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 its capabilities 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, 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 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 a 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 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 of free fall, and 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 vessel 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.

It would seem that there is nothing simpler than Archimedes' law. But once upon a time Archimedes himself really puzzled over his discovery. How it was?

There is an interesting story connected with the discovery of the fundamental law of hydrostatics.

Interesting facts and legends from the life and death of Archimedes

In addition to such a gigantic breakthrough as the discovery of Archimedes’ law itself, the scientist has a whole list of merits and achievements. In general, he was a genius who worked in the fields of mechanics, astronomy, and mathematics. He wrote such works as a treatise “on floating bodies”, “on the ball and cylinder”, “on spirals”, “on conoids and spheroids” and even “on grains of sand”. The latest work attempted to measure the number of grains of sand needed to fill the Universe.

Role of Archimedes in the Siege of Syracuse

In 212 BC, Syracuse was besieged by the Romans. 75-year-old Archimedes designed powerful catapults and light short-range throwing machines, as well as the so-called “Archimedes claws”. With their help it was possible to literally turn over enemy ships. Faced with such powerful and technological resistance, the Romans were unable to take the city by storm and were forced to begin a siege. According to another legend, Archimedes, using mirrors, managed to set fire to the Roman fleet, focusing the sun's rays on the ships. The veracity of this legend seems doubtful, because None of the historians of that time mentioned this.

Death of Archimedes

According to many testimonies, Archimedes was killed by the Romans when they finally took Syracuse. Here is one of the possible versions of the death of the great engineer.

On the porch of his house, the scientist was thinking about the diagrams that he drew with his hand right in the sand. A passing soldier stepped on the drawing, and Archimedes, deep in thought, shouted: “Get away from my drawings.” In response to this, a soldier hurrying somewhere simply pierced the old man with a sword.

Well, now about the sore point: about the law and power of Archimedes...

How Archimedes' law was discovered and the origin of the famous "Eureka!"

Antiquity. Third century BC. Sicily, where there is still no mafia, but there are ancient Greeks.

An inventor, engineer and theoretical scientist from Syracuse (a Greek colony in Sicily), Archimedes served under King Hiero II. One day, jewelers made a golden crown for the king. The king, being a suspicious person, summoned the scientist to his place and instructed him to find out whether the crown contained silver impurities. Here it must be said that at that distant time no one had resolved such issues and the case was unprecedented.

Archimedes thought for a long time, came up with nothing, and one day decided to go to the bathhouse. There, sitting down in a basin of water, the scientist found a solution to the problem. Archimedes drew attention to a completely obvious thing: a body, immersed in water, displaces a volume of water equal to the body’s own volume. It was then that, without even bothering to get dressed, Archimedes jumped out of the bathhouse and shouted his famous “Eureka,” which means “found.” Appearing to the king, Archimedes asked to give him ingots of silver and gold, equal in weight to the crown. By measuring and comparing the volume of water drawn out by the crown and the ingots, Archimedes discovered that the crown was not made of pure gold, but had admixtures of silver. This is the story of the discovery of Archimedes' law.

The essence of Archimedes' law

If you are asking yourself how to understand Archimedes' principle, we will answer. Just sit down, think, and understanding will come. Actually, this law says:

A body immersed in a gas or liquid is subject to a buoyancy force equal to the weight of the liquid (gas) in the volume of the immersed part of the body. This force is called the Archimedes force.

As we can see, the Archimedes force acts not only on bodies immersed in water, but also on bodies in the atmosphere. The force that makes a balloon rise up is the same Archimedes force. The Archimedean force is calculated using the formula:

Here the first term is the density of the liquid (gas), the second is the acceleration of gravity, the third is the volume of the body. If the force of gravity is equal to the force of Archimedes, the body floats, if it is greater, it sinks, and if it is less, it floats until it begins to float.

In this article we looked at Archimedes' law for dummies. If you want to learn how to solve problems where Archimedes' law is found, please contact. The best authors will be happy to share their knowledge and break down the solution to the most difficult problem “on the shelves.”

Let's continue our study of Archimedean force. Let's do some experiments. We hang two identical balls from the balance beam. Their weight is the same, so the rocker is in balance (Fig. “a”). Place an empty glass under the right ball. This will not change the weight of the balls, so the equilibrium will remain (Fig. “b”).

Second experience. Let's hang a large potato from the dynamometer. You see that its weight is 3.5 N. Let's immerse the potato in water. We will find that its weight has decreased and become equal to 0.5 N.

