How buoyant force works. How to calculate buoyancy (buoyancy). Fluid media and static pressure in them

Liquids and gases, according to which, on any body immersed in a liquid (or gas), a buoyant force acts from this liquid (or gas), equal to the weight of the liquid (gas) displaced by the body and directed vertically upwards.

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

The following are the findings from Archimedes' law.

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

If you submerge an air-filled ball in water and release it, it will float. The same will happen with wood chips, cork and many other bodies. What force makes them float?

A body immersed in water is subjected to water pressure from all sides (Fig. A). At each point of the body, these forces are directed perpendicular to its surface. If all these forces were the same, 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 turn out to be greater than the pressure forces acting on the body from above.

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

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

Due to the buoyancy force, the weight of any body in water (or in any other liquid) is less than in air, and less in air than in airless space. It is easy to verify this by weighing the weight with the help of a training spring dynamometer, first in the air, and then lowering it into a vessel with water.

Weight reduction also occurs when a body is transferred from vacuum to air (or some other gas).

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

Where F´ A is the Archimedean force acting on a given body in 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 the body in liquid decreases much more than in air. If the weight of the body in the air P air =P 0, then the weight of the body in the fluid is P liquid \u003d P 0 - F A. Here F A is the Archimedean force acting in the fluid. Hence it follows that

Therefore, in order to find the Archimedean force acting on a body in any liquid, this body must be weighed in air and in the 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, suppose that a body immersed in a fluid consists of the same fluid 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 f is the mass of liquid in the volume of a given body):

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

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

Where Vand is the volume of the displaced fluid. 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 the liquid, then it coincides with the volume V of the whole body; if the body is partially immersed in the liquid, then the volume Vand volume of displaced fluid V bodies (Fig. 1.39).

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

In view of the foregoing, Archimedes' law can be formulated as follows:

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

Despite the obvious differences in the properties of liquids and gases, in many cases their behavior is determined by the same parameters and equations, which makes it possible to use a unified approach to studying the properties of these substances.

In mechanics, gases and liquids are considered as continuous media. It is assumed that the molecules of a substance are distributed continuously in the part of space they occupy. In this case, the density of a gas depends significantly on pressure, while the situation is different for a liquid. Usually, when solving problems, this fact is neglected, using the generalized concept of an incompressible fluid, the density of which is uniform and constant.

Definition 1

Pressure is defined as the normal force $F$ acting from the side of the fluid per unit area $S$.

$ρ = \frac(\Delta P)(\Delta S)$.

Remark 1

Pressure is measured in pascals. One Pa is equal to a force of 1 N acting on a unit area of 1 sq. m.

In a state of equilibrium, the pressure of a liquid or gas is described by Pascal's law, according to which the pressure on the surface of the liquid, produced by external forces, is transferred by the liquid equally in all directions.

In mechanical equilibrium, the horizontal pressure of a fluid is always the same; consequently, the free surface of a static fluid is always horizontal (except in cases of contact with the walls of the vessel). If we take into account the condition of incompressibility of the liquid, then the density of the considered medium does not depend on pressure.

Imagine a certain volume of fluid bounded by a vertical cylinder. We denote the cross section of the liquid column $S$, its height $h$, the liquid density $ρ$, and the weight $P=ρgSh$. Then the following is true:

$p = \frac(P)(S) = \frac(ρgSh)(S) = ρgh$,

where $p$ is the pressure on the bottom of the vessel.

It follows that the pressure varies linearly with altitude. In this case, $ρgh$ is the hydrostatic pressure, the change of which explains the emergence of the Archimedes force.

Formulation of the Law of Archimedes

The law of Archimedes, one of the basic laws of hydrostatics and aerostatics, states: a body immersed in a liquid or gas is subjected to a buoyant or lifting force equal to the weight of the volume of liquid or gas displaced by the part of the body immersed in the liquid or gas.

Remark 2

The emergence of the Archimedean force is due to the fact that the medium - liquid or gas - tends to occupy the space taken away by the body immersed in it; while the body is pushed out of the medium.

Hence the second name for this phenomenon is buoyancy or hydrostatic lift.

