends were made about Archimedes. One day the ruler of this glorious city doubted the honesty of his jeweler. The crown made for the ruler had to contain a certain amount of gold. Archimedes was assigned to check this fact.

Archimedes established that bodies in air and water have different weights, and the difference is directly proportional to the density of the body being measured. By measuring the weight of the crown in air and water, and conducting a similar experiment with a whole piece of gold, Archimedes proved that there was an admixture of a lighter metal in the manufactured crown.

According to legend, Archimedes made this discovery in the bathtub, watching the water splash out. History is silent about what happened next to the dishonest jeweler, but the conclusion of the Syracuse scientist formed the basis of one of the most important laws of physics, which is known to us as Archimedes’ law.

Formulation

Archimedes presented the results of his experiments in his work “On Floating Bodies,” which, unfortunately, has survived to this day only in the form of fragments. Modern physics describes Archimedes' law as a cumulative force acting on a body immersed in a liquid. The buoyant force of a body in a liquid is directed upward; her absolute value equal to the weight of the displaced fluid.

The action of liquids and gases on a submerged body

Any object immersed in a liquid experiences pressure forces. At each point on the surface of the body, these forces are directed perpendicular to the surface of the body. If they were the same, the body would only experience compression. But pressure forces increase in proportion to depth, so the lower surface of the body experiences more compression than the upper. You can consider and add up all the forces acting on a body in water. The final vector of their direction will be directed upward, and the body will be pushed out of the liquid. The magnitude of these forces is determined by Archimedes' law. The floating of bodies is entirely based on this law and on various consequences from it. Archimedean forces also act in gases. It is thanks to these buoyancy forces that airships fly in the sky and Balloons: Air displacement makes them lighter than air.

Physical formula

The power of Archimedes can be clearly demonstrated by simple weighing. Weighing a training weight in a vacuum, in air and in water, you can see that its weight changes significantly. In a vacuum the weight of the weight is the same, in air it is slightly lower, and in water it is even lower.

If we take the weight of a body in a vacuum as P o, then its weight in the air can be described by the following formula: P in = P o - F a;

here P o - weight in vacuum;

As can be seen from the figure, any action involving weighing in water significantly lightens the body, so in such cases the Archimedes force must be taken into account.

For air this difference is negligible, so usually the weight of a body immersed in air environment, is described by the standard formula.

Density of the medium and Archimedes' force

Analyzing the simplest experiments with body weight in different environments, we can come to the conclusion that the weight of a body in various environments depends on the mass of the object and the density of the immersion environment. Moreover, the denser the medium, the greater the Archimedes force. Archimedes' law linked this relationship and the density of a liquid or gas is reflected in its final formula. What else affects given power? In other words, on what characteristics does Archimedes' law depend?

Formula

The Archimedean force and the forces that influence it can be determined using simple logical deductions. Let us assume that a body of a certain volume immersed in a liquid consists of the same liquid in which it is immersed. This assumption does not contradict any other premises. After all, the forces acting on a body in no way depend on the density of this body. In this case, the body will most likely be in equilibrium, and the buoyant force will be compensated by gravity.

Thus, the equilibrium of a body in water will be described as follows.

But the force of gravity, from the condition, is equal to the weight of the liquid that it displaces: the mass of the liquid is equal to the product of density and volume. By substituting known quantities, you can find out the weight of a body in a liquid. This parameter is described as ρV * g.

Substituting known values, we get:

This is Archimedes' law.

The formula we derived describes the density as the density of the body under study. But in initial conditions it was stated that body density identical to the density of the surrounding liquid. Thus, in this formula You can safely substitute the density value of the liquid. The visual observation that in a denser medium the buoyancy force is greater has received theoretical justification.

Application of Archimedes' Law

The first experiments demonstrating Archimedes' law have been known since school. Metal plate sinks in water, but, folded in the form of a box, it can not only stay afloat, but also carry a certain load. This rule is the most important conclusion from Archimedes' rule; it determines the possibility of constructing river and sea vessels taking into account their maximum capacity (displacement). After all, the density of sea and fresh water is different, and ships and submarines must take into account changes in this parameter when entering river mouths. An incorrect calculation can lead to disaster - the ship will run aground and significant efforts will be required to raise it.

