The pressure on the surface of a liquid produced by external forces is transmitted by the liquid equally in all directions.
The nature of the pressure of a liquid, gas and solid is different. Although the pressures of liquids and gases are of different natures, their pressures have one similar effect that distinguishes them from solids. This effect, or rather a physical phenomenon, describes Pascal's law.
Pascal's law The pressure produced by external forces at some point in a liquid or gas is transmitted through the liquid or gas without change to any point.
Pascal's law was discovered by the French scientist B. Pascal in 1653, this law is confirmed by various experiments.
Pressure is a physical quantity equal to the modulus of the force F acting perpendicular to the surface, which is per unit area S of this surface.
Pascal's law formula Pascal's law is described by the pressure formula:
\(p = \dfrac(F)(S)\)
where p is the pressure (Pa), F is the applied force (N), S is the surface area (m2).
Pressure is a scalar quantity It is important to understand that pressure is a scalar quantity, that is, it has no direction.
Ways to reduce and increase pressure:
In order to increase the pressure, it is necessary to increase the applied force and/or reduce the area of its application.
Conversely, to reduce pressure, it is necessary to reduce the applied force and/or increase the area of its application.
The following types of pressure are distinguished:
- atmospheric (barometric)
- absolute
- excess (gauge)
Gas pressure depends on:
- from the mass of gas - the more gas in the vessel, the greater the pressure;
- on the volume of the vessel - the smaller the volume with a gas of a certain mass, the greater the pressure;
- from temperature - with increasing temperature, the speed of movement of molecules increases, which interact more intensely and collide with the walls of the vessel, and therefore the pressure increases.
Liquids and gases transmit in all directions not only the pressure exerted on them, but also the pressure that exists inside them due to the weight of their own parts. The upper layers press on the middle ones, and the middle ones on the lower ones, and the lower ones on the bottom.
There is pressure inside the liquid. At the same level, it is the same in all directions. With depth, pressure increases.
Pascal's law means that if, for example, you press on a gas with a force of 10 N, and the area of this pressure is 10 cm2 (i.e. (0.1 * 0.1) m2 = 0.01 m2), then the pressure in the place where the force is applied will increase by p = F/S = 10 N / 0.01 m2 = 1000 Pa, and the pressure in all places of the gas will increase by this amount. That is, the pressure will be transmitted without changes to any point in the gas.
The same is true for liquids. But for solids - no. This is due to the fact that the molecules of liquid and gas are mobile, and in solids, although they can vibrate, they remain in place. In gases and liquids, molecules move from an area of higher pressure to an area of lower pressure, so that the pressure throughout the volume quickly equalizes.
Unlike solids, liquids and gases in a state of equilibrium do not have elastic shape. They have only volumetric elasticity. In a state of equilibrium, the voltage in a liquid and gas is always normal to the area on which it acts. Tangential stresses cause only changes in the shape of elementary volumes of the body (shears), but not the magnitude of the volumes themselves. For such deformations in liquids and gases, no effort is required, and therefore, in these media at equilibrium, tangential stresses do not arise.
law of communicating vessels in communicating vessels filled with a homogeneous liquid, the pressure at all points of the liquid located in the same horizontal plane is the same, regardless of the shape of the vessels.
In this case, the surfaces of the liquid in communicating vessels are installed at the same level
The pressure that appears in a liquid due to the gravitational field is called hydrostatic. In a liquid at a depth \(H\), counting from the surface of the liquid, the hydrostatic pressure is equal to \(p=\rho g H\) . The total pressure in a liquid is the sum of the pressure at the surface of the liquid (usually atmospheric pressure) and hydrostatic pressure.
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Liquids and gases transmit the pressure that is applied to them equally in all directions.
This law was discovered in the middle of the 14th century by the French scientist B. Pascal and subsequently received his name.
The fact that liquids and gases transmit pressure is explained by the high mobility of the particles from which they are composed; this significantly distinguishes them from solid bodies, whose particles are inactive and can only oscillate around their equilibrium positions. Let's say a gas is in a closed vessel with a piston; its molecules evenly fill the entire volume provided to it. Let us move the piston, reducing the volume of the vessel, the layer of gas adjacent to the piston will compress, the gas molecules will be located more densely than at some distance from the piston. But after some time, the gas particles, participating in chaotic movement, will mix with other particles, the gas density will level out, but will become greater than before the piston moved. In this case, the number of impacts on the bottom and walls of the vessel increases, therefore, the pressure of the piston is transmitted by the gas in all directions equally and at each point increases by the same amount. Similar reasoning can be applied to liquids.
