Take a cylindrical vessel with a horizontal bottom and vertical walls, filled with liquid to a height (Fig. 248).
Rice. 248. In a vessel with vertical walls, the pressure force on the bottom is equal to the weight of the entire liquid poured
Rice. 249. In all the depicted vessels, the force of pressure on the bottom is the same. In the first two vessels, it is greater than the weight of the poured liquid, in the other two it is less.
The hydrostatic pressure at each point on the bottom of the vessel will be the same:
If the bottom of the vessel has an area, then the pressure force of the liquid on the bottom of the vessel, i.e., is equal to the weight of the liquid poured into the vessel.
Let us now consider vessels that differ in shape, but with the same bottom area (Fig. 249). If the liquid in each of them is poured to the same height, then the pressure is on the bottom. the same in all vessels. Therefore, the pressure force on the bottom, equal to
also the same in all vessels. It is equal to the weight of a column of liquid with a base equal to the area of the bottom of the vessel and a height equal to the height of the poured liquid. On fig. 249 this pillar is shown near each vessel with dashed lines. Please note that the force of pressure on the bottom does not depend on the shape of the vessel and can be either greater or less than the weight of the poured liquid.
Rice. 250. Pascal's instrument with a set of vessels. The cross sections are the same for all vessels
Rice. 251. Pascal's barrel experiment
This conclusion can be verified experimentally using the device proposed by Pascal (Fig. 250). Vessels of various shapes that do not have a bottom can be fixed on the stand. Instead of a bottom, a plate suspended from the balance beam is pressed tightly against the vessel from below. In the presence of liquid in the vessel, a pressure force acts on the plate, which tears off the plate when the pressure force begins to exceed the weight of the weight standing on the other scale pan.
In a vessel with vertical walls (cylindrical vessel), the bottom opens when the weight of the poured liquid reaches the weight of the weight. In vessels of a different shape, the bottom opens at the same height of the liquid column, although the weight of the poured water can be both greater (vessel expanding upwards) and less (tapering vessel) than the weight of the weight.
This experience leads to the idea that with the proper shape of the vessel, with the help of a small amount of water, huge forces of pressure on the bottom can be obtained. Pascal attached a long thin vertical tube to a tightly caulked barrel filled with water (Fig. 251). When the tube is filled with water, the force of hydrostatic pressure on the bottom becomes equal to the weight of the column of water, the base area of which is equal to the area of the bottom of the barrel, and the height is equal to the height of the tube. Accordingly, the pressure forces on the walls and the upper bottom of the barrel also increase. When Pascal filled the tube to a height of several meters, for which it took only a few cups of water, the resulting pressure forces ruptured the barrel.
How to explain that the force of pressure on the bottom of the vessel can be, depending on the shape of the vessel, more or less than the weight of the liquid contained in the vessel? After all, the force acting from the side of the vessel on the liquid must balance the weight of the liquid. The fact is that not only the bottom, but also the walls of the vessel act on the liquid in the vessel. In an upwardly expanding vessel, the forces with which the walls act on the liquid have components directed upwards: thus, part of the weight of the liquid is balanced by the pressure forces of the walls and only a part must be balanced by the pressure forces from the bottom. On the contrary, in a vessel tapering upwards, the bottom acts on the liquid upwards, and the walls - downwards; therefore, the force of pressure on the bottom is greater than the weight of the liquid. The sum of the forces acting on the liquid from the bottom of the vessel and its walls is always equal to the weight of the liquid. Rice. 252 clearly shows the distribution of forces acting from the side of the walls on the liquid in vessels of various shapes.
Rice. 252. Forces acting on a liquid from the side of the walls in vessels of various shapes
Rice. 253. When pouring water into the funnel, the cylinder rises.
In a vessel tapering upwards, a force directed upwards acts on the walls from the side of the liquid. If the walls of such a vessel are made movable, then the liquid will lift them. Such an experiment can be made on the following device: the piston is fixedly fixed, and a cylinder is put on it, turning into a vertical tube (Fig. 253). When the space above the piston is filled with water, the pressure forces on the sections and walls of the cylinder lift the cylinder up.
Plumbing, it would seem, does not give much reason to delve into the jungle of technologies, mechanisms, to engage in scrupulous calculations to build the most complex schemes. But such a vision is a superficial look at plumbing. The real plumbing industry is in no way inferior in terms of the complexity of the processes and, like many other industries, requires a professional approach. In turn, professionalism is a solid store of knowledge on which plumbing is based. Let's plunge (albeit not too deeply) into the plumbing training stream in order to get one step closer to the professional status of a plumber.