Let's calculate the change in potato weight:

DW = 3.5 N – 0.5 N = 3 N

Why did the weight of the potato decrease by exactly 3 N? Obviously because in water the potatoes were subjected to a buoyant force of the same magnitude. In other words, Archimedes' force is equal to the change in weight t ate:

This formula expresses method for measuring Archimedean force: you need to measure your body weight twice and calculate its change. The resulting value is equal to the Archimedes force.

To derive the following formula let's do an experiment with the “Archimedes bucket” device. Its main parts are as follows: spring with arrow 1, bucket 2, body 3, casting vessel 4, cup 5.

First, the spring, bucket and body are suspended from a tripod (Fig. “a”) and the position of the arrow is marked with a yellow mark. The body is then placed in a casting vessel. As the body sinks, it displaces a certain volume of water, which is poured into a glass (Fig. “b”). The body weight becomes lighter, the spring compresses, and the arrow rises above the yellow mark.

Let's pour the water displaced by the body from the glass into the bucket (Fig. “c”). The most amazing thing is that when the water is poured (Figure “d”), the arrow will not just go down, but will point exactly to the yellow mark! Means, the weight of water poured into the bucket balanced the Archimedean force. In the form of a formula, this conclusion will be written as follows:

Summarizing the results of two experiments, we obtain Archimedes' law: the buoyant force acting on a body in a liquid (or gas) is equal to the weight of the liquid (gas) taken in the volume of this body and is directed opposite to the weight vector.

In § 3-b we indicated that the Archimedes force usually directed upwards. Since it is opposite to the weight vector, and it is not always directed downward, the Archimedean force also does not always act upward. For example, in rotating centrifuge in a glass of water, air bubbles will not float up, but deviate towards the axis of rotation.

Pressure forces act on the surface of a body that is in a liquid or gas. It is known that pressure increases with increasing diving depth. This means that the pressure forces that act on the lower part of the body and are directed upward are greater in magnitude than the forces that act on the upper part of the body and are directed downward.

Definition and formula of buoyancy force

Definition

The resultant force of pressure on a body that is immersed in a liquid or gas is called buoyant force. The buoyant force can be greater than the force of gravity that acts on the body. Buoyancy forces also appear if the body is partially in a liquid or gas.

If a body in a liquid is left alone, it sinks, is in equilibrium, or floats to the surface. This depends on the ratio of gravity and buoyant force (F A) acting on the body. In the first case (the body sinks) mg>F A . If mg=F A, then the body is in equilibrium. At mg

Archimedes' Law

A body immersed in a liquid or gas is subject to a buoyant force (Archimedes' force F A), equal to the weight of the liquid or gas displaced by it. In mathematical form, this law looks like:

where is the density of the liquid (gas) in which the body is immersed, g=9.8 m/s 2 is the acceleration of gravity, V is the volume of the body (part of it) that is in the liquid (gas). The Archimedes force is applied to the center of gravity of the volume of a body part that is in a liquid (gas).

Archimedes' law can be used to calculate the density of a homogeneous body irregular shape. In this case, the body is weighed twice: once in the air, and a second time by immersing the body in a liquid whose density is known.

Units of buoyancy force

The basic unit of measurement for the Archimedes force, like any force in the SI system, is: =N

In GHS: F A ]=din

1Н= (kg m)/s 2

Examples of problem solving

Example

Exercise. What is the buoyancy force that acts on a cube immersed in a system of liquids. The vessel is filled with water, kerosene is poured on top of the water. The interface between the liquids runs through the middle of the cube's face. Consider the density of water equal to 1 = 10 3 kg/m 3, the density of kerosene equal to 2 = 0.81 10 3 kg/m 3. The side of the cube is a=0.1 m.

Solution. Let's make a drawing.

The buoyancy force that acts on the side of water per half cube is equal to:

The buoyancy force that acts on the side of kerosene per half cube is equal to:

Both forces are directed upward. They are applied to different points (the centers of mass of the volumes of bodies immersed in the corresponding liquids); upon summation, the vectors can be transferred to one point parallel to themselves. We obtain that the resulting buoyancy force is equal to:

Let us substitute force components (1.2), (1.3) into expression (1.1), we have:

Let's carry out the calculations:

Answer. Answer: F A =8.8 N

Example

Exercise. What is the density of a stone if its weight in air is 3.2 N and its weight in water is 1.8 N.

Solution. Weight of stone in air:

where is the density of the stone, V is the volume of the stone. Weighing a stone in water, we obtain the weight of the stone in the liquid equal to.