Start in science. Archimedes' Law: definition and formula Where is the Archimedes force acting on the plane less?

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 and balloons fly in the sky: thanks to air displacement, they become 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 is described by the standard formula.

Density of the medium and Archimedes' force

Analyzing the simplest experiments with body weight in various 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 indicated that the density of a body is 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. A metal plate sinks in water, but, folded into 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 fact is that the density of 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.

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) displaced by this body. The force is called by the power of Archimedes:

where is the density of the liquid (gas), is the acceleration free fall, a is the volume of the submerged body (or the part of the volume of the body located below the surface). If a body floats on the surface or moves uniformly up or down, then the buoyant 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.

A body floats if the Archimedes force balances the force of gravity of the body.

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.

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.

18. Equilibrium of a body in a fluid at rest

A body immersed (fully or partially) in a liquid experiences a total pressure from the liquid, directed from bottom to top 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 Pogr

For a homogeneous body floating on the surface, the relation is true

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

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

The ability of a floating body, removed from a state of equilibrium, to return to this state again is called stability. The weight of liquid taken in the volume of the immersed part of the ship is called displacement, and the point of application of the resultant pressure (i.e., the center of pressure) is displacement center. In the normal position of the ship, the center of gravity WITH and center of displacement 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 tilt 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, we received a new position for the center of displacement d". Let's apply it to the point d" lift R and continue the line of its 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 point m lies above the point C, and negative - otherwise.

Rice. 2.5. Cross profile of the vessel

Now consider the equilibrium conditions 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.

Consequently, the lower the center of gravity and the greater the metacentric height, the greater will be the stability of the vessel.

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 for a liquid the situation is different. 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 on the part of the fluid per unit area $S$.

$ρ = \frac(\Delta P)(\Delta S)$.

Note 1

Pressure is measured in pascals. One Pa is equal to a force of 1 N acting per unit area of ​​1 square. 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 a liquid produced by external forces is transmitted by the liquid equally in all directions.

In mechanical equilibrium, the horizontal fluid pressure is always the same; therefore, the free surface of a static liquid 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 medium under consideration does not depend on pressure.

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

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

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

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

Formulation of Archimedes' law

Archimedes' law, one of the basic laws of hydrostatics and aerostatics, states: a body immersed in a liquid or gas is acted upon by 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.

Note 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; in this case the body is pushed out of the medium.

Hence the second name for this phenomenon – 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 Archimedean force is due to the difference in environmental pressure at different depths. For example, the pressure on the lower layers of water is always greater than on the upper layers.

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

The emergence of Archimedes' Force

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

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

Note 3

If a body immersed in a liquid is in mechanical equilibrium, then the buoyant force of hydrostatic pressure acts on it from the environment surrounding it, which is 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 Archimedes’ law, the buoyancy force holds:

$F_A = ρgV$, where:

• $V_A$ - buoyancy force, H;
• $ρ$ - density of liquid or gas, $kg/m^3$;
• $V$ - volume of a 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 gravity moduli $F_T$ and the Archimedean force $F_A$. There are three possible cases here:

1. $F_T$ > $F_A$. The force of gravity exceeds the buoyant force, therefore the body sinks/falls;
2. $F_T$ = $F_A$. The force of gravity is equalized with the buoyant force, so the body “hangs” in the liquid;
3. $F_T$

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 in 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 hull shell and the air inside it, it turns out that the density of the ship (as physical body) is less than the density of water, so the buoyant force acting on it as a result of the upward impulses of the impact of water molecules turns out to be higher than the gravitational force of attraction of the Earth, pulling the ship 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.

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

In weightlessness there is no Archimedean force. If we imagine that the force of gravity on Earth suddenly disappeared, then all the ships in the seas, oceans and rivers will go to any depth at the slightest push. But something independent of gravity will not allow them to rise up. surface tension water, so they won’t be able to take off, they’ll all drown.

How does the power of Archimedes manifest itself?

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 modern idea: a body immersed in a liquid or gaseous medium in a gravity field is acted upon 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 – density of the medium; g – free fall acceleration; V is the volume of liquid (gas) displaced by the body or an immersed part of it.

If in fresh water a buoyancy force of 1 kg (9.81 N) acts on each liter of volume of a submerged body, then in sea ​​water, the density of which is 1.025 kg * cubic. 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 across 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 and measure its force from the water. Then roll out this cylinder into a sheet, immerse it flat and edge-on in water. In all three cases, the power of Archimedes will be the same.

It may seem strange at first glance, but if a sheet is immersed flat, the decrease in pressure difference for a thin sheet is compensated by an increase in its area perpendicular to the surface of the water. And when immersed with an edge, on the contrary, the small area of ​​the edge is compensated by the larger height of the sheet.

If the water is very highly saturated with salts, causing its density to become higher than the density of the human body, then even a person who cannot swim will not drown in it. At 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 support area is small, the person falls into the water up to his neck, until the weight of the submerged 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 create a legend about walking on water. But in kerosene, the density of which is only 0.815 kg*cubic. dm, even a very experienced swimmer will not be able to stay on the surface.

Archimedean force in dynamics

Everyone knows that ships float thanks to the power of Archimedes. But fishermen know that Archimedean force can also be used in dynamics. If you come across a large and strong fish (taimen, for example), then there is no point in slowly pulling it to the net (fishing for it): it will break the fishing line and leave. You need to tug lightly first when it goes away. Feeling the hook, the fish, trying to free itself from it, rushes 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 and inertia are preserved. With this method of fishing, the Archimedean force will seem to kick the fish in the tail, and the prey itself will plop down at the angler’s feet or into his boat.

Archimedes' power in the air

Archimedes' force acts not only in liquids, but also in gases. Thanks to it, hot air balloons and airships (zeppelins) fly. 1 cu. m air at normal conditions(20 degrees Celsius at sea level) weighs 1.29 kg, and 1 kg of helium weighs 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 has a diameter of 10 m, then its volume will be 523 cubic meters. m. Having made it from light synthetic material, we get a lifting force of about half a ton. Aeronauts call Archimedes' force in the air fusion force.

If you pump out the air from the balloon without allowing it to shrink, then each cubic meter of it will pull up the entire 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 appear from time to time. But materials that can withstand large amounts (about 1 kg per sq. cm) Atmosphere pressure outside onto the shell, modern technology not yet able to create.