Miscellaneous items behave differently in liquids. 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. Specific gravity 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 shape. 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 – liquid 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 same sizes and shapes, 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, which 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 .
Often scientific discoveries are the result of simple chance. But only people with a trained mind can appreciate the importance of a simple coincidence and draw far-reaching conclusions from it. It is thanks to the chain random events Archimedes' law appeared in physics, explaining the behavior of bodies in water.
Tradition
In Syracuse, legends were made about Archimedes. One day 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. Archimedes was assigned to check this fact.
Archimedes established that in air and water bodies have different weight, and the difference is directly proportional to the density of the measured body. By measuring the weight of the crown in air and water, and conducting a similar experiment with a whole piece of gold, Archimedes proved that there was an admixture of a lighter metal in the manufactured crown.
According to legend, Archimedes made this discovery in the bathtub, watching the water splash out. History is silent about what happened next to the dishonest jeweler, but the conclusion of the Syracuse scientist formed the basis of one of most important laws physics, which is known to us as Archimedes' law.
Formulation
Archimedes presented the results of his experiments in his work “On Floating Bodies,” which, unfortunately, has survived to this day only in the form of fragments. Modern physics describes Archimedes' law as a cumulative force acting on a body immersed in a liquid. The buoyant force of a body in a liquid is directed upward; her absolute value 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 only experience 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 upward, and the body will be pushed out of the liquid. The magnitude of these forces is determined by Archimedes' law. The floating 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 fly in the sky and Balloons: Air displacement makes them lighter than air.
Physical formula
The power of Archimedes can be clearly demonstrated by simple weighing. Weighing a training weight in a vacuum, in air and in water, you can see that its weight changes significantly. In a vacuum the weight of the weight is the same, in air it is slightly lower, and in water it is even lower.
If we take the weight of a body in a vacuum as P o, then its weight in the air can be described by the following formula: P in = P o - F a;
here P o - weight in vacuum;
As can be seen from the figure, any actions involving weighing in water significantly lighten the body, so 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 the standard formula.
Density of the medium and Archimedes' force
Analyzing the simplest experiments with body weight in various environments, we can come to the conclusion that the weight of a body in various environments depends on the mass of the object and the density of the immersion environment. Moreover, the denser the medium, the greater the Archimedes force. Archimedes' law linked this relationship and the density of a liquid or gas is reflected in its final formula. What else influences this force? In other words, on what characteristics does Archimedes' law depend?
Formula
The Archimedean force and the forces that influence it can be determined using simple logical deductions. Let us assume 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 premises. 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 equilibrium, 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. By substituting known quantities, you can find out the weight of a body in a liquid. This parameter is described as ρV * g.
Substituting known values, we get:
This is Archimedes' law.
The formula we derived describes the density as the density of the body under study. But in initial conditions it was indicated that the density of a body is identical to the density of the surrounding liquid. Thus, in this formula You can safely substitute the density value of the liquid. The visual observation that in a denser medium the buoyancy force is greater has received theoretical justification.
Application of Archimedes' Law
The first experiments demonstrating Archimedes' law have been known since school. A metal plate sinks in water, but, folded into a box, it can not only stay afloat, but also carry a certain load. This rule is the most important conclusion from Archimedes’ rule; 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 changes 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 immersion. Correct calculation of density will allow submariners to correctly calculate the air pressure inside the suit, which will affect the diver’s maneuverability and ensure his safe diving and ascent. Archimedes' law must also be taken into account when deep-sea drilling; huge drilling rigs lose up to 50% of their weight, which makes their transportation and operation less expensive.
Despite the obvious differences in the properties of liquids and gases, in many cases their behavior is determined by the same parameters and equations, which makes it possible to use a unified approach to studying the properties of these substances.
In mechanics, gases and liquids are considered as continuous media. It is assumed that the molecules of a substance are distributed continuously in the part of space they occupy. In this case, the density of a gas depends significantly on pressure, while for a liquid the situation is different. Usually, when solving problems, this fact is neglected, using the generalized concept of an incompressible fluid, the density of which is uniform and constant.
Definition 1
Pressure is defined as the normal force $F$ acting on the part of the fluid per unit area $S$.
$ρ = \frac(\Delta P)(\Delta S)$.
