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

Background

"Eureka!" (“Found!”) - this exclamation, according to legend, was issued by the ancient Greek scientist and philosopher Archimedes, having discovered the principle of displacement. 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 for Archimedes to weigh the crown, but it 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 to determine whether it was pure gold. Further, 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 it is lowered into a basin filled to the brim, will displace from it a volume of water equal to its volume. The 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 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 fluid will now act on the solid that displaced it. And, if the buoyant force acting vertically upwards is greater than the gravity pulling the body vertically downwards, the body will float; otherwise it will go to the bottom (drown). talking modern language, a body floats if its average density is less than the density of the fluid 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 by a solid body, the upward momentum of molecular impacts 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 buoyancy force will still 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 one, from which the buoyancy force arises. This is the explanation of the buoyancy force at the molecular level.

This buoyancy pattern explains why a ship made of steel, which is much denser than water, stays afloat. The fact is that the volume of water displaced by the 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, so the buoyancy 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 bottom, and the ship sails.

Wording 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. Swimming in a river or in the sea, you can easily lift and move very heavy stones along the bottom - those that cannot be lifted on land. At the same time, light bodies resist being submerged in water: it takes both strength and dexterity to sink a ball the size of a small watermelon; most likely it will 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 action of a fluid on a body immersed in it; one cannot be satisfied with the answer that light bodies float, and heavy bodies sink: a steel plate, of course, will sink in water, but if you make a box out of it, then it can float; while her weight did not change.

The existence of hydrostatic pressure leads to the fact that a buoyant force acts on any body in a liquid or gas. For the first time, the value of this force in liquids was determined experimentally by Archimedes. Archimedes' law is formulated as follows: a body immersed in a liquid or gas is subjected to a buoyant force equal to the weight of the amount of liquid or gas 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 w gV fri,

where ρzh is the density of the liquid,

g is the free fall acceleration,

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

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

1) Ft > FA - the body sinks;

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

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

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

Add-ons

The buoyant or lifting force in the direction is opposite to the force of gravity, it is applied to the center of gravity of the volume displaced by the body from a liquid or gas.

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 an inhomogeneous field. For example, this refers to the field of inertia forces (for example, to the field of centrifugal force) - centrifugation is based on this. An example for a field of non-mechanical nature: a diamagnet in a vacuum is displaced from a region of a magnetic field of greater intensity to a region of lesser intensity.

Derivation of the law of Archimedes for a body of arbitrary shape

hydrostatic pressure p (\displaystyle p) at a depth h (\displaystyle h), rendered by the liquid density ρ (\displaystyle \rho ) on the body, there p = ρ g h (\displaystyle p=\rho gh). Let the fluid density ( ρ (\displaystyle \rho )) and tension gravitational field (g (\displaystyle g)) - constants, a h (\displaystyle h)- parameter. Let's take an arbitrary-shaped body with 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 coinciding with the direction of the vector g → (\displaystyle (\vec (g))). Zero along the z axis is set on the surface of the liquid. Let us single out an elementary area on the surface of the body d S (\displaystyle dS). It will be acted upon by the fluid pressure force directed inside 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, we 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 passing from the integral over the surface to the integral over the volume, we use the generalized

Archimedes- Greek mechanic, physicist, mathematician, engineer. Born in Syracuse (Sicily). His father Phidias was an astronomer and mathematician. The father was engaged in the upbringing and education of his son. From him, Archimedes inherited the ability to mathematics, astronomy and mechanics. Archimedes studied in Alexandria (Egypt), which at that time was a cultural and scientific center. There he met Eratosthenes- a Greek mathematician, astronomer, geographer and poet, who became the mentor of Archimedes and patronized him for a long time.

Archimedes combined the talents of an engineer-inventor and a theoretical scientist. He became the founder theoretical mechanics and hydrostatics, developed methods for finding surface areas and volumes of various figures and bodies.

According to legend, Archimedes owns many amazing technical inventions who won him fame among his contemporaries. It is believed that Archimedes, using mirrors and reflecting the sun's rays, was able to set fire to the Roman fleet, which laid siege to Alexandria. This case is a clear example of excellent command of optics.

