Hydraulics inverted vessel with water experiments. What is the hydrostatic paradox? Basics of the theory of floating bodies

In this section we will consider a law of nature that applies only to liquids and gases and does not apply to solids.
Let's mentally imagine that inside the liquid at a given point there is a small platform. The liquid produces pressure on this area. It is important that the fluid pressure on this small area does not depend on the orientation of the area. To prove the validity of this statement, we will use the so-called solidification principle. According to this principle, any volume of liquid or gas in the static case, when the elements of the liquid do not move relative to each other, can be considered as a solid body and the equilibrium conditions of a solid body can be applied to this volume.
Let us select a small volume in the liquid in the form of a long triangular prism (Fig. 9.23, a), one of the faces of which (the OBCD face) is located horizontally. The areas of the prism bases will be considered small compared to the area of ​​the side faces. The volume of the prism will be small, and therefore the force of gravity acting on this prism will be small. This force can be neglected in comparison with the pressure forces acting on the edge of the prism1.

1 The surface area is proportional to the square of the linear dimensions of the body, and the volume is proportional to the cube. Therefore, for a small prism, the force of gravity, proportional to the volume, can always be neglected in comparison with the force of pressure, proportional to the surface area.
Figure 9.23, b shows the cross section of the prism. The forces Flt F2, F3 act on the lateral faces of the prism. We do not take into account the pressure forces on the bases of the prism, since they are balanced. Then according to the equilibrium condition
Fi + F2 + F3 = o.
The vectors of these forces form a triangle similar to triangle AOB, since the angles in these two triangles are respectively equal (Fig. 9.23, c). From the similarity of triangles it follows that
?i = = ї±
OA OB AB"
Let us multiply the denominators of these fractions by OD, BC and KA, respectively (OD = BC = KA):
F1 F2 F3
OA OD OB BC AV KA"
From Figure 9.23a it is clear that the denominator of each fraction is equal to the area of ​​the corresponding lateral face of the prism. Denoting the areas of these prism faces as S2, S3, we obtain:
F±==F_2=F3 S2 "3
or
Рі=Рг=Рз- (9.6.1)
So, the pressure in a stationary liquid (or gas) does not depend on the orientation of the area inside the liquid.
According to formula (9.5.3), the pressure is the same at all points lying at a given level. This pressure on the underlying layers of liquid is created by a column of liquid of height h. Therefore, we can conclude that the pressure of the upper layers of liquid on the layers of liquid located below them is transmitted by the underlying layers equally in all directions.
But pressure on the liquid can be produced by external forces, for example using a piston. Taking this into account, we arrive at Pascal's law: the pressure produced by external forces on a fluid at rest is transmitted equally by the fluid in all directions.
In this formulation, Pascal's law remains true for the general case, that is, for the case when we take gravity into account. If the force of gravity creates a pressure inside a fluid at rest, depending on the depth of immersion, then the applied

New external (surface) forces increase the pressure at each point of the liquid by the same amount.
Rice. 9.24
Pascal's law can be confirmed experimentally. If, for example, you fill a metal ball with several holes in it with water, and then compress the water with a piston, then identical jets of water will spray out of all the holes (Fig. 9.24, a). Pascal's law is also valid for gases (Fig. 9.24, b). Hydrostatic paradox
Let's take three vessels of different shapes (Fig. 9.25). Water weighing 3 N is poured into vessel A, water weighing 3 N is poured into vessel B, and weighing 1 N is poured into vessel C. The water level in all three vessels is at a height of 0.1 m. The bottom area of ​​each vessel is 20 cm2 = 0.002 m2. Using the formula p = pgh, we find that the pressure at the bottom of each vessel is 1000 Pa. Knowing the pressure, we use the formula F = pS to find that the pressure force on the bottom of the vessel in all three cases is equal to 2 N. It can’t be, you say. How can water weighing 1 N in the third vessel create a force of 2 N on the bottom? This position, which seems to contradict common sense, is known as the “hydrostatic paradox”, or “Pascal’s paradox”.