The buoyancy force does not depend on the shape of the body, as well as on the composition of the body and its other characteristics.

The emergence of the Archimedean force is due to the difference in pressure of the medium at different depths. For example, the pressure on the lower layers of water is always greater than on the upper layers.

The manifestation of the Archimedes force is possible only in the presence of gravity. So, for example, on the Moon the buoyancy force will be six times less than on Earth for bodies of equal volumes.

The Emergence of the Force of Archimedes

Imagine any liquid medium, for example, ordinary water. Mentally select an arbitrary volume of water by a closed surface $S$. Since the entire liquid is in mechanical equilibrium by condition, the volume allocated by us is also static. This means that the resultant and the moment of external forces acting on this limited volume take on zero values. External forces in this case are the weight of a limited volume of water and the pressure of the surrounding fluid on the outer surface $S$. In this case, it turns out that the resultant $F$ of the forces of hydrostatic pressure experienced by the surface $S$ is equal to the weight of the volume of liquid that was bounded by the surface $S$. In order for the total moment of the external forces to vanish, the resultant $F$ must be directed upwards and pass through the center of mass of the selected liquid volume.

Now we denote that instead of this conditional limited liquid, any solid body of the corresponding volume was placed in the medium. If the condition of mechanical equilibrium is observed, then no changes will occur from the side of the environment, including the pressure acting on the surface $S$ will remain the same. Thus, we can give a more precise formulation of the law of Archimedes:

Remark 3

If a body immersed in a liquid is in mechanical equilibrium, then from the side of the environment surrounding it, the buoyant force of hydrostatic pressure acts on it, numerically equal to the weight of the medium in the volume displaced by the body.

The buoyant force is directed upward and passes through the center of mass of the body. So, according to the law of Archimedes for the buoyant force, the following is true:

$F_A = ρgV$, where:

$V_A$ - buoyancy force, H;
$ρ$ - liquid or gas density, $kg/m^3$;
$V$ - volume of the body immersed in the medium, $m^3$;
$g$ - free fall acceleration, $m/s^2$.

The buoyant force acting on the body is opposite in direction to the force of gravity, therefore the behavior of the immersed body in the medium depends on the ratio of the modules of gravity $F_T$ and Archimedean force $F_A$. There are three possible cases here:

$F_T$ > $F_A$. The force of gravity exceeds the buoyant force, hence the body sinks/falls;
$F_T$ = $F_A$. The force of gravity equalizes with the buoyant force, so the body "hangs" in the fluid;
$F_T$

The reason for the emergence of the Archimedean force is the difference in pressure of the medium at different depths. Therefore, the Archimedes force arises only in the presence of gravity. On the Moon, it will be six times, and on Mars - 2.5 times less than on Earth.

There is no Archimedean force in weightlessness. If we imagine that gravity on Earth suddenly disappeared, then all the ships in the seas, oceans and rivers from the slightest push will go to any depth. But the surface tension of water, which does not depend on gravity, will not let them rise up, so they will not be able to take off, they will all drown.

How is the power of Archimedes manifested?

The magnitude of the Archimedean force depends on the volume of the immersed body and the density of the medium in which it is located. Its exact in the modern view: a body immersed in a liquid or gaseous medium in the field of gravity is affected by a buoyant force exactly equal to the weight of the medium displaced by the body, that is, F = ρgV, where F is the Archimedes force; ρ is the density of the medium; g is the free fall acceleration; V is the volume of the liquid (gas) displaced by the body or part of it immersed.

If in fresh water a buoyancy force of 1 kg (9.81 n) acts on each liter of the volume of an immersed body, then in sea water, the density of which is 1.025 kg * cu. dm, the Archimedes force of 1 kg 25 g will act on the same liter of volume. For a person of average build, the difference in the support force of sea and fresh water will be almost 1.9 kg. Therefore, swimming in the sea is easier: imagine that you need to swim at least a pond without a current with a two-kilogram dumbbell in your belt.

The Archimedean force does not depend on the shape of the immersed body. Take an iron cylinder, measure its strength from the water. Then roll this cylinder into a sheet, immerse in water flat and edgewise. In all three cases, the strength of Archimedes will be the same.