Archimedes' Law is also necessary for submariners. The point is that the density sea water changes its value depending on the depth of immersion. Correct calculation of density will allow submariners to correctly calculate the air pressure inside the suit, which will affect the diver’s maneuverability and ensure his safe diving and ascent. Archimedes' law must also be taken into account when deep-sea drilling; huge drilling rigs lose up to 50% of their weight, which makes their transportation and operation less expensive.

Liquids and gases, according to which any body immersed in a liquid (or gas) is acted upon by this liquid (or gas) by a buoyant force equal to the weight of the liquid (gas) displaced by the body and directed vertically upward.

This law was discovered by the ancient Greek scientist Archimedes in the 3rd century. BC e. Archimedes described his research in his treatise “On Floating Bodies,” which is considered one of his last scientific works.

Below are the conclusions drawn from Archimedes' law.

The action of liquid and gas on a body immersed in them.

If you immerse a ball filled with air in water and release it, it will float up. The same thing will happen with a piece of wood, with a cork and many other bodies. What force makes them float?

A body immersed in water is affected by water pressure forces from all sides (Fig. A). At every point of the body these forces are directed perpendicular to its surface. If all these forces were equal, the body would experience only all-round compression. But at different depths the hydrostatic pressure is different: it increases with increasing depth. Therefore, the pressure forces applied to the lower parts of the body are greater than the pressure forces acting on the body from above.

If we replace all the pressure forces applied to a body immersed in water by one (resultant or resultant) force that has the same effect on the body as all these individual forces together, then the resultant force will be directed upward. This is what makes the body float. This force is called buoyant force, or Archimedean force(named after Archimedes, who first pointed out its existence and established what it depends on). On the image b it is designated as F A.

The Archimedean (buoyant) force acts on a body not only in water, but also in any other liquid, since in any liquid there is hydrostatic pressure, different at different depths. This force also acts in gases, which is why balloons and airships fly.

Thanks to the buoyant force, the weight of any body located in water (or any other liquid) turns out to be less than in air, and in air less than in airless space. This can be easily verified by weighing a weight using a training spring dynamometer, first in the air, and then lowering it into a vessel with water.

A decrease in weight also occurs when a body is transferred from a vacuum to air (or some other gas).

If the weight of a body in a vacuum (for example, in a vessel from which air has been pumped out) is equal to P0, then its weight in the air is:

Where F´A- Archimedean force acting on a given body in the air. For most bodies this force is negligible and can be neglected, i.e. we can assume that P air =P 0 =mg.

The weight of a body in liquid decreases much more than in air. If the body's weight is in the air P air =P 0, then the weight of the body in the liquid is equal to P liquid = P 0 - F A. Here F A- Archimedean force acting in a liquid. It follows that

Therefore, in order to find the Archimedean force acting on a body in any liquid, you need to weigh this body in air and in liquid. The difference between the obtained values will be the Archimedean (buoyant) force.

In other words, taking into account formula (1.32), we can say:

The buoyant force acting on a body immersed in a liquid is equal to the weight of the liquid displaced by this body.

The Archimedean force can also be determined theoretically. To do this, assume that a body immersed in a liquid consists of the same liquid in which it is immersed. We have the right to assume this, since the pressure forces acting on a body immersed in a liquid do not depend on the substance from which it is made. Then the Archimedean force applied to such a body F A will be balanced by the downward force of gravity mandg(Where m- mass of liquid in the volume of a given body):

But gravity is equal to the weight of the displaced fluid R. Thus.

Considering that the mass of a liquid is equal to the product of its density ρ on volume, formula (1.33) can be written as:

Where Vand— volume of displaced liquid. This volume is equal to the volume of that part of the body that is immersed in the liquid. If the body is completely immersed in liquid, then it coincides with the volume V of the whole body; if the body is partially immersed in liquid, then the volume Vand the displaced liquid is less than the volume V bodies (Fig. 1.39).