Trying to solve the riddle of the hydrostatic paradox, Pascal placed vessels similar to those shown in Figure 9.25 on special scales that made it possible to measure the pressure force on the bottom of each vessel (Figure 9.26, a, b, c). The bottom of the vessel, standing on the scales, was not rigidly connected to the vessel, and the vessel itself was fixed motionless on a special stand. The readings of the scales confirmed the calculations. Thus, contrary to common sense, the force of pressure on the bottom of the vessel does not depend on the shape of the vessel, but depends only on the height of the liquid column, its density and the area of ​​the bottom.
This experience leads to the idea that with the proper shape of the vessel it is possible, with the help of a very small amount of liquid, to - 300 cm3
100 cm3

V)
10 cm
shhhhhhhh,
A)
200 cm3
10 cm
ъшшшяшШЯШ, b)
Rice. 9.26 create very large pressure forces on the bottom. Pascal attached a tube with a cross-sectional area of ​​1 cm to a tightly sealed barrel and poured water into it to a height of 4 m (weight of water in the tube P = mg = 4 N). The resulting pressure forces tore the barrel (Fig. 9.27). Taking the area of ​​the bottom of the barrel to be 7500 cm2, we obtain a pressure force on the bottom of 30,000 N, and this enormous force is caused by just one mug of water (400 cm3) poured into the tube.

How to explain Pascal's paradox? The force of gravity creates pressure inside a liquid at rest, which, according to Pascal’s law, is transmitted to both the bottom and the walls of the vessel. If a liquid presses on the bottom and walls of a vessel, then the walls of the vessel also produce pressure on the liquid (Newton’s third law).
If the walls of the vessel are vertical (Fig. 9.28, a), then the pressure forces of the vessel walls on the liquid are directed horizontally. Consequently, these forces do not have a vertical component. Therefore, the force of liquid pressure on the bottom of the vessel is equal in this case to the weight of the liquid in the vessel. If the vessel expands upward (Fig. 9.28, b) or narrows (Fig. 9.28, c), then the pressure force of the vessel walls on the liquid has a vertical component, directed upward in the first case, and downward in the second. Therefore, in a vessel expanding upward, the pressure force on the bottom is equal to the difference between the weight of the liquid and the vertical component of the pressure force Fig. 9.27 walls. Therefore, the pressure force on

Rice. 9.28
the bottom in this case is less than the weight of the liquid. In a vessel that tapers upward, on the contrary, the pressure force on the bottom is equal to the sum of the weight of the liquid and the vertical component of the pressure force of the walls on the liquid. Now the force of pressure on the bottom is greater than the weight of the liquid.
Of course, if you put various vessels on the scales without a separable bottom and not fixed on stands, then the scale readings will be different (2 N, 3 N and 1 N, if the mass of the vessels can be neglected). In this case, the vertical component of the forces of liquid pressure on the side surface will be added to the force of liquid pressure on the bottom in an expanding vessel. In a narrowing vessel, the corresponding component of the pressure forces will be subtracted from the pressure force on the bottom.
Hydraulic Press
Pascal's law allows us to explain the action of a device common in technology - a hydraulic press.
A hydraulic press consists of two cylinders of different diameters, equipped with pistons and connected by a tube (Fig. 9.29). The space under the pistons and the tube are filled with liquid (mineral oil). Let us denote the area of ​​the first piston by S1, and the area of ​​the second by S2. Let us apply force F2 to the second piston. Let's find what force F2 must be applied to the first piston to maintain equilibrium.
According to Pascal's law, the pressure at all points of the liquid must be the same (we neglect the effect of gravity on the liquid). But the pressure under the first piston is equal
Fi
-x-, and under the second.
shhhhhhhh,: Fig. 9.29 Therefore,

shhhhhhhhh, Fig. 9.30
i 2
2s:
і
(9.6.2)
F^F,
Hence the force module Fy is the same number of times greater than the force module F2, how many times the area of ​​the first piston is greater than the area of ​​the second. Thus, with the help of a hydraulic press, it is possible, by means of a small force applied to a piston of a small cross-section, to obtain enormous forces acting on a piston of a large cross-section. The hydraulic press principle is used in hydraulic jacks to lift heavy loads.
Thanks to Pascal's law, paradoxical situations are possible when a mug of water added to a barrel leads to its rupture. The same Pascal's law underlies the design of hydraulic presses.
A vessel with water is installed on the edge of the board (Fig. 9.30). Will the balance be disrupted if a plank is placed on the surface of the water and a weight is placed on it so that the plank and the weight float on the surface of the water not in the middle of the vessel?

Discovery of the fundamental law of hydrostatics. Pascal's experiment.

Date of: 1647–1653.

Methods: qualitative and semi-quantitative research.

Directness of the experiment: direct observation.