Note 1
Pressure is measured in pascals. One Pa is equal to a force of 1 N acting per unit area of 1 square. m.
In a state of equilibrium, the pressure of a liquid or gas is described by Pascal's law, according to which the pressure on the surface of a liquid produced by external forces is transmitted by the liquid equally in all directions.
In mechanical equilibrium, the horizontal fluid pressure is always the same; therefore, the free surface of a static liquid is always horizontal (except in cases of contact with the walls of the vessel). If we take into account the condition of incompressibility of the liquid, then the density of the medium under consideration does not depend on pressure.
Let's imagine a certain volume of liquid bounded by a vertical cylinder. Cross section Let us denote the liquid column as $S$, its height as $h$, liquid density as $ρ$, and weight as $P=ρgSh$. Then the following is true:
$p = \frac(P)(S) = \frac(ρgSh)(S) = ρgh$,
where $p$ is the pressure at the bottom of the vessel.
It follows that pressure varies linearly with altitude. In this case, $ρgh$ is the hydrostatic pressure, the change in which explains the emergence of the Archimedes force.
Formulation of Archimedes' law
Archimedes' law, one of the basic laws of hydrostatics and aerostatics, states: a body immersed in a liquid or gas is acted upon by a buoyant or lifting force equal to the weight of the volume of liquid or gas displaced by the part of the body immersed in the liquid or gas.
Note 2
The emergence of the Archimedean force is due to the fact that the medium - liquid or gas - tends to occupy the space taken away by the body immersed in it; in this case the body is pushed out of the environment.
Hence the second name for this phenomenon – buoyancy or hydrostatic lift.
The buoyancy force does not depend on the shape of the body, as well as on the composition of the body and its other characteristics.
The emergence of Archimedean force is due to the difference in environmental pressure at different depths. For example, the pressure on the lower layers of water is always greater than on the upper layers.
The manifestation of Archimedes' force is possible only in the presence of gravity. So, for example, on the Moon the buoyant force will be six times less than on Earth for bodies of equal volumes.
The emergence of Archimedes' Force
Let's imagine any liquid medium, for example, ordinary water. Let us mentally select an arbitrary volume of water by a closed surface $S$. Since all liquid is in mechanical equilibrium, the volume we have allocated is also static. This means that the resultant and moment external forces, affecting this limited volume, take zero values. External forces in in this case– the weight of a limited volume of water and the pressure of the surrounding liquid on the outer surface $S$. It turns out that the resultant $F$ of the forces of hydrostatic pressure experienced by the surface $S$ is equal to the weight of the volume of liquid that was limited by the surface $S$. In order for the total moment of external forces to vanish, the resultant $F$ must be directed upward and pass through the center of mass of the selected volume of liquid.
Now let us denote that instead of this conditional limited liquid, any solid body of the appropriate volume was placed in the medium. If the condition of mechanical equilibrium is met, then from the side environment no changes will occur, including the pressure acting on the surface $S$ will remain the same. Thus we can give a more precise formulation of Archimedes' law:
Note 3
If a body immersed in a liquid is in mechanical equilibrium, then the buoyant force of hydrostatic pressure acts on it from the environment surrounding it, which is numerically equal to the weight of the medium in the volume displaced by the body.
The buoyant force is directed upward and passes through the center of mass of the body. So, according to Archimedes’ law, the buoyancy force holds:
$F_A = ρgV$, where:
- $V_A$ - buoyancy force, H;
- $ρ$ - density of liquid or gas, $kg/m^3$;
- $V$ - volume of a body immersed in the medium, $m^3$;
- $g$ - free fall acceleration, $m/s^2$.
The buoyant force acting on the body is opposite in direction to the force of gravity, therefore the behavior of the immersed body in the medium depends on the ratio of the gravity moduli $F_T$ and the Archimedean force $F_A$. There are three possible cases here:
- $F_T$ > $F_A$. The force of gravity exceeds the buoyant force, therefore the body sinks/falls;
- $F_T$ = $F_A$. The force of gravity is equalized with the buoyant force, so the body “hangs” in the liquid;
- $F_T$
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). Speaking modern language, the body floats if it average density 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 solid body, the upward impulse of the collisions of the 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.