The fundamental basis of modern hydraulics was formed when Blaise Pascal was able to discover that the action of fluid pressure is invariable in any direction. The action of liquid pressure is directed at right angles to the surface area.
If a measuring device (manometer) is placed under a layer of liquid at a certain depth and its sensitive element is directed in different directions, the pressure readings will remain unchanged in any position of the manometer.
That is, the pressure of the liquid does not depend on the change of direction. But the fluid pressure at each level depends on the depth parameter. If the pressure gauge is moved closer to the surface of the liquid, the reading will decrease.
Accordingly, when immersed, the measured readings will increase. Moreover, under conditions of doubling the depth, the pressure parameter will also double.
Pascal's law clearly demonstrates the effect of water pressure in the most familiar conditions for modern life.
Therefore, whenever the velocity of the fluid is given, a part of its initial static pressure is used to organize this velocity, which later exists as a pressure velocity.
Volume and flow rate
The volume of liquid passing through a certain point at a given time is considered as the volume flow or flow rate. The flow volume is usually expressed in liters per minute (L/min) and is related to the relative pressure of the fluid. For example, 10 liters per minute at 2.7 atm.
The flow rate (fluid velocity) is defined as the average speed at which the fluid moves past a given point. Typically expressed in meters per second (m/s) or meters per minute (m/min). Flow rate is an important factor in sizing hydraulic lines.
Volume and fluid flow rate are traditionally considered "related" indicators. With the same amount of transmission, the speed may vary depending on the cross section of the passage Volume and flow rate are often considered simultaneously. Other things being equal (with the injection volume unchanged), the flow rate increases as the section or size of the pipe decreases, and the flow rate decreases as the section increases.
Thus, a slowdown in the flow rate is noted in the wide parts of the pipelines, and in narrow places, on the contrary, the speed increases. At the same time, the volume of water passing through each of these control points remains unchanged.
Bernoulli principle
The widely known Bernoulli principle is built on the logic that the rise (fall) in the pressure of a fluid fluid is always accompanied by a decrease (increase) in speed. Conversely, an increase (decrease) in fluid velocity leads to a decrease (increase) in pressure.
This principle is the basis of a number of familiar plumbing phenomena. As a trivial example, Bernoulli's principle is "guilty" of causing the shower curtain to "pull in" when the user turns on the water.
The difference in pressure outside and inside causes a force on the shower curtain. With this force, the curtain is pulled inward.
Another good example is a spray bottle of perfume, where a low pressure area is created by high air velocity. Air carries liquid with it.
Bernoulli's principle for an aircraft wing: 1 - low pressure; 2 - high pressure; 3 - fast flow; 4 - slow flow; 5 - wing Bernoulli's principle also shows why windows in a house tend to spontaneously break in hurricanes. In such cases, the extremely high speed of the air outside the window causes the pressure outside to become much less than the pressure inside, where the air remains virtually motionless.
The significant difference in force simply pushes the windows outward, causing the glass to break. So when a major hurricane approaches, one should essentially open the windows as wide as possible to equalize the pressure inside and outside the building.
And a couple more examples when the Bernoulli principle works: the rise of an airplane with the subsequent flight due to the wings and the movement of “curved balls” in baseball.
In both cases, a difference in the speed of air passing past the object from above and below is created. For aircraft wings, the difference in speed is created by the movement of the flaps, in baseball, by the presence of a wavy edge.
home plumbing practice
Pressure is a physical quantity that plays a special role in nature and human life. This phenomenon, imperceptible to the eye, not only affects the state of the environment, but is also very well felt by everyone. Let's figure out what it is, what types of it exist and how to find the pressure (formula) in different environments.
What is called pressure in physics and chemistry
This term refers to an important thermodynamic quantity, which is expressed in the ratio of the perpendicularly exerted pressure force to the surface area on which it acts. This phenomenon does not depend on the size of the system in which it operates, and therefore refers to intensive quantities.
In a state of equilibrium, the pressure is the same for all points in the system.
In physics and chemistry, this is denoted by the letter "P", which is an abbreviation for the Latin name of the term - pressūra.
If we are talking about the osmotic pressure of a liquid (the balance between the pressure inside and outside the cell), the letter "P" is used.
Pressure units
According to the standards of the International SI system, the physical phenomenon under consideration is measured in pascals (in Cyrillic - Pa, in Latin - Ra).
Based on the pressure formula, it turns out that one Pa is equal to one N (newton - divided by one square meter (a unit of area).
However, in practice, it is rather difficult to use pascals, since this unit is very small. In this regard, in addition to the standards of the SI system, this value can be measured in a different way.
