If we place a heavy stack of books on the table, we will increase the pressure not only on the table, but also, accordingly, on the floor under the table. Walls, ceilings, windows and doors will not feel this pressure.
Even if we put all the clothes from the closets, food from the refrigerator, TV, dumbbells on the table and, in addition, we perch ourselves with our feet, the walls and ceiling will not feel any changes. Unless they might be hit by a splinter from the table flying apart under the weight of all this stuff, but the changes in pressure on them will be zero. The situation is different with gases and liquids. If in a closed vessel we change the pressure on the liquid or gas filling the vessel, then the change in pressure will be felt by absolutely all the walls of this vessel.
What is Pascal's law?
You can independently carry out an experiment that clearly confirms this phenomenon. To do this, you need to take a thick rubber ball and fill it with water, and then tie it or seal it in some other way. Carefully, so as not to tear, use a needle to make several holes in different places of the water-filled ball. Water begins to seep through the holes. And now, if we squeeze the ball in our hands, we will see that water begins to pour out much more actively through absolutely all the holes. That is, by increasing the pressure at the compression points, we see that the pressure also increased equally in all directions, on all the walls of the vessel, that is, in this case, the ball.
The same thing will happen if you fill a balloon with smoke. This occurs due to the fact that actively moving particles of liquid and gas are mixed throughout the entire volume, and the pressure that reduces the volume for their free movement in one place will cause the same reduction in volume in all directions. This is Pascal's law: Liquids and gases transmit the pressure exerted on them in all directions equally. This law was discovered in the 17th century by the French scientist Pascal and therefore bears his name.
Formula of Pascal's law and its application
Pascal's law is described by the pressure formula:
where p is the pressure,
F - applied force,
S is the area of the vessel.
From the formula we see that with increasing force of influence for the same area of the vessel, the pressure on its walls will increase. Pressure is measured in newtons per square meter or in pascals (Pa), in honor of the scientist who discovered Pascal's law. Its use underlies many devices and is quite common in manufacturing. These include, in particular, hydraulic presses, pneumatic brakes and tools, and much more.
Pressure is a scalar quantity equal to the ratio of the normal component of the force acting on an elementary area inside the liquid to the area of this elementary area.
Tangential components of force D F not significant, because lead to liquid fluidity, i.e. imbalance.
Units of pressure. In SI – Pa (pascal): 1 Pa = 1 N/m 2 ;
in GHS – dyn/cm2.
Extra-systemic units: physical (normal) atmosphere (atm) is equal to the pressure of a column of mercury 760 mm high;
millimeter of mercury (mmHg).
1mm. rt. Art. = r Hg gh = (13.6 × 10 3 kg/m 3) × (9.81 m/s 2) × (10 -3 m) = 133 Pa.
1 atm = 760 mm. rt. Art. = 1.01×10 5 Pa.
Properties of a liquid (gas) at rest.
1. The force caused by the pressure of a fluid at rest always acts perpendicular to the surface with which this medium is in contact.
2. Liquids and gases create pressure in all directions.
Forces acting on particles of a liquid or gas are of one of two types.
1) Volume forces- these are long-range forces that act on each element of the volume of a liquid or gas. An example of such a force is gravity.
2) Surface forces- these are short-range forces that arise as a result of direct contact between the interacting elements of a liquid, gas and solid at their common boundary. An example of a surface force is the force of atmospheric pressure.
Pascal's law. Surface forces acting on a stationary liquid (or gas) create pressure that is equal at all points of the liquid (gas). The magnitude of pressure at any point in a liquid (gas) does not depend on the direction (i.e., on the orientation of the elementary area).
Proof.
1. Let us prove that the pressure at a given point in the liquid is the same in all directions.
Rice. 5.1.1.a Fig. 5.1.1.b
To prove this we will use hardening principle: Any element of a fluid can be treated as a solid and the equilibrium conditions of a solid can be applied to that element.
Let us mentally select in the vicinity of a given point of the liquid an infinitely small solidified volume in the form of a trihedral prism (Fig. 5.1.1), one of the faces of which (the OBCD face) is located horizontally. The areas of the bases AOB and KDC will be considered small compared to the areas of the side faces. Then the volume of the prism will be small, and, consequently, the force of gravity acting on this prism will be small.
Surface forces act on each face of the prism F 1 , F 2 and F 3. From the equilibrium of the liquid it follows that, i.e. vectors F 1 , F 2 and F 3 form a triangle (in Fig. 5.1.1.b), similar to triangle. Then
Let's multiply the denominators of these fractions by OD = BC = AK, Þ
Thus, pressure in a stationary liquid does not depend on the orientation of the area inside the liquid.