At first glance, it is strange, but if the sheet is immersed flat, then the decrease in the pressure difference for a thin sheet is compensated by an increase in its area perpendicular to the water surface. And when immersed by an edge, on the contrary, the small area of \u200b\u200bthe edge is compensated by the greater height of the sheet.

If the water is very strongly saturated with salts, which is why its density has become higher than the density of the human body, then a person who cannot swim will not drown in it. In the Dead Sea in Israel, for example, tourists can lie on the water for hours without moving. True, it is still impossible to walk on it - the area of \u200b\u200bsupport turns out to be small, a person falls into the water up to his throat until the weight of the immersed part of the body is equal to the weight of the water displaced by him. However, if you have a certain amount of imagination, you can add up the legend of walking on water. But in kerosene, the density of which is only 0.815 kg * cu. dm, will not be able to stay on the surface and a very experienced swimmer.

Archimedean force in dynamics

The fact that ships float thanks to the power of Archimedes is known to everyone. But fishermen know that Archimedean force can also be used in dynamics. If a large and strong fish (taimen, for example) has caught on, then slowly pulling it up to the net (pulling it out) is not: it will break the line and leave. You need to first pull lightly when she leaves. Feeling the hook at the same time, the fish, trying to get rid of it, will rush towards the fisherman. Then you need to pull very hard and sharply so that the fishing line does not have time to break.

In water, the body of a fish weighs almost nothing, but its mass is preserved with inertia. With this method of fishing, the Archimedean force, as it were, will give the fish a tail, and the prey itself will flop at the feet of the fisherman or into his boat.

Archimedean force in the air

Archimedean force acts not only in liquids, but also in gases. Thanks to her, balloons and airships (zeppelins) fly. 1 cu. m of air under normal conditions (20 degrees Celsius at sea level) weighs 1.29 kg, and 1 kg of helium - 0.21 kg. That is, 1 cubic meter of a filled shell is capable of lifting a load of 1.08 kg. If the shell is 10 m in diameter, then its volume will be 523 cubic meters. m. Having done it from a lightweight synthetic material, we get a lifting force of about half a ton. Aeronauts call the Archimedean force in the air the floating force.

If air is pumped out of the balloon without letting it wrinkle, then each cubic meter of it will pull up all 1.29 kg. An increase of more than 20% in lift is technically very tempting, but helium is expensive, and hydrogen is explosive. Therefore, projects of vacuum airships are born from time to time. But materials capable of withstanding a large (about 1 kg per sq. cm) atmospheric pressure from the outside on the shell, modern technology is not yet able to create.

Archimedes is one of the most famous and great scientists who laid the foundation for modern science. Not all of his discoveries are known to the general public. Usually everyone remembers only what they taught at school, although his other experiments are no less interesting and useful to society.

Sovereign's crown

There is a very famous legend about the crown of Emperor Hieron, some historians call it a sacrificial crown. It is authentically known that the sovereign asked Archimedes to figure out whether his master jeweler turned out to be a deceiver, whether he spent all the gold on the crown or stole something for himself. At that time, it was a very difficult task, and it took the great scientist a lot of time and luck to solve this riddle. One day he was taking a bath. When he descended into it, he did not notice that it was too full and some water poured out of the bath, after which Archimedes "Eureka!" This word in Greek means "found". The Greek philosopher really found a solution, because today every child knows that with any element in a vessel filled with water, the volume of displaced water will be equal to the volume of the immersed element.

Thanks to this hunch, Archimedes helped the Greek king to identify the liar jeweler and find out the truth. Since the jeweler was given a whole piece of gold, it was placed in a full vessel with water, and then the same experiment was carried out and it turned out that different amounts of water poured out. Thanks to this discovery, a whole science will eventually arise -. The same discovery of Archimedes explains why a ball with a lighter gas than air can rise up, why a steel ball sinks, but a tree does not.

Other experiments of Archimedes

It is authentically known that Archimedes invented a screw pump, which served for a very long time in mines and in various devices for pumping water. This pump is called a kohlya. The principle of operation is that a screw with large blades is placed in a hollow tube, the tube must be at an angle. After that, with the help of the labor force, the screw is untwisted, and water flows upward from the well through the blades.

