Archimedes' law: a body immersed in water. Academy of Entertaining Sciences. Physics. Video. Does the egg float?

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.

Add-ons

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 liquid 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:

When passing from the surface integral to the volume integral, we use the generalized

And static gases.

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  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) in the volume of the immersed part of the body. The force is called by the power of Archimedes:

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

  Where ρ (\displaystyle \rho )- density of liquid (gas), g (\displaystyle (g)) is the acceleration of free fall, and V (\displaystyle V)- the volume of the submerged part of the body (or the part of the volume of the body located below the surface). If a body floats on the surface (uniformly moves up or down), then the buoyancy 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.

  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.

  P B − P A = ρ g h (\displaystyle P_(B)-P_(A)=\rho gh) F B − F A = ​​ρ g h S = ρ g V , (\displaystyle F_(B)-F_(A)=\rho ghS=\rho gV,)

  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.

  In theoretical physics, Archimedes' law is also used in integral form:

  F A = ​​∬ S p d S (\displaystyle (F)_(A)=\iint \limits _(S)(p(dS))),

  Where S (\displaystyle S)- surface area, p (\displaystyle p)- 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 inertia forces (for example, 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 of fluid at depth h (\displaystyle h) There is p = ρ g h (\displaystyle p=\rho gh). At the same time we consider ρ (\displaystyle \rho ) liquids and the gravitational field strength are constant values, 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; ∗ ∗ g r a d (h) = ∇ h = e → z (\displaystyle ()^(*)h(x,y,z)=z;\quad ^(**)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 it is directed in the direction opposite to the direction of the gravitational field strength vector.

  Another wording (where ρ t (\displaystyle \rho _(t))- body density, ρ s (\displaystyle \rho _(s))- density of the medium in which it is immersed).

  Issue 8

  In a physics video lesson from the Academy of Entertaining Sciences, Professor Daniil Edisonovich will talk about the ancient Greek scientist Archimedes and some of his amazing discoveries. How do you know if gold is pure? How do multi-ton ships manage to float on ocean waves? Our life is full of mysterious phenomena and tricky puzzles. Physics can provide clues to some of them. After watching the eighth physics video lesson, you will become acquainted with Archimedes' law and Archimedes' force, as well as the history of their discovery.

  Archimedes' Law

  Why do objects weigh less in water than on land? For a person, being in water is comparable to being in a state of weightlessness. Astronauts use this in their training. But why does this happen? The fact is that bodies immersed in water are subject to a buoyancy force, discovered by the ancient Greek philosopher Archimedes. Archimedes' law goes like this: a body immersed in a liquid loses as much weight as the volume of water it displaces weighs. The buoyancy force was called Archimedes, in honor of the discoverer. Archimedes was one of the greatest scientists of Ancient Greece. This brilliant mathematician and mechanic lived in Syracuse in the 3rd century BC. e. At this time, King Hiero ruled in Syracuse. One day, Hieron, having received the golden crown he had ordered from the craftsmen, doubted their honesty. It seemed to him that they had hidden part of the gold given for its production and replaced it with silver. But how can jewelers be caught counterfeiting? Hiero instructed Archimedes to determine whether there was an admixture of silver in the golden crown. Archimedes was constantly looking for a solution to the problem, never stopping to think about it when he was doing other things. And the solution was found... in the bathhouse. Archimedes soaped himself with ash and climbed into the bathtub. And something happened that happens every time when any person, not even a scientist, sits down in any bathtub, not even a marble one - the water in it rises. But something that Archimedes usually did not pay any attention to suddenly interested him. He stood up - the water level dropped, he sat down again - the water rose; and it rose as the body sank. And at that moment it dawned on Archimedes. He saw in the experiment performed a dozen times a hint of how the volume of a body is related to its weight. And I realized that King Hieron’s task was solvable. And he was so happy about his accidental discovery that as he was - naked, with the remains of ash on his body - he ran home through the city, filling the street with shouts: “Eureka! Eureka!". This is how Archimedes, according to legend, found the solution to Hiero’s problem. Archimedes asked the king for two ingots - silver and gold. The weight of each ingot was equal to the weight of the crown. Having placed first a silver and then a gold ingot into a vessel filled to the brim with water, the scientist measured the volume of water displaced by each ingot. Gold displaced less water than silver. And all because the volume of a piece of gold was less than a piece of silver of the same weight. After all, gold is heavier than silver. Archimedes then immersed the crown in the vessel and measured the volume of water it displaced. The crown displaced less water than an ingot of silver. but more than a bar of gold. So the jeweler's fraud was exposed. Thanks to Archimedes' power, giant ships weighing hundreds of thousands of tons are able to sail. This is due to the fact that they have a large displacement. That is, their volume is such that it displaces a huge amount of water. And as you remember, the larger the volume of the body, the stronger the Archimedes force acts on it.

  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 .