Artificiality of the conditions being studied: natural and artificial.

Fundamental principles explored: basic law of hydrostatics.

The French scientist Blaise Pascal (1623–1662) became famous in mathematics, physics, and philosophy - the mother of all sciences. One of his most important contributions to physics is related to the study of hydrostatics, i.e. science of liquid (gas) in a state of equilibrium (i.e. rest).

Pascal's experiment is the talk of physics: the engraving shown below will probably seem familiar to everyone. In it, Pascal, standing on the second floor balcony of his house, pours a couple of mugs of water into a thin long tube inserted into a barrel of water - and the barrel cracks, unable to withstand the pressure of a large column of liquid. Experience clearly demonstrates hydrostatic paradox: the force of liquid pressure on the bottom of the vessel turns out to be much greater than the weight of this liquid.

So, according to Pascal's instructions, the oak barrel was filled to the brim with water and hermetically sealed. After that, a hole was made in its top cover and a thin long tube was inserted into it, so that the latter was located vertically. After this tube is filled with water, topped up by Pascal from a mug, the pressure in the barrel will increase by

where is the density of water, is the acceleration of gravity, and is the height of the tube. If we substitute the last value into the formula for pressure, we get . Such excess pressure acting from the inside of the barrel to the outside, as it turns out, is quite enough to break a strong tree.

Let us note, firstly, that O greater strength atmospheric pressure presses on the barrel when the latter is open and a thin tube is not inserted into it - but this force presses on both the outer and inner surfaces of the walls of the barrel. Thus, atmospheric pressure only compresses the wooden walls. If the latter were fragile in relation to compression, then the barrel would not withstand even simple atmospheric pressure. Secondly, if we take the diameter of the barrel equal to one meter, then it turns out that a force equal to

which corresponds to the weight of a solid body with a mass of about three tons. The formula for hydrostatic pressure itself (see above) is an expression of Pascal's law. However, this law also states that the pressure inside a fluid at rest is the same in all directions. In particular, water presses on the lowest parts of the side walls of Pascal's barrel with the same force (per unit area) as on its bottom. Pascal was convinced of the independence of pressure from direction with the help of another experiment, namely the experiment with a tube, which was later named in his honor (see figure on the right). A Pascal tube consists of a cylindrical glass vessel into which a piston is inserted, and a hollow ball with small holes placed at the end of the vessel. If the tube is filled with water (see the right tube in the figure) and pressed on the piston, then water streams will flow from the holes in the ball. By the power of these streams (as well as their average length of their straight sections) one can judge the pressure with which water is thrown out through a given hole in the ball. Since these streams have approximately the same parameters, we can conclude that pressure in the liquid is transmitted evenly in all directions. A similar experiment was carried out with smoke (see the left tube in the figure), which led to the conclusion that Pascal’s law is also valid for gases. Naturally, the shape of the streams is distorted by the gravitational field of the Earth - but turning the tube upside down, we will see that, despite the fact that the piston is now located below the ball, the same picture is observed: the streams shooting upward “curve” a little faster than the streams hitting down. This suggests that the difference in their length is caused mainly by the Earth's gravity.

Finally, the formula for hydrostatic pressure given at the beginning of this article suggests that atmospheric pressure decreases with altitude, approximately linearly. Pascal was also convinced of this using Torricelli's mercury barometer, conducting experiments in Paris at the Tower of Saint-Jacques, and also comparing their results with the results of experiments carried out by his son-in-law Florent Perrier at the Puy de Dome mountain. In honor of these experiments, a monument to Pascal was subsequently erected on the Tower of Saint-Jacques.

Also, a simple experiment (see the figure on the right) allows you to verify that the pressure of the liquid decreases linearly with height. Water is poured into a cylinder, in the side walls of which small holes are made at different heights, and, just as in the experiment with Pascal’s tube, the lengths of the streams from these holes are observed. As is easy to see, the stronger the jet, the lower the hole is located, which indicates an increase in hydrostatic pressure under the weight of the liquid column.

In addition to his theoretical conclusions, Pascal's experiments and his basic law of hydrostatics led to his invention of the hydraulic press (see figure on the right), which is widely used in modern technology. If the pistons, as in the figure, are at the same height, then the fluid pressure under them is the same and equal. The pistons can be balanced if we place weights on them with masses and , accordingly, so that the forces of gravity acting on these weights are balanced by the force of fluid pressure (see figure):

The same principle allows, with the help of a small force applied to a smaller piston, to create a force that exceeds and acts on a larger one, which is actively used in technology. Of course, such amplification does not violate the law of conservation of energy in any way, since when the smaller rod moves downward, the upper one (due to the conservation of the volume of liquid in the press) moves up a smaller distance, and the work performed is equal.