Oddly enough, the first, most primitive lever was also classified and singled out by Archimedes. Everyone knows his famous phrase: "Give me a foothold and I will move the world." The levers created by the great scientist were the most productive at that time. A large number of his studies and achievements have come down to us from other philosophers. Like many other scientists of that time, he rarely wrote down his thoughts or his texts were lost in time.

To say that he had other ideas, military developments allow, which, as is known from history, made it possible to resist for a very long time before the Romans still managed to take Syracuse.

18. Equilibrium of a body in a fluid at rest

A body immersed (completely or partially) in a liquid experiences a total pressure from the side of the liquid directed upwards and equal to the weight of the liquid in the volume of the immersed part of the body. P you are t = ρ and gV burial

For a homogeneous body floating on the surface, the relation

Where: V- the volume of the floating body; p m is the density of the body.

The existing theory of a floating body is quite extensive, so we will confine ourselves to considering only the hydraulic essence of this theory.

The ability of a floating body, taken out of equilibrium, to return to this state again is called stability. The weight of the liquid taken in the volume of the submerged part of the ship is called displacement, and the point of application of the resultant pressure (i.e. the center of pressure) - displacement center. In the normal position of the vessel, the center of gravity WITH and displacement center d lie on the same vertical line O"-O", representing the axis of symmetry of the vessel and called the axis of navigation (Fig. 2.5).

Let, under the influence of external forces, the ship tilted at a certain angle α, part of the ship KLM came out of the liquid, and part K"L"M" on the contrary, plunged into it. At the same time, a new position of the center of displacement was obtained d". Apply to a point d" lifting force R and continue its line of action until it intersects with the axis of symmetry O"-O". Received point m called metacenter, and the segment mC = h called metacentric height. We assume h positive if the point m lies above the point C, and negative otherwise.

Rice. 2.5. Vessel transverse profile

Now consider the conditions for the equilibrium of the ship:

1) if h> 0, then the ship returns to its original position; 2) if h= 0, then this is a case of indifferent equilibrium; 3) if h<0, то это случай неостойчивого равновесия, при котором продолжается дальнейшее опрокидывание судна.

Therefore, the lower the center of gravity and the greater the metacentric height, the greater the stability of the vessel.

The dependence of pressure in a liquid or gas on the depth of immersion of the 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 gravity, so 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 the law of Archimedes.

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

The discovery of the basic law of hydrostatics is the greatest achievement of ancient science. Most likely you already know the legend about how Archimedes discovered his law: "The Syracusan king Hieron once called him and said .... And what happened next? ...

The law of Archimedes was first mentioned by him in his treatise On Floating Bodies. Archimedes wrote: "bodies heavier than a 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 the Archimedean force:

Interestingly, the Archimedes force is zero when a body immersed in a liquid is dense, with its entire base pressed to the bottom.

WEIGHT OF A BODY IMMERSED IN A LIQUID (OR GAS)

body weight in vacuum Po=mg.
If a body is immersed in a liquid or gas,
That P \u003d Po - Fa \u003d Po - Pzh

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

Or otherwise:

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

BOOKSHELF

TURNS OUT

The density of organisms living in water is almost the same as the density of water, so they do not need strong skeletons!

Fish regulate their diving depth by changing their average body density. 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 causes the fagaka to quickly swallow water. In this case, in the esophagus of the fish there is a rapid decomposition of food with the release of a significant amount of gases. Gases fill not only the existing cavity of the esophagus, but also the blind outgrowth present with it. As a result, the body of the fagaka swells strongly, and, in accordance with the law of Archimedes, it quickly floats to the surface of the reservoir. Here he swims, hanging upside down, until the gases released in his body evaporate. After that, gravity lowers it to the bottom of the reservoir, where it takes refuge among the bottom algae.

Chilim (water chestnut) after flowering gives heavy fruits under water. These fruits are so heavy that they may well carry the whole plant to the bottom. However, at this time, the chilim, growing in deep water, develops swellings on the petioles of the leaves, giving it the necessary lifting force, and it does not sink.