During this lesson, using mathematical transformations and logical deductions, 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: Calculation of liquid pressure 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 a vessel, it is most convenient to use a vessel in the shape 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. Let us 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 liquid P, located in the vessel
Since the liquid in the container is motionless, its weight is equal to the force of gravity, which can be calculated if the mass of the liquid is known m
Let us recall that the symbol g indicates the acceleration of gravity.
In order to find the mass of a liquid, you need to know its density ρ and volume V
We obtain 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 into the above formulas, then to calculate the pressure we obtain the following expression:
In this expression, the numerator and denominator contain the same quantity S- area of the bottom of the vessel. If we shorten it, we get the required formula for calculating the pressure of the liquid at the bottom of the vessel:
So, to find the pressure, it is necessary to multiply the density of the liquid by the magnitude of the acceleration due to gravity and the height of the liquid column.
The formula obtained above 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 the pressure at any point in the liquid at a given depth h can be found according to 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 column of liquids 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 different shaped vessels. They are filled with the same liquid to the same level. Will the pressure at the bottom of the vessels be the same? After all, the mass, and therefore the weight, of liquids in 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. 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 volume of liquid at a given depth. It can be used to determine the depth of a scuba diver's dive, when calculating the design of bathyscaphes, submarines, and to solve many other scientific and engineering problems.
Bibliography
- Peryshkin A.V. Physics. 7th grade - 14th ed., stereotype. - M.: Bustard, 2010.
- Peryshkin A.V. Collection of problems in physics, grades 7-9: 5th ed., stereotype. - M: Publishing House “Exam”, 2010.
- Lukashik V.I., Ivanova E.V. Collection of problems in physics for grades 7-9 of educational institutions. - 17th ed. - M.: Education, 2004.
- Unified collection of digital educational resources ().
Homework
- Lukashik V.I., Ivanova E.V. Collection of problems in physics for grades 7-9 No. 504-513.
Hydrostatics is the branch of hydraulics that studies the laws of equilibrium of fluids and considers the practical application of these laws. In order to understand hydrostatics, it is necessary to define some concepts and definitions.
Pascal's law for hydrostatics.
In 1653, the French scientist B. Pascal discovered a law that is commonly called the fundamental law of hydrostatics.
It sounds like this:
The pressure on the surface of a liquid produced by external forces is transmitted into the liquid equally in all directions.
Pascal's law is easily understood if you look at the molecular structure of matter. In liquids and gases, molecules have relative freedom; they are able to move relative to each other, unlike solids. In solids, molecules are assembled into crystal lattices.
The relative freedom that the molecules of liquids and gases have allows the pressure exerted on the liquid or gas to be transferred not only in the direction of the force, but also in all other directions.
Pascal's law for hydrostatics is widely used in industry. The work of hydraulic automation, which controls CNC machines, cars and airplanes, and many other hydraulic machines, is based on this law.
Definition and formula of hydrostatic pressure
From Pascal’s law described above it follows that:
Hydrostatic pressure is the pressure exerted on a fluid by gravity.
The magnitude of hydrostatic pressure does not depend on the shape of the vessel in which the liquid is located and is determined by the product
P = ρgh, where
ρ – liquid density
g – free fall acceleration
h – depth at which pressure is determined.
To illustrate this formula, let's look at 3 vessels of different shapes.
In all three cases, the pressure of the liquid at the bottom of the vessel is the same.
The total pressure of the liquid in the vessel is equal to
P = P0 + ρgh, where
P0 – pressure on the surface of the liquid. In most cases it is assumed to be equal to atmospheric pressure.
Hydrostatic pressure force