2. Let us prove that the pressure at any two points of the liquid is the same.
Let us consider two arbitrary points A and B of the fluid, separated from each other by a distance DL. Let us select an arbitrarily oriented cylinder in the liquid, at the centers of whose bases are the points A and B we have chosen (Fig. 5.1.2). We will assume that the areas of the bases of the cylinder DS are small, then the volumetric forces will also be small compared to the surface forces.
Let's assume that the pressures at points A and B are different: , then , which means the selected volume will begin to move. The resulting contradiction proves that the pressure at any two points in a liquid is the same.
An example of surface forces for which Pascal's law holds is the force of atmospheric pressure.
Atmosphere pressure- this is the pressure that atmospheric air exerts on all bodies; it is equal to the force of gravity acting on a column of air with a unit base area.
Torricelli experience demonstrated the presence of atmospheric pressure and made it possible to measure it for the first time. This experience was described in 1644.
Rice. 5.1.3. Rice. 5.1.4.
In this experiment, a long glass tube, sealed at one end, is filled with mercury; then its open end is clamped, after which the tube is turned over, the clamped end is lowered into a vessel with mercury and the clamp is removed. The mercury in the tube drops somewhat, i.e. Some of the mercury is poured into the vessel. Volume of space above mercury in a tube called a torrichel void. (The vapor pressure of mercury in a torrichel void at 0°C is 0.025 Pa.)
The mercury level in the tube is the same regardless of how the tube is installed: vertically or at an angle to the horizontal (Fig. 5.1.3). Under normal normal conditions, the vertical height of mercury in the tube is h= 760 mm. If instead of mercury the tube were filled with water, then the height h= 10.3 m.
Instruments used to measure atmospheric pressure are called barometers. The simplest mercury barometer is a Torricelli tube.
In order to explain why the Torricelli tube really allows you to measure atmospheric pressure, we turn to a consideration of volumetric forces and the calculation of the dependence of pressure in a liquid on depth h.
Pressure in a liquid created by volumetric forces, i.e. gravity is called hydrostatic pressure.
Let us obtain the formula for fluid pressure at depth h. To do this, we select a solidified parallelepiped in the liquid, one of the bases of which is located on the surface of the liquid, and the other at depth h(Fig. 5.1.4). At this depth, the forces shown in the figure act on the parallelepiped.
Forces acting on the parallelepiped along the axis x balanced. Let us write down the condition of equilibrium of forces along the axis y.
Where p 0 – atmospheric pressure, – mass of the parallelepiped, r – density of the liquid. Then
The first term in formula (5.1.3) is associated with surface forces, and the second term, called hydrostatic pressure, is associated with volumetric forces.
If a container of liquid moves with acceleration a, directed downward, then condition (5.1.2) takes the form: , Þ
In the state of zero-gravity ( a = g) hydrostatic pressure is zero.
Examples of application of Pascal's law.
1. Hydraulic press (Fig. 5.1.5).
3. Hydrostatic paradox . (Fig. 5.1.8).
Let's take three vessels of different shapes, but with the same cross-sectional area of the bottom. Let's assume this area is S = 20 cm 2 = 0.002 m 2. The water level in all vessels is the same and equal to h = 0.1 m. However, due to the different shapes of the vessels, they contain different amounts of water. In particular, vessel A contains water weighing 3 N, vessel B contains water weighing 2 N, and vessel C contains water weighing 1 N.
The hydrostatic pressure at the bottom in all vessels is equal to Pa. The force of water pressure on the bottom of the vessels N is also the same. How can water weighing 1 N in the third vessel create a pressure force of 2 N?
The famous 17th-century French philosopher, mathematician and physicist Blaise Pascal made an important contribution to the development of modern science. One of his main achievements was the formulation of the so-called Pascal's law, which is associated with the properties of fluid substances and the pressure created by them. Let's take a closer look at this law.
Brief biography of the scientist
Blaise Pascal was born on June 19, 1623 in the French city of Clermont-Ferrand. His father was a vice president for tax collection and a mathematician, and his mother belonged to the bourgeois class. From a young age, Pascal began to show interest in mathematics, physics, literature, languages and religious teachings. He invented a mechanical calculator that could perform addition and subtraction operations. He devoted a lot of time to studying the physical properties of fluid bodies, as well as developing the concepts of pressure and vacuum. One of the scientist’s important discoveries was the principle that bears his name - Pascal’s law. Blaise Pascal died in 1662 in Paris due to paralysis of his legs, an illness that had accompanied him since 1646.
Concept of pressure
Before considering Pascal's law, let's look at such a physical quantity as pressure. It is a scalar physical quantity that denotes the force that acts on a given surface. When a force F begins to act on a surface of area A perpendicular to it, then the pressure P is calculated using the following formula: P = F/A. Pressure is measured in the International System of Units SI in pascals (1 Pa = 1 N/m2), that is, in honor of Blaise Pascal, who devoted many of his works to the issue of pressure.
