Archimedes' Law- one of the main laws of hydrostatics and gas statics.
Formulation and explanations
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 gravity, 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 PA, PB- 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.
In theoretical physics, Archimedes' law is also used in integral form:
,
where is the surface area, is the pressure at an arbitrary point, integration is carried out over the entire surface of the body.
In the absence of a gravitational field, that is, in a state of weightlessness, Archimedes' law does not work. Astronauts are quite familiar with this phenomenon. In particular, in zero gravity there is no phenomenon of (natural) convection, therefore, for example, air cooling and ventilation of the living compartments of spacecraft is carried out forcibly by fans.
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 a non-uniform field. For example, this refers to the field of inertial forces (for example, centrifugal force) - centrifugation is based on this. An example for a field of a non-mechanical nature: a conducting body is displaced from a region of a magnetic field of higher intensity to a region of lower intensity.
Derivation of Archimedes' law for a body of arbitrary shape
There is hydrostatic pressure of the fluid at depth. In this case, we consider the fluid pressure and the gravitational field strength to be constant values, and - a parameter. Let's take a body of arbitrary shape that has a non-zero volume. Let's introduce a right-handed orthonormal coordinate system, and choose the direction of the z axis to coincide with the direction of the vector. We set zero along the z axis on the surface of the liquid. Let us select an elementary area on the surface of the body. It will be acted upon by the fluid pressure force directed into the body, 
. To get the force that will act on the body, take the integral over the surface:
When passing from the surface integral to the volume integral, we use the generalized Ostrogradsky-Gauss theorem.
We find that the modulus of the Archimedes force is equal to , and it is directed in the direction opposite to the direction of the gravitational field strength vector.
Condition of floating bodies
The behavior of a body located in a liquid or gas depends on the relationship between the modules of gravity and the Archimedes force, which act on this body. The following three cases are possible:
Another formulation (where is the density of the body, is the density of the medium in which it is immersed).
Different objects in liquid behave differently. Some drown, others remain on the surface and float. Why this happens is explained by Archimedes' law, which he discovered under very unusual circumstances and became the basic law of hydrostatics.
How Archimedes discovered his law
Legend tells us that Archimedes discovered his law by accident. And this discovery was preceded by the following event.
King Hiero of Syracuse, who reigned 270-215. BC, suspected his jeweler of mixing a certain amount of silver into the gold crown he ordered. To dispel doubts, he asked Archimedes to confirm or refute his suspicions. As a true scientist, Archimedes was fascinated by this task. To solve it, it was necessary to determine the weight of the crown. After all, if silver was mixed into it, then its weight would be different from what it would be if it were made of pure gold. The specific gravity of gold was known. But how to calculate the volume of the crown? After all, it had an irregular geometric shape.
According to legend, one day Archimedes, while taking a bath, was thinking about a problem that he had to solve. Suddenly, the scientist noticed that the water level in the bathtub became higher after he immersed himself in it. As it rose, the water level dropped. Archimedes noticed that with his body he was displacing a certain amount of water from the bath. And the volume of this water was equal to the volume of his own body. And then he realized how to solve the problem with the crown. It is enough just to immerse it in a vessel filled with water and measure the volume of displaced water. They say that he was so happy that he shouted “Eureka!” (“Found it!”) jumped out of the bath without even getting dressed.
Whether this really happened or not does not matter. Archimedes found a way to measure the volume of bodies with complex geometric shapes. He first drew attention to the properties of physical bodies, which are called density, comparing them not with each other, but with the weight of water. But most importantly, it was open to them buoyancy principle .
Archimedes' Law
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.
And the body itself in the liquid is acted upon by a force that pushes it to the surface. Its value is equal to the weight of the fluid displaced by it. This force is called by the power of Archimedes .
For a liquid, Archimedes' law looks like this: a body immersed in a liquid is acted upon by a buoyant force directed upward and equal to the weight of the liquid displaced by this body.
The magnitude of the Archimedes force is calculated as follows:
F A = ρ ɡ V ,
Where ρ – fluid density,
ɡ - acceleration of gravity
V – the volume of a body immersed in a liquid, or the part of the volume of a body located below the surface of the liquid.