Below are its most famous analogues. Most of them are widely used in the former USSR.
- bars. One bar is equal to 105 Pa.
- Torres, or millimeters of mercury. Approximately one Torr corresponds to 133.3223684 Pa.
- millimeters of water column.
- Meters of water column.
- technical atmospheres.
- physical atmospheres. One atm is equal to 101,325 Pa and 1.033233 at.
- Kilogram-force per square centimeter. There are also ton-force and gram-force. In addition, there is an analog pound-force per square inch.
General pressure formula (7th grade physics)
From the definition of a given physical quantity, one can determine the method of finding it. It looks like the photo below.
In it, F is force, and S is area. In other words, the formula for finding pressure is its force divided by the surface area on which it acts.
It can also be written as follows: P = mg / S or P = pVg / S. Thus, this physical quantity is related to other thermodynamic variables: volume and mass.
For pressure, the following principle applies: the smaller the space affected by the force, the greater the amount of pressing force it has. If, however, the area increases (with the same force) - the desired value decreases.
Hydrostatic pressure formula
Different aggregate states of substances provide for the presence of their properties that are different from each other. Based on this, the methods for determining P in them will also be different.
For example, the formula for water pressure (hydrostatic) looks like this: P = pgh. It also applies to gases. At the same time, it cannot be used to calculate atmospheric pressure, due to the difference in altitudes and air densities.
In this formula, p is the density, g is the gravitational acceleration, and h is the height. Based on this, the deeper the object or object sinks, the higher the pressure exerted on it inside the liquid (gas).
The variant under consideration is an adaptation of the classical example P = F / S.
If we recall that the force is equal to the derivative of the mass by the free fall velocity (F = mg), and the mass of the liquid is the derivative of the volume by the density (m = pV), then the pressure formula can be written as P = pVg / S. In this case, the volume is area multiplied by height (V = Sh).
If you insert this data, it turns out that the area in the numerator and denominator can be reduced and the output is the above formula: P \u003d pgh.
Considering the pressure in liquids, it is worth remembering that, unlike solids, the curvature of the surface layer is often possible in them. And this, in turn, contributes to the formation of additional pressure.
For such situations, a slightly different pressure formula is used: P \u003d P 0 + 2QH. In this case, P 0 is the pressure of a non-curved layer, and Q is the liquid tension surface. H is the average curvature of the surface, which is determined by Laplace's Law: H \u003d ½ (1 / R 1 + 1 / R 2). The components R 1 and R 2 are the radii of the main curvature.
Partial pressure and its formula
Although the P = pgh method is applicable to both liquids and gases, it is better to calculate the pressure in the latter in a slightly different way.
The fact is that in nature, as a rule, absolutely pure substances are not very common, because mixtures predominate in it. And this applies not only to liquids, but also to gases. And as you know, each of these components exerts a different pressure, called partial pressure.
It's pretty easy to define. It is equal to the sum of the pressure of each component of the mixture under consideration (ideal gas).
From this it follows that the partial pressure formula looks like this: P \u003d P 1 + P 2 + P 3 ... and so on, according to the number of constituent components.
There are often cases when it is necessary to determine the air pressure. However, some mistakenly carry out calculations only with oxygen according to the scheme P = pgh. But air is a mixture of different gases. It contains nitrogen, argon, oxygen and other substances. Based on the current situation, the air pressure formula is the sum of the pressures of all its components. So, you should take the aforementioned P \u003d P 1 + P 2 + P 3 ...
The most common instruments for measuring pressure
Despite the fact that it is not difficult to calculate the thermodynamic quantity under consideration using the above formulas, sometimes there is simply no time to carry out the calculation. After all, you must always take into account numerous nuances. Therefore, for convenience, a number of devices have been developed over several centuries to do this instead of people.
In fact, almost all devices of this kind are varieties of a pressure gauge (it helps to determine the pressure in gases and liquids). However, they differ in design, accuracy and scope.
- Atmospheric pressure is measured using a pressure gauge called a barometer. If it is necessary to determine the vacuum (that is, pressure below atmospheric pressure), another version of it, a vacuum gauge, is used.
- In order to find out the blood pressure in a person, a sphygmomanometer is used. To most, it is better known as a non-invasive tonometer. There are many varieties of such devices: from mercury mechanical to fully automatic digital. Their accuracy depends on the materials from which they are made and the place of measurement.
- Pressure drops in the environment (in English - pressure drop) are determined using or difnamometers (not to be confused with dynamometers).
Types of pressure
Considering the pressure, the formula for finding it and its variations for different substances, it is worth learning about the varieties of this quantity. There are five of them.