If the force F acts on a given surface A not perpendicularly, but at a certain angle α to it, then the expression for pressure will take the form: P = F*sin(α)/A, in this case F*sin(α) is the perpendicular component force F to surface A.
Pascal's law
In physics, this law can be formulated as follows:
Pressure applied to a practically incompressible fluid substance, which is in equilibrium in a vessel having non-deformable walls, is transmitted in all directions with the same intensity.
You can verify the correctness of this law in the following way: you need to take a hollow sphere, make holes in it in various places, equip this sphere with a piston and fill it with water. Now, by creating pressure on the water using a piston, you can see how it pours out of all the holes at the same speed, which means that the water pressure in the area of each hole is the same.
Liquids and gases
Pascal's law was formulated for fluid substances. Liquids and gases fall under this concept. However, unlike gases, the molecules that form a liquid are located close to each other, which causes liquids to have such a property as incompressibility.
Due to the incompressibility property of a liquid, when a finite pressure is created in a certain volume, it is transmitted in all directions without loss of intensity. This is exactly what we are talking about in Pascal’s principle, which is formulated not only for fluid, but also for incompressible substances.
Considering the question of “gas pressure and Pascal’s law” in this light, it should be said that gases, unlike liquids, are easily compressed without retaining volume. This leads to the fact that when a certain volume of gas is exposed to external pressure, it is also transmitted in all directions and directions, but at the same time loses intensity, and its loss will be stronger, the lower the gas density.
Thus, Pascal's principle is valid only for liquid media.
Pascal's principle and the hydraulic machine
Pascal's principle is used in various hydraulic devices. In order to use Pascal's law in these devices, the formula is as follows: P = P 0 +ρ*g*h, here P is the pressure that acts in the liquid at a depth h, ρ is the density of the liquid, P 0 is the pressure applied to the surface of the liquid, g (9.81 m/s 2) - acceleration of free fall near the surface of our planet.
The operating principle of a hydraulic machine is as follows: two cylinders, which have different diameters, are connected to each other. This complex vessel is filled with some liquid, such as oil or water. Each cylinder is equipped with a piston in such a way that no air remains between the cylinder and the surface of the liquid in the vessel.
Suppose that a piston in a cylinder with a smaller cross-section is affected by a certain force F 1, then it creates a pressure P 1 = F 1 / A 1. According to Pascal's law, pressure P 1 will be instantly transmitted to all points in space inside the liquid in accordance with the above formula. As a result, a piston with a large cross-section will also be subject to pressure P 1 with a force F 2 = P 1 * A 2 = F 1 * A 2 / A 1 . The force F2 will be directed opposite to the force F1, that is, it will tend to push the piston upward, and it will be greater than the force F1 exactly as many times as the cross-sectional area of the machine’s cylinders differs.
Thus, Pascal's law allows you to lift large loads with the help of small balancing forces, which is a kind of similarity to the Archimedes lever.
Other applications of Pascal's principle
The law under consideration is used not only in hydraulic machines, but is more widely used. Below are examples of systems and devices whose operation would be impossible if Pascal’s law were not valid:
- In the braking systems of cars and in the well-known anti-lock ABS system, which prevents the wheels of the car from locking during braking, which helps to avoid skidding and sliding of the vehicle. In addition, the ABS system allows the driver to maintain control of the vehicle when the latter performs emergency braking.
- In any type of refrigerators and cooling systems where the working substance is a liquid substance (freon).
Blaise Pascal was a French mathematician, physicist and philosopher who lived in the mid-seventeenth century. He studied the behavior of liquids and gases and studied pressure.
He noticed that the shape of the vessel had no effect on the pressure of the liquid inside it. He also formulated the principle: liquids and gases transmit the pressure exerted on them equally in all directions.
This principle is called Pascal's law for liquids and gases.
It is necessary to understand that this law did not take into account the force of gravity acting on the liquid. In reality, fluid pressure increases with depth due to gravity towards the Earth, and this is hydrostatic pressure.
To calculate its value, use the formula:
- pressure of the liquid column.
- ρ - liquid density;
- g - free fall acceleration;
- h - depth (height of the liquid column).
The total fluid pressure at any depth is the sum of hydrostatic pressure and pressure associated with external compression:
where p0 is the external pressure, for example, of a piston in a vessel with water.
Application of Pascal's law in hydraulics
Hydraulic systems use incompressible fluids, such as oil or water, to transfer pressure from one point to another within the fluid with a gain in force. Hydraulic devices are used to crush solids in presses. Aircraft have hydraulics installed in the brake systems and landing gear.