Formula (1.33) is also valid for the Archimedean force acting in a gas. Only in this case should the density of the gas and the volume of the displaced gas, and not the liquid, be substituted into it.

Taking into account the above, Archimedes' law can be formulated as follows:

Any body immersed in a liquid (or gas) at rest is acted upon by a buoyant force from this liquid (or gas), equal to the product liquid (or gas) density, acceleration free fall and the volume of that part of the body that is immersed in liquid (or gas).

The buoyant force acting on a body immersed in a liquid is equal to the weight of the liquid displaced by it.

"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 in royal palace without even bothering to get dressed.

However, what is true is true: it was Archimedes who discovered buoyancy principle. 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, the body floats if it average density less than the density of the liquid in which it is immersed.

Archimedes' principle can be interpreted in terms of 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 Bottom part The body is subjected to greater pressure than the upper one, 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 hull shell 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 gravitational force the gravity of the Earth pulling the ship to the bottom - and the ship floats.

The buoyant force acting on a body immersed in a liquid is equal to the weight of the liquid displaced by it.

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). In modern language, a body floats if its average density is less than the density of the liquid in which it is immersed.

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 turns out to be washed up on the shore - out aquatic environment the animal cannot move - 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).

A cube with edge a is 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 opposite side faces, equal and opposite directions - they try to compress the cube, but cannot influence its balance or movement. There remain forces acting on the upper and lower faces. Let be the depth of immersion of the upper face, be the density of the liquid, be the acceleration of gravity; then the pressure is on top edge equals

And on the bottom

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

,
,

where is the edge of the cube, and the force is directed downward, and the force is directed upward. Thus, the action of the liquid on the cube is reduced to two forces - and and is determined by their difference, which is the buoyancy force:
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. The value is equal to the volume of the body (cube) multiplied by the weight of one cubic centimeter liquid (if we take 1 cm as a unit of length). In other words, the buoyant force, which is often called the 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 ancient Greek scientist Archimedes, one of the greatest scientists on Earth.

If the body free form(Fig. 2) occupies a volume inside the liquid, then the effect of the liquid on the 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 a body, one must mentally remove a given body from volume V and fill (mentally) this volume with the same liquid. On the one hand, there is a vessel with a liquid at rest, on the other hand, inside the volume there is a body consisting of this 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.

Since the weight of the liquid in the volume of the body is equal to and balanced by the resultant pressure forces, its value is equal to the weight of the liquid in the volume, i.e. .

Having mentally made the reverse replacement - having placed this body in the volume and noting that this replacement will not affect the distribution of pressure forces on the surface of the volume, 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 within the volume of a 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 " specific gravity

Archimedes' law can be interpreted from the point of view of 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 by a solid body, the upward impulse of the collisions of 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.

Archimedes of Syracuse, c. 287–212 BC e.

Ancient Greek mathematician, inventor and natural philosopher. Little is known about his life. He proved a number of fundamental mathematical theorems and became famous thanks to the invention of various mechanisms that are still widely used both in everyday life and in the defense industry. Legend says that Archimedes died violent death, having fallen at the hands of a Roman soldier during the siege of Syracuse, not wanting to take refuge in the house, since he was completely absorbed geometric problem, drawn by him on the coastal sand.

Comments: 1

Nikolai Gorkavy

It began with the fact that King Hieron II invited Archimedes to his palace, poured him the best wine, asked about his health, and then showed him a golden crown made for the ruler by the court jeweler. “I don’t understand jewelry, but I understand people,” said Hieron. - And I think that the jeweler is deceiving me. The king took a gold bar from the table. - I gave him exactly the same ingot, and he made a crown out of it. The weight of the crown and the ingot are the same, my servant checked this. But I still have doubts: is there silver mixed into the crown? You, Archimedes, are the greatest scientist of Syracuse, and I ask you to check this, because if the king puts on a false crown, even the street boys will laugh at him...