- Absolute.
- barometric
- Excess.
- Vacuum.
- Differential.
Absolute
This is the name of the total pressure under which a substance or object is located, without taking into account the influence of other gaseous components of the atmosphere.
It is measured in pascals and is the sum of excess and atmospheric pressure. It is also the difference between barometric and vacuum types.
It is calculated by the formula P = P 2 + P 3 or P = P 2 - P 4.
For the reference point for absolute pressure under the conditions of the planet Earth, the pressure inside the container from which air is removed (that is, classical vacuum) is taken.
Only this type of pressure is used in most thermodynamic formulas.
barometric
This term refers to the pressure of the atmosphere (gravity) on all objects and objects found in it, including the surface of the Earth itself. Most people also know it under the name atmospheric.
It is referred to and its value varies with the place and time of measurement, as well as weather conditions and being above / below sea level.
The value of barometric pressure is equal to the modulus of the force of the atmosphere per unit area along the normal to it.
In a stable atmosphere, the magnitude of this physical phenomenon is equal to the weight of a column of air on a base with an area equal to one.
The norm of barometric pressure is 101,325 Pa (760 mm Hg at 0 degrees Celsius). Moreover, the higher the object is from the surface of the Earth, the lower the air pressure on it becomes. Every 8 km it decreases by 100 Pa.
Thanks to this property, in the mountains, water in kettles boils much faster than at home on the stove. The fact is that pressure affects the boiling point: with its decrease, the latter decreases. And vice versa. The work of such kitchen appliances as a pressure cooker and an autoclave is built on this property. The increase in pressure inside them contributes to the formation of higher temperatures in the dishes than in ordinary pans on the stove.
The barometric altitude formula is used to calculate atmospheric pressure. It looks like the photo below.
P is the desired value at height, P 0 is the density of air near the surface, g is the free fall acceleration, h is the height above the Earth, m is the molar mass of the gas, t is the temperature of the system, r is the universal gas constant 8.3144598 J⁄ ( mol x K), and e is the Eclair number, equal to 2.71828.
Often in the above formula for atmospheric pressure, instead of R, K is used - Boltzmann's constant. The universal gas constant is often expressed in terms of its product by the Avogadro number. It is more convenient for calculations when the number of particles is given in moles.
When making calculations, it is always worth taking into account the possibility of changes in air temperature due to a change in the meteorological situation or when climbing above sea level, as well as geographic latitude.
Gauge and vacuum
The difference between atmospheric and measured ambient pressure is called overpressure. Depending on the result, the name of the value changes.
If it is positive, it is called gauge pressure.
If the result obtained is with a minus sign, it is called a vacuum gauge. It is worth remembering that it cannot be more than barometric.
differential
This value is the pressure difference at different measuring points. As a rule, it is used to determine the pressure drop on any equipment. This is especially true in the oil industry.
Having figured out what kind of thermodynamic quantity is called pressure and with the help of what formulas it is found, we can conclude that this phenomenon is very important, and therefore knowledge about it will never be superfluous.
During this lesson, using mathematical transformations and logical conclusions, a formula will be obtained for calculating the pressure of a liquid on the bottom and walls of a vessel.
Topic: Pressure of solids, liquids and gases
Lesson: Calculating the Pressure of a Liquid on the Bottom and Walls of a Vessel
In order to simplify the derivation of the formula for calculating the pressure on the bottom and walls of the vessel, it is most convenient to use a vessel in the form of a rectangular parallelepiped (Fig. 1).
Rice. 1. Vessel for calculating liquid pressure
The area of the bottom of this vessel is S, his high - h. Assume that the vessel is filled with liquid to its full height h. To determine the pressure on the bottom, you need to divide the force acting on the bottom by the area of the bottom. In our case, the force is the weight of the fluid P located in the vessel
Since the liquid in the vessel is stationary, its weight is equal to the force of gravity, which can be calculated if the mass of the liquid is known m
Recall that the symbol g free fall acceleration.
To find the mass of a liquid, you need to know its density. ρ and volume V
We get the volume of liquid in the vessel by multiplying the bottom area by the height of the vessel
These values are initially known. If we substitute them in turn in the above formulas, then to calculate the pressure we get the following expression:
In this expression, the numerator and denominator contain the same value S is the area of the bottom of the vessel. If you reduce it, you get the desired formula for calculating the pressure of the liquid on the bottom of the vessel:
So, to find the pressure, it is necessary to multiply the density of the liquid by the value of the acceleration of free fall and the height of the liquid column.