Pascal's experiments on the pressure of liquids and gases once again debunked the theory of “fear of emptiness” dating back to Aristotle, which, in particular, explained the retention of a column of mercury in the tube of Torricelli’s barometer. Pascal showed that such phenomena are a consequence of pressure, and not a reverse force, a suction force acting from the side of the void, and also that all physical phenomena fall into place if we only take into account the pressure of the air around us.

Finally, the fundamental law of hydrostatics (Pascal’s law), formulated on the basis of Pascal’s experiments, is of fundamental importance for hydrostatics and expresses isotropy(independence of direction) internal stresses arising in liquids and gases.

Hydrostatic paradox

lies in the fact that the weight of a liquid poured into a vessel may differ from the pressure it exerts on the bottom of the vessel. Thus, in vessels expanding upward ( rice. ) the force of pressure on the bottom is less than the weight of the liquid, and in converging areas it is greater. In a cylindrical vessel both forces are equal.

If the same liquid is poured to the same height into vessels of different shapes, but with the same bottom area, then, despite the different weight of the poured liquid, the pressure force on the bottom is the same for all vessels and is equal to the weight of the liquid in a cylindrical vessel. This follows from the fact that the pressure of a fluid at rest depends only on the depth below the free surface and on the density of the fluid. G. p. is explained by the fact that since hydrostatic pressure R always normal to the walls of the vessel, the pressure force on the inclined walls has a vertical component p 1, which compensates for the excess weight against the cylinder 1 volume of liquid in the vessel 3 and the weight of the missing one against the cylinder 1 volume of liquid in the vessel 2 . G. p. was discovered by the French physicist B. Pascal (See Pascal).

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what the “Hydrostatic paradox” is in other dictionaries:

The weight of a liquid poured into a vessel may differ from the force of pressure it exerts on the bottom of the vessel. Thus, in vessels that expand upward, the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In a cylindrical vessel both forces are equal.... ... Big Encyclopedic Dictionary

The point is that the weight of a liquid poured into a vessel may differ from the force of pressure it exerts on the bottom of the vessel. Thus, in vessels that expand upward (Fig.), the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In cylindrical... ... Physical encyclopedia

Hydrostatic paradox is a phenomenon in which the weight of liquid poured into a vessel may differ from the force of pressure on the bottom. Causes Scheme of Pascal's experiment The reason for the hydrostatic paradox is that the liquid gives ... Wikipedia

Phys. the law by virtue of which the pressure on the bottom in vessels of different shapes, but with the same size, is filled. with the same liquid to the same height, in the same way, despite the difference in the amount of liquid. Dictionary of foreign words,... ... Dictionary of foreign words of the Russian language

The weight of the liquid poured into the vessel may differ from the force of pressure of the liquid on the bottom of the vessel. Thus, in vessels that expand upward (Fig.), the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In a cylindrical vessel both forces are equal.... ... encyclopedic Dictionary

The weight of the liquid poured into the vessel may differ from the force of pressure of the liquid on the bottom of the vessel. Thus, in vessels that expand upward (Fig.), the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In cylindrical in the vessel both forces are equal.... ... Natural science. encyclopedic Dictionary- (Pascal’s law) is formulated as follows: The pressure exerted on a liquid (or gas) in any one place on its boundary, for example, by a piston, is transmitted without change to all points of the liquid (or gas). The law is named after the French scientist Blaise... ... Wikipedia

Get lost! Get lost!

He's had bad luck all his life. As a child, an unexplained illness nearly ended his life. Fate spared him, but not for long. In his youth, sudden paralysis made him crippled - his legs refused to work, he could barely move. But all the more immeasurable is his feat in science. Overcoming physical suffering, he worked with tenacity and ecstasy, characteristic only of a brilliant thinker,

At the age of 16, Blaise Pascal became no less famous a mathematician than his contemporaries such as Fermat and Descartes. At the age of 18, he invented a calculating machine - the predecessor of the adding machine and the great-grandmother of the computer.