The movement of a physical body in one dimension does not depend on its movement in the other two dimensions. For example, the flight path of a cannonball is a combination of two independent motion trajectories: uniform motion horizontally at the speed imparted to the cannonball, and uniformly accelerated motion vertically under the influence of gravity.

You may have experienced strange physical sensations in high-speed elevators: when the elevator starts going up (or slows down when moving down), you are pressed to the floor, and it seems to you that you are momentarily heavier; and at the moment of braking when moving up (or starting when moving down), the floor of the elevator literally disappears from under your feet. You yourself, perhaps without realizing it, are experiencing the effect of the principle of equivalence of inert and gravitational mass. When the elevator starts upward, it moves with an acceleration that is added to the acceleration due to gravity in the non-inertial (accelerating) frame associated with the elevator, and your weight increases. However, as soon as the elevator reaches “cruising speed”, it begins to move evenly, the “gain” in weight disappears, and your weight returns to your usual value. Thus, acceleration produces the same effect as gravity.

In classical Newtonian mechanics, any force is just an attractive or repulsive force, causing change the nature of the movement of the physical body. In modern quantum theories, however, the concept of force (now interpreted as the interaction between elementary particles) is interpreted somewhat differently. Force interaction is now considered to be the result of the exchange of an interaction carrier particle between two interacting particles. With this approach, the electromagnetic interaction between, for example, two electrons is due to the exchange of a photon between them, and similarly, the exchange of other intermediary particles leads to the emergence of three other types of interactions.

Newton's laws, depending on how you look at them, represent either the end of the beginning or the beginning of the end of classical mechanics. Either way, this is a turning point in history. physical science- a brilliant compilation of all the accumulated historical moment knowledge about movement physical bodies within physical theory, which is now commonly called classical mechanics. We can say that history began with Newton's laws of motion. modern physics and natural sciences in general.

Once the body begins to move, it tends to continue moving. Newton's first law of mechanics states: if a body moves, then in the absence external influences it will continue to move in a straight line and evenly until it is exposed to external force. This trend is called linear momentum. Similarly, a body rotating around its axis, in the absence of forces braking rotation, will continue to rotate, since the rotating body has a certain amount of motion, expressed in the form angular momentum momentum or, briefly, angular momentum or torque.

Today the standard model is called the theory the best way reflecting our ideas about the source material from which the Universe was originally built. It also describes how exactly matter is formed from these basic components, and the forces and mechanisms of interaction between them.

Galileo Galilei is one of the people who became famous for something that was not for which they should have enjoyed the well-deserved fame. Everyone remembers how this Italian naturalist, at the end of his life, was tried by the Inquisition on suspicion of heresy and forced to renounce the belief that the Earth revolves around the Sun. In fact, this trial practically did not influence the development of science - in contrast to the experiments previously carried out by Galileo and the conclusions he made on the basis of these experiments, which actually predetermined further development mechanics as a branch of physical science

Magnetic field at a point in space created by a small segment of a conductor along which flows electricity, proportional to the strength of the current, inversely proportional to the square of the distance from this point to the conductor and directed perpendicular to both the current and the direction to the conductor.

In a rotating reference frame, the observer experiences a force that moves him away from the axis of rotation. You've probably experienced unpleasant sensations when the car you're driving in takes a sharp turn. It seemed that now you would be thrown to the sidelines. And if you remember Newton’s laws of mechanics, it turns out that since you were literally pressed into the door, it means that some force was acting on you. It is usually called "centrifugal force". It is because of the centrifugal force that it is so breathtaking on sharp turns, when this force presses you against the side of the car. (Incidentally, this term, which comes from Latin words centrum (“center”) and fugus (“running”), introduced into scientific use in 1689 by Isaac Newton.)

And static gases.

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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 area cross section bodies, 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 in 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 toρ g V (\displaystyle \rho gV)

, and it is directed in the direction opposite to the direction of the gravitational field intensity vector. Another wording (whereρ t (\displaystyle \rho _(t)) - body density,ρ s (\displaystyle \rho _(s))