The above formula is called the hydrostatic pressure formula. It allows you to find the pressure to the bottom vessel. How to calculate the pressure lateralwalls vessel? To answer this question, remember that in the last lesson we established that the pressure at the same level is the same in all directions. This means that the pressure at any point in the fluid at a given depth h can be found by the same formula.
Let's look at a few examples.
Let's take two vessels. One of them contains water, and the other contains sunflower oil. The liquid level in both vessels is the same. Will the pressure of these liquids be the same at the bottom of the vessels? Certainly not. The formula for calculating hydrostatic pressure includes the density of the liquid. Since the density of sunflower oil is less than the density of water, and the height of the liquid column is the same, the oil will exert less pressure on the bottom than water (Fig. 2).
Rice. 2. Liquids with different densities at the same column height exert different pressures on the bottom
One more example. There are three vessels of different shapes. The same liquid is poured into them up to the same level. Will the pressure at the bottom of the vessels be the same? After all, the mass, and hence the weight of the liquids in the vessels is different. Yes, the pressure will be the same (Fig. 3). Indeed, in the formula for hydrostatic pressure there is no mention of the shape of the vessel, the area of its bottom and the weight of the liquid poured into it. The pressure is determined solely by the density of the liquid and the height of its column.
Rice. 3. Liquid pressure does not depend on the shape of the vessel
We have obtained a formula for finding the pressure of a liquid on the bottom and walls of a vessel. This formula can also be used to calculate the pressure in a liquid volume at a given depth. It can be used to determine the diving depth of a scuba diver, when calculating the design of bathyscaphes, submarines, and to solve many other scientific and engineering problems.
Bibliography
- Peryshkin A. V. Physics. 7 cells - 14th ed., stereotype. - M.: Bustard, 2010.
- Peryshkin A. V. Collection of problems in physics, 7-9 cells: 5th ed., stereotype. - M: Exam Publishing House, 2010.
- Lukashik V. I., Ivanova E. V. Collection of problems in physics for grades 7-9 of educational institutions. - 17th ed. - M.: Enlightenment, 2004.
- A single collection of digital educational resources ().
Homework
- Lukashik V. I., Ivanova E. V. Collection of problems in physics for grades 7-9 No. 504-513.
Liquids and gases transmit in all directions not only the external pressure exerted on them, but also the pressure that exists inside them due to the weight of their own parts. The upper layers of the liquid press on the middle ones, those on the lower ones, and the last ones on the bottom.
The pressure exerted by a fluid at rest is called hydrostatic.
We obtain a formula for calculating the hydrostatic pressure of a liquid at an arbitrary depth h (in the vicinity of point A in Figure 98). The pressure force acting in this place from the overlying narrow vertical column of liquid can be expressed in two ways:
firstly, as the product of the pressure at the base of this column and its cross-sectional area:
F = pS ;
secondly, as the weight of the same liquid column, i.e. the product of the mass of the liquid (which can be found by the formula m = ρV, where the volume is V = Sh) and the gravitational acceleration g:
F = mg = ρShg .
Let us equate both expressions for the pressure force:
pS = ρShg .
Dividing both sides of this equation by the area S, we find the fluid pressure at depth h:
p = rgh. (37.1)
We got hydrostatic pressure formula. The hydrostatic pressure at any depth inside a liquid does not depend on the shape of the vessel in which the liquid is located, and is equal to the product of the density of the liquid, the gravitational acceleration and the depth at which the pressure is considered.
The same amount of water, being in different vessels, can exert different pressure on the bottom. Since this pressure depends on the height of the liquid column, it will be greater in narrow vessels than in wide ones. Thanks to this, even a small amount of water can create a very large pressure. In 1648, B. Pascal demonstrated this very convincingly. He inserted a narrow tube into a closed barrel filled with water and, going up to the balcony of the second floor of the house, poured a mug of water into this tube. Due to the small thickness of the tube, the water in it rose to a great height, and the pressure in the barrel increased so much that the fastenings of the barrel could not stand it, and it cracked (Fig. 99).
Our results are valid not only for liquids, but also for gases. Their layers also press on each other, and therefore they also have hydrostatic pressure. 
1. What pressure is called hydrostatic? 2. On what quantities does this pressure depend? 3. Derive the formula for hydrostatic pressure at an arbitrary depth. 4. How can you create a lot of pressure with a small amount of water? Tell us about Pascal's experience.
Experimental task. Take a tall vessel and make three small holes in its wall at different heights. Close the holes with plasticine and fill the vessel with water. Open the holes and follow the jets of flowing water (Fig. 100). Why does water leak from holes? What does it mean that water pressure increases with depth?