The Archimedes force is always applied to the center of gravity of the volume and is directed opposite to the force of gravity.
It should be said that in order for this law to be fulfilled, one condition must be met: the body either intersects with the boundary of the liquid or is surrounded on all sides by this liquid. For a body that lies on the bottom and touches it hermetically, Archimedes' law does not apply. So, if we put a cube on the bottom, one of the faces of which is in close contact with the bottom, we will not be able to apply Archimedes’ law to it.
Archimedes' force is also called buoyant force .
This force, by its nature, is the sum of all pressure forces acting from the liquid on the surface of a body immersed in it. The buoyant force arises from the difference in hydrostatic pressure at different levels of the liquid.
Let's consider this force using the example of a body shaped like a cube or parallelogram.
P 2 – P 1 = ρ ɡ h
F A = F 2 – F 1 = ρɡhS = ρɡhV
Archimedes' law also applies to gases. But in this case, the buoyant force is called lifting force, and to calculate it, the density of the liquid in the formula is replaced by the density of the gas.
Body floating condition
The ratio of the values of gravity and the Archimedes force determines whether the body will float, sink or float.
If the Archimedes force and the force of gravity are equal in magnitude, then a body in a liquid is in a state of equilibrium when it neither floats up nor sinks. It is said to float in liquid. In this case F T = F A .
If the force of gravity is greater than the force of Archimedes, the body sinks or sinks.
Here F T˃ F A .
And if the value of gravity is less than the force of Archimedes, the body floats up. This happens when F T˂ F A .
But it does not float up indefinitely, but only until the moment when the force of gravity and the force of Archimedes become equal. After this, the body will float.
Why don't all bodies drown?
If you put two bars of the same shape and size into water, one of which is made of plastic and the other of steel, you can see that the steel bar will sink, while the plastic bar will remain afloat. The same will happen if you take any other objects of the same size and shape, but different in weight, for example, plastic and metal balls. The metal ball will sink to the bottom, and the plastic ball will float.
But why do plastic and steel bars behave differently? After all, their volumes are the same.
Yes, the volumes are the same, but the bars themselves are made of different materials that have different densities. And if the density of the material is higher than the density of water, then the block will sink, and if it is less, it will float until it reaches the surface of the water. This is true not only for water, but also for any other liquid.
If we denote the density of the body P t , and the density of the medium in which it is located is as P s , then if
P t ˃ Ps (the density of the body is higher than the density of the liquid) – the body sinks,
Pt = Ps (the density of the body is equal to the density of the liquid) – the body floats in the liquid,
P t ˂ Ps (the density of the body is less than the density of the liquid) – the body floats up until it reaches the surface. After which it floats.
Archimedes' law is not fulfilled even in a state of weightlessness. In this case, there is no gravitational field, and, therefore, no acceleration of gravity.
The property of a body immersed in a liquid to remain in equilibrium without floating or sinking further is called buoyancy .
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 reflected on the diagrams that he drew with his hand directly 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.”
Why can we lie on the surface of the sea without sinking to the bottom? Why do heavy ships float on the surface of the water?
There is probably some kind of force that pushes people and boats, that is, all bodies out of the water and allows them to float on the surface.
The dependence of pressure in a liquid or gas on the depth of immersion of a body leads to the appearance of a buoyancy force, or otherwise the Archimedes force, acting on any body immersed in a liquid or gas. Let's take a closer look at the Archimedes force using an example.
We all launched boats through puddles. What's a boat without a captain? What did we observe? The ship sinks deeper under the weight of the captain. What if we placed five or eight captains on our boat? Our boat sank to the bottom.
What can we learn useful from this experience? When the weight of the boat increased, we saw that the boat sank lower into the water. That is, the body weight increased the pressure on the water, but the buoyancy force remained the same.
When the weight of the body exceeded the magnitude of the buoyant force, the boat, under the influence of this force, sank to the bottom. That is, there is a buoyancy force that is the same for a particular body, but different for different bodies.
The buoyancy force, also known as the Archimedes force, acting on a body immersed in a liquid is equal to the weight of the liquid displaced by this body.