The time came when he invaded that area of ​​\u200b\u200bknowledge in which the great Galileo failed. He began with a discrepancy between the mass of water poured into a vessel and the force with which this mass presses on the bottom. Wanting to obtain a visual proof of the “hydrostatic paradox,” Pascal performs an experiment called “Pascal’s barrel.”

According to his instructions, a strong oak barrel was filled to the brim with water and tightly closed with a lid. The end of a vertical glass tube of such length was inserted into a small hole in the lid that its end was at the level of the second floor.

Going out onto the balcony, Pascal began filling the tube with water (Fig. 2). He had not even managed to pour out a dozen glasses when suddenly, to the amazement of the onlookers who surrounded the barrel, the barrel burst with a crash. She was torn apart by an incomprehensible force.

Pascal is convinced: yes, the force that breaks the barrel does not at all depend on the amount of water in the tube. It's all about the height to which the tube was filled. Next, the amazing property of water manifests itself - to transmit the pressure created on its surface (in the barrel) throughout the entire volume, to each point of the wall or bottom of the barrel.

So he comes to the discovery of the law that received his name, the name of Blaise Pascal: “Pressure applied to the surface of a liquid is transferred to each of its particles without changing its original value.”

On the surface of the water in the barrel under the lid, this pressure is P = ρgh, where ρ is the density of water; g - free fall acceleration; h is the height of the water column in the tube. By multiplying the resulting pressure by the diametrical cross-sectional area of ​​the barrel (S = DH), we get the force that crushed its strong oak walls:

P= ρg(h+H/2)(DH)

If we take the height of the water in the tube to be 4 m (second floor balcony), the diameter of the barrel is 0.8 m and the height of the barrel is 0.8 m, then no matter how small the amount of water in the tube is, the force breaking the barrel will be 27.6 kN.

Already relying on the law he discovered, Pascal obtains a corollary: “If a full vessel, closed on all sides, has two holes, one of which is 100 times larger than the other, then, placing in each hole a piston corresponding to this hole, the person pressing a small piston will create a force equal to the force of 100 people pressing on a piston with an area 100 times larger." Thus, Pascal substantiated the possibility of obtaining arbitrarily large forces from arbitrarily small ones using a liquid. It is difficult to overestimate the importance of this consequence for modern mechanical engineering. It led to the creation of superpresses with a pressure of (65-75) * 10 7 Pa. It formed the basis of the hydraulic drive, which in turn led to the emergence of hydraulic automation that controls modern jetliners, spaceships, computer-controlled machines, powerful dump trucks, mining combines, excavators...

But what about Pascal himself? Did he foresee that his law would mark an era in technological progress?

Suddenly, Pascal stopped all research activities and, leaving Paris, settled in a cell at the Port-Royal monastery. He cut off all ties with people of science, renounced everything that just yesterday constituted the meaning of his existence and completely devoted himself to religion. If the great Galileo, even the most cruel tortures in the dungeons of the Inquisition, did not force him to change science, then Pascal did it himself, without any coercion.

He ended his days dressed in hair shirt with a Bible on his lap. He mortified his flesh in order to atone for the most terrible, from the point of view of religion, sin - curiosity, passion for knowledge. And he died when he was only 39 years old.

But why did he recant? Perhaps he was frightened by his truly anti-divine discoveries, which promised the world such power, in comparison with which the divine power paled, or he lacked that single step from ignorance to knowledge that Archimedes was able to take, and which would have allowed him to reveal the paradoxical properties of water. In the bright chronicle of the history of science, the tragedy of Blaise Pascal became the only dark spot.

The hydrostatic paradox lies in the fact that the weight of a liquid poured into a vessel may differ from the pressure it exerts on the bottom of the vessel. Thus, in vessels that expand upward, the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In a cylindrical vessel both forces are equal. If the same liquid is poured to the same height into vessels of different shapes, but with the same bottom area, then, despite the different weight of the poured liquid, the pressure force on the bottom is the same for all vessels and is equal to the weight of the liquid in a cylindrical vessel. This follows from the fact that the pressure of a fluid at rest depends only on the depth below the free surface and on the density of the fluid. The hydrostatic paradox is explained as follows. Since hydrostatic pressure is always normal to the walls of the vessel, the pressure force on the inclined walls has a vertical component, which compensates for the weight of the volume of liquid that is excessive against the cylinder in a vessel expanding upward and the weight of the volume of liquid that is missing against the cylinder in a vessel that narrows upward. The hydrostatic paradox was discovered by the French physicist Blaise Pascal (1623–1662).