Since Pascal's law is also valid for gases, there are pneumatic systems in technology that use air pressure.
Archimedes' power. Condition of floating bodies
Knowing Archimedean force (also known as buoyant force) is important when trying to understand why some bodies float while other bodies sink.
Let's look at an example. A man is in the pool. When he is completely submerged under water, he can easily perform a somersault, somersault, or jump very high. On land, performing such stunts is much more difficult.
This situation in the pool is possible due to the fact that the Archimedean force acts on a person in the water. In a liquid, pressure increases with depth (this is also true for gas). When the body is completely under water, the pressure of the liquid from below the body prevails over the pressure from above, and the body begins to float.
Archimedes' Law
A body in a liquid (gas) is subject to a buoyant force equal in magnitude to the weight of the amount of liquid (gas) that is displaced by the immersed part of the body.
- Ft - gravity;
- Fa - Archimedean force;
- ρl - density of liquid or gas;
- Vv. and. - the volume of displaced liquid (gas) equal to the volume of the immersed part of the body;
- Pv. and. - weight of displaced liquid.
Sailing condition
- FT>FA - the body is drowning;
- FT< FA - тело поднимается к поверхности до тех пор, пока не окажется в положении равновесия и не начнёт плыть;
- FT = FA - the body is in equilibrium in an aqueous or gaseous environment (floats).
Pascal's law is formulated as follows:
The pressure exerted on a liquid or gas is transmitted to any point without changes in all directions.
The law was formulated by the French scientist Blaise Pascal.
It should be noted that Pascal’s law is not about pressures at different points, but about disturbances pressure, therefore the law is also valid for liquid in the field of gravity. When moving incompressible fluid, we can conditionally speak of the validity of Pascal's law, because adding an arbitrary constant value to the pressure does not change the form of the equation of motion of the fluid (Euler's equation or, if the action of viscosity is taken into account, the Navier-Stokes equation), however in this case the term Pascal's law as a rule not applied.
Pascal's law is a consequence of the law of conservation of energy and is also valid for compressible liquids (gases).
Formula of Pascal's law and its application
Various hydraulic devices operate on the basis of Pascal’s law: brake systems, hydraulic presses, etc.
see also
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Notes
An excerpt characterizing Pascal's Law
-Where is Lise? – he asked, only answering her question with a smile.“She was so tired that she fell asleep in my room on the sofa. Ax, Andre! Que! tresor de femme vous avez,” she said, sitting down on the sofa opposite her brother. “She’s a perfect child, such a sweet, cheerful child.” I loved her so much.
Prince Andrei was silent, but the princess noticed the ironic and contemptuous expression that appeared on his face.
– But one must be lenient towards small weaknesses; who doesn't have them, Andre! Don't forget that she was brought up and grew up in the world. And then her situation is no longer rosy. You have to put yourself in everyone's position. Tout comprendre, c "est tout pardonner. [Whoever understands everything will forgive everything.] Think about what it must be like for her, poor thing, after the life to which she is accustomed, to part with her husband and remain alone in the village and in her situation? This very hard.
Prince Andrei smiled, looking at his sister, as we smile when listening to people whom we think we see right through.
“You live in a village and don’t find this life terrible,” he said.
- I'm different. What to say about me! I don’t wish for another life, and I cannot wish for it, because I don’t know any other life. And just think, Andre, for a young and secular woman to be buried in the best years of her life in the village, alone, because daddy is always busy, and I... you know me... how poor I am in ressources, [in interests.] for a woman accustomed to the best to society. M lle Bourienne is one...
“I don’t like her very much, your Bourienne,” said Prince Andrei.
- Oh no! She is very sweet and kind, and most importantly, she is a pitiful girl. She has no one, no one. To tell the truth, I not only don’t need her, but she’s shy. You know, I have always been a savage, and now I’m even more so. I love being alone... Mon pere [Father] loves her very much. She and Mikhail Ivanovich are two persons to whom he is always affectionate and kind, because they are both blessed by him; as Stern says: “we love people not so much for the good they have done to us, but for the good we have done to them.” Mon pere took her as an orphan sur le pavé, [on the pavement], and she is very kind. And mon pere loves her reading style. She reads aloud to him in the evenings. She reads great.
- Well, to be honest, Marie, I think it’s sometimes hard for you because of your father’s character? - Prince Andrei suddenly asked.
Princess Marya was at first surprised, then frightened by this question.
– ME?... Me?!... Is it hard for me?! - she said.
– He has always been cool; and now it’s getting hard, I think,” said Prince Andrei, apparently on purpose to puzzle or test his sister, speaking so easily about his father.