A brick, as everyone knows, will sink to the bottom in any case, but a wooden door will not only float on the surface, but can also hold a couple of passengers. This force is called Archimedean force and is expressed by the formula:
Fout = g*m f = g* ρ f * V f = P f,
where m is the mass of the liquid,
and Pf is the weight of the fluid displaced by the body.
And since our mass is equal to: m f = ρ f * V f, then from the formula of the Archimedean force we see that it does not depend on the density of the immersed body, but only on the volume and density of the fluid displaced by the body.
Archimedean force is a vector quantity. The reason for the existence of the buoyant force is the difference in pressure on the upper and lower parts of the body. The pressure indicated in the figure is P 2 > P 1 due to the greater depth. For the Archimedes force to arise, it is enough that the body is at least partially immersed in the liquid.
So, if a body floats on the surface of a liquid, then the buoyant force acting on the part of this body immersed in the liquid is equal to the gravitational force of the entire body. If the density of the body is greater than the density of the liquid, then the body sinks, if less, then it floats.
A body immersed in a liquid loses its weight exactly as much as the weight of the water it displaces. Therefore, it is natural to assume that if the weight of a body is less than the weight of water of the same volume, then it will float on the surface, and if it is more, it will drown.
If the weight of the body and the water is equal, then the body can swim remarkably well in the water, as all aquatic inhabitants do. The density of organisms living in water is almost no different from the density of water, so they don’t need strong skeletons!
Fish regulate their diving depth by changing the average density of their body. 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 forces the fagak to quickly swallow water. At the same time, rapid decomposition of food products occurs in the fish esophagus with the release of a significant amount of gases. Gases fill not only the active cavity of the esophagus, but also the blind outgrowth attached to it. As a result, the phagak's body swells greatly, and, in accordance with Archimedes' law, it quickly floats to the surface of the reservoir. Here he swims, hanging upside down, until the gases released in his body disappear. After this, gravity lowers it to the bottom of the reservoir, where it takes refuge among the bottom algae.
F A = ρ g V , (\displaystyle F_(A)=\rho gV,)
Description
A buoyant or lifting force in the direction opposite to the force of gravity is applied to the center of gravity of the volume displaced by a 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 a non-uniform field. For example, this refers to the field of inertial forces (for example, to the field of centrifugal force) - centrifugation is based on this. An example for a field of a non-mechanical nature: a diamagnetic material in a vacuum is displaced from a region of a magnetic field of higher intensity to a region of lower intensity.
Derivation of Archimedes' law for a body of arbitrary shape
Hydrostatic pressure p (\displaystyle p) at a depth h (\displaystyle h), exerted by the liquid density ρ (\displaystyle \rho ) on the body, there is p = ρ g h (\displaystyle p=\rho gh). Let the fluid density ( ρ (\displaystyle \rho )) and gravitational field strength ( g (\displaystyle g)) are constants, and h (\displaystyle h)- parameter. Let's take a body of arbitrary shape that has 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 to coincide with the direction of the vector g → (\displaystyle (\vec (g))). We set zero along the z axis on the surface of the liquid. Let us select an elementary area on the surface of the body d S (\displaystyle dS). It will be acted upon by the fluid pressure force directed into 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, 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 moving from the surface integral to the volume integral, we use the generalized Ostrogradsky-Gauss theorem.
∗ h (x, y, z) = z; (\displaystyle ()^(*)h(x,y,z)=z;) ∗ ∗ g r a d (h) = ∇ h = e → z . (\displaystyle ^(**)grad(h)=\nabla h=(\vec (e))_(z).)We find that the modulus of the Archimedes force is equal to ρ g V (\displaystyle \rho gV), and the Archimedes force is directed in the direction opposite to the direction of the gravitational field intensity vector.
Comment. Archimedes' principle can also be derived from the law of conservation of energy. The work of the force acting on the fluid from the immersed body leads to a change in its potential energy:
A = F Δ h = m f g Δ h = Δ E p (\displaystyle \ A=F\Delta h=m_(\text(g))g\Delta h=\Delta E_(p))
Where m f − (\displaystyle m_(\text(f))-) mass of the displaced part of the liquid, Δ h (\displaystyle \Delta h)- movement of its center of mass. Hence the modulus of the displacement force:
F = m f g (\displaystyle \F=m_(\text(g))g)