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 displaced by the crown and the ingots, Archimedes discovered that the crown was not made of pure gold, but was mixed with 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, contact to our specialists. The best authors will be happy to share their knowledge and break down the solution to the most difficult problem “on the shelves.”
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 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.
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 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.
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.
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 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.
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 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.
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.
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 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.
One of the first physical laws studied by high school students. Any adult remembers at least approximately this law, no matter how far he is from physics. But sometimes it is useful to return to the exact definitions and formulations - and understand the details of this law that may have been forgotten.
What does Archimedes' law say?
There is a legend that the ancient Greek scientist discovered his famous law while taking a bath. Having plunged into a container filled to the brim with water, Archimedes noticed that the water splashed out - and experienced an epiphany, instantly formulating the essence of the discovery.
Most likely, in reality the situation was different, and the discovery was preceded by long observations. But this is not so important, because in any case, Archimedes managed to discover the following pattern:
- plunging into any liquid, bodies and objects experience several multidirectional forces at once, but directed perpendicular to their surface;
- the final vector of these forces is directed upward, so any object or body, finding itself in a liquid at rest, experiences pushing;
- in this case, the buoyancy force is exactly equal to the coefficient that is obtained if the product of the volume of the object and the density of the liquid is multiplied by the acceleration of free fall.
So, Archimedes established that a body immersed in a liquid displaces a volume of liquid that is equal to the volume of the body itself. If only part of a body is immersed in a liquid, then it will displace the liquid, the volume of which will be equal to the volume of only the part that is immersed.
The same principle applies to gases - only here the volume of the body must be correlated with the density of the gas.
You can formulate a physical law a little more simply - the force that pushes an object out of a liquid or gas is exactly equal to the weight of the liquid or gas displaced by this object during immersion.
The law is written in the form of the following formula:
What is the significance of Archimedes' law?
The pattern discovered by the ancient Greek scientist is simple and completely obvious. But at the same time, its importance for everyday life cannot be overestimated.
It is thanks to the knowledge of the pushing of bodies by liquids and gases that we can build river and sea vessels, as well as airships and balloons for aeronautics. Heavy metal ships do not sink due to the fact that their design takes into account Archimedes' law and numerous consequences from it - they are built so that they can float on the surface of the water, and do not sink. Aeronautics operate on a similar principle - they use the buoyancy of air, becoming, as it were, lighter in the process of flight.
Homeric period (XI-VIII centuries)
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