Formulation of Pascal's law
The pressure produced by external forces on a liquid (gas) at rest is transmitted by the substance in all directions without change to any point of the liquid (gas) and the walls of the vessel.
Pascal's law holds for incompressible and compressible liquids and gases if compressibility is neglected. This law is a consequence of the law of conservation of energy.
Hydrostatic pressure of liquids and gases
Liquids and gases transmit not only external pressure, but also pressure that arises due to the existence of gravity. This force creates pressure inside the liquid (gas), which depends on the depth of immersion, while applied external forces increase this pressure at any point in the substance by the same amount.
The pressure exerted by a liquid (gas) at rest is called hydrostatic. Hydrostatic pressure ($p$) at any depth inside a liquid (gas) does not depend on the shape of the vessel in which it (he) is located and is equal to:
where $h$ is the height of the liquid (gas) column; $\rho$ is the density of the substance. From formula (1) for hydrostatic pressure it follows that in all places of liquid (gas) that are at the same depth, the pressure is the same. As depth increases, hydrostatic pressure increases. Thus, at a depth of 10 km, the water pressure is approximately $(10)^8Pa$.
A corollary of Pascal's law: the pressure at any point on the same horizontal level of a liquid (gas) in a state of equilibrium has the same value.
Examples of problems with solutions
Example 1
Exercise. Three vessels of different shapes are given (Fig. 1). The area of the bottom of each vessel is $S$. In which of the vessels is the pressure of the same liquid on the bottom greatest?
Solution. This problem deals with the hydrostatic paradox. A consequence of Pascal's law is that the pressure of a liquid does not depend on the shape of the vessel, but is determined by the height of the liquid column. Since, according to the conditions of the problem, the area of the bottom of each vessel is equal to S, from Fig. 1 we see that the height of the liquid columns is the same, despite the different weight of the liquid, the force of “weight” pressure on the bottom in all vessels is the same and is equal to the weight of the liquid in a cylindrical vessel. The explanation for this paradox lies in the fact that the force of liquid pressure on inclined walls has a vertical component, which is directed downwards in a vessel that narrows towards the top and directed upwards in an expanding one.
Example 2
Exercise. Figure 2 shows two communicating vessels with liquid. The cross section of one of the vessels is $n\$ times smaller than the second. The vessels are closed with pistons. A force $F_2 is applied to the small piston.\ $What force must be applied to the large piston for the system to be in a state of equilibrium?
Solution. The problem presents a diagram of a hydraulic press that operates on the basis of Pascal's law. The pressure that the first piston creates on the liquid is:
The second piston exerts pressure on the liquid:
If the system is in equilibrium, $p_1$ and $p_2$ are equal, we write:
\[\frac(F_1)(S_1)=\frac(F_2)(S_2)\left(2.3\right).\]
Let's find the magnitude of the force applied to the large piston:
Answer.$F_1=nF_(2)$
Let us consider a liquid that is in a vessel under a piston (Fig. 1), when the forces acting on the free surface of the liquid are significantly greater than the weight of the liquid or the liquid is in weightlessness, i.e. we can assume that only surface forces act on the liquid, and the weight of the liquid can be neglected. Let us mentally select some small cylindrical arbitrarily oriented volume of liquid. Pressure forces and the rest of the liquid act on the bases of this volume of liquid, and pressure forces and on the side surface. The equilibrium condition for a small volume released in a liquid:
In projection onto the axis Ox:
those. the pressure at all points of a weightless stationary fluid is the same.
When the surface force changes, the values will change p 1 and p 2, but their equality will remain. This was first established by B. Pascal.
Pascal's law: liquid (gas) transfers the external pressure produced on top of it by lean forces in all directions without change.
The pressure exerted on a liquid or gas is transmitted not only in the direction of the force, but also to each point of the liquid (gas) due to the mobility of the molecules of the liquid (gas).
This law is a direct consequence of the absence of static friction forces in liquids and gases.
Pascal's law is not applicable in the case of a moving liquid (gas), as well as in the case when the liquid (gas) is in a gravitational field; Thus, it is known that atmospheric and hydrostatic pressure decreases with altitude
Archimedes' Law: 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 (called by the power of Archimedes)
F A = ρ gV,
where ρ is the density of the liquid (gas), g is the acceleration of free fall, and V- 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 .
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.
In the absence of gravity, 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.
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:
The body drowns;
A body floats in a liquid or gas;
The body floats up until it begins to float.
Another formulation (where is the density of the body, is the density of the medium in which it is immersed):
· - the body drowns;
· - the body floats in liquid or gas;
· - the body floats up until it begins to float.
Bernoulli's equation.
Bernoulli's law is a consequence of the law of conservation of energy for a stationary flow of an ideal (that is, without internal friction) incompressible fluid: 
, here is the density of the liquid, is the flow velocity, is the height at which the liquid element in question is located, is the pressure at the point in space where the center of mass of the liquid element in question is located, is the acceleration of gravity. The constant on the right side is usually called pressure, or total pressure, as well as Bernoulli integral. The dimension of all terms is the unit of energy per unit volume of liquid.
According to Bernoulli's law, the total pressure in a steady fluid flow remains constant along the flow. Total pressure consists of weight (ρ gh), static ( p) and dynamic pressure.
From Bernoulli's law it follows that as the flow cross-section decreases, due to an increase in speed, that is, dynamic pressure, the static pressure decreases. Bernoulli's law is valid in its pure form only for liquids whose viscosity is zero, that is, liquids that do not stick to the surface of the pipe. In fact, it has been experimentally established that the velocity of a liquid on the surface of a solid is almost always exactly zero (except in cases of jet separation under certain rare conditions). Bernoulli's law can be applied to the flow of an ideal incompressible fluid through a small hole in the side wall or bottom of a wide vessel.
For a compressible ideal gas 
, (constant along the streamline or vortex line) where is the Adiabatic constant of the gas, p- gas pressure at a point, ρ - gas density at a point, v- gas flow speed, g- acceleration of gravity, h- height relative to the origin. When moving in a non-uniform field gh is replaced by the gravitational field potential.
This law was discovered by the French scientist B. Pascal in 1653. It is sometimes called the fundamental law.
Pascal's law can be explained in terms of the molecular structure of matter. In solids, molecules form a crystal lattice and vibrate around their own. In liquids and gases, molecules have relative freedom; they can move relative to each other. It is this feature that allows the pressure produced on a liquid (or gas) to be transmitted not only in the direction of the force, but in all directions.
Pascal's law has found wide application in modern technology. The work of modern superpresses is based on Pascal's law, which allows creating pressures of about 800 MPa. Also, the work of all hydraulic automation that controls spaceships, jet airliners, numerically controlled machines, excavators, dump trucks, etc. is based on this law.
Hydrostatic fluid pressure
The hydrostatic pressure inside a liquid at any depth does not depend on the shape of the vessel in which the liquid is located and is equal to the product of the liquid and the depth at which the pressure is determined:
In a homogeneous fluid at rest, the pressures at points lying in the same horizontal plane (at the same level) are the same. In all cases shown in Fig. 1, the liquid pressure at the bottom of the vessels is the same.
Fig.1. Independence of hydrostatic pressure from the shape of the vessel
At a given depth, the liquid presses equally in all directions, so the pressure on the wall at a given depth will be the same as on a horizontal platform located at the same depth.
The total pressure in a liquid poured into a vessel is the sum of the pressure at the surface of the liquid and hydrostatic pressure:
The pressure at the surface of a liquid is often equal to atmospheric pressure.
Examples of problem solving
EXAMPLE 1
| Exercise | Water is poured into a hollow cube with an edge of 40 cm. Find the force of water pressure on the bottom and walls of the cube. |
| Solution | Let's do the drawing.
1) Hydrostatic pressure at depth The force of water pressure on the bottom of the cube: where is the bottom area; , 2) The average pressure on the side face is equal to half the sum of the pressures at the surface level and at the bottom level:
pressure force on the cube wall:
From the tables, the density of water is kg/m. Let's convert the units to the SI system: cube edge length cm m. Let's calculate: 1) pressure force on the bottom: 2) pressure force on the wall: |
| Answer | The water pressure forces on the bottom and walls of the cube are 627 and 314 N, respectively. |
EXAMPLE 2
| Exercise | Two elbows of a U-shaped tube are filled with water and oil, separated by mercury. The interfaces between mercury and liquids in both elbows are at the same height. Determine the height of the water column if the height of the oil column is 20 cm. |
| Solution | Let's do the drawing.
According to Pascal's law, the pressure in both bends of the tube is equal: Water pressure level oil pressure level Substituting expressions for liquid pressures into the first equality, we obtain: |