Archimedes is also credited with inventing the catapult, a military throwing machine, constructing a planetarium in which the planets moved. The scientist created a screw for lifting water (Archimedean screw), which is still in use and is a water-lifting machine, a shaft with a helical surface located in an inclined pipe immersed in water. During rotation, the helical surface of the shaft moves water through the pipe to different heights.

Archimedes wrote a lot scientific papers: "About spirals", "About conoids and spheroids", "About a ball and a cylinder", "About levers", "About floating bodies". And in the treatise "On grains of sand" he counted the number of grains of sand in the volume of the globe.

Archimedes discovered his famous law under interesting circumstances. King Gireon II, whom Archimedes served, wanted to know if jewelers mixed gold with silver when they made the crown. To do this, it is necessary to determine not only the mass, but the volume of the crown in order to calculate the density of the metal. Determine the volume of the product irregular shape – not an easy task, over which Archimedes pondered for a long time.

The solution came to Archimedes' mind as he plunged into the bath: the water level in the bath rose after the scientist's body was lowered into the water. That is, the volume of his body displaced an equal volume of water. With a cry of "Eureka!" Archimedes ran to the palace without even bothering to get dressed. He lowered the crown into the water and determined the volume of the displaced liquid. The problem has been solved!

Thus, Archimedes 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. A body can float in water if its average density is less than the density of the liquid in which it is placed.

Archimedes' principle states that any body immersed in a liquid or gas is subjected to an upward buoyancy force equal to the weight of the liquid or gas displaced by it.

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A body immersed in a liquid or gas is subjected to a buoyant force equal to the weight of the liquid or gas displaced by this body.

In integral form

Archimedean force 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.

If a body floats on a surface or moves up or down uniformly, then the buoyant force (also called Archimedean force) is equal in absolute value (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.

As for bodies that are in a gas, for example, in air, to find the lifting force (Archimedes Force), you need to replace the density of the liquid with the density of the gas. For example, a balloon with helium flies upwards due to the fact that the density of helium is less than the density of air.

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

In the formula we used

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

Tradition

In Syracuse, Archimedes was legendary. Once 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. Check this fact instructed Archimedes.

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

According to legend, Archimedes made this discovery in a bathtub, watching the splashed water. What happened next with the dishonest jeweler, history is silent, 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 the law of Archimedes.

Wording

Archimedes outlined the results of his experiments in the work “On Floating Bodies”, which, unfortunately, has survived to this day only in the form of fragments. Modern physics describes the law of Archimedes as the total force acting on a body immersed in a liquid. The buoyant force of a body in a fluid is directed upwards; its absolute value is 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 experience only 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 upwards, the body is pushed out of the liquid. The magnitude of these forces is determined by the law of Archimedes. The navigation 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

Visually, the power of Archimedes can be demonstrated by simple weighing. When weighing a training weight in vacuum, in air and in water, one can see that its weight changes significantly. In a vacuum, the weight of the weight is one, in air - a little lower, and in water - even lower.

If we take the weight of a body in vacuum as P o, then its weight in air can be described by the following formula: P in \u003d P o - F a;

here P about - weight in vacuum;

As can be seen from the figure, any actions with weighing in water significantly lighten the body, therefore, 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 a standard formula.

The density of the medium and the force of Archimedes

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

Formula

Archimedean force and the forces that affect it can be determined using simple logical reasoning. Suppose 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 assumptions. 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 balance, 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. Substituting the known values, you can find out the weight of the body in the liquid. This parameter is described as ρV * g.

Substituting known values, we get:

This is the law of Archimedes.

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

Application of the law of Archimedes

The first experiments demonstrating the law of Archimedes have been known since school days. A 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 the rule of Archimedes, 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 the differences 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 the dive. The correct calculation of the density will allow divers to correctly calculate the air pressure inside the suit, which will affect the diver's maneuverability and ensure his safe diving and ascent. Archimedes' principle must also be taken into account in deep-water drilling, huge drilling rigs lose up to 50% of their weight, which makes their transportation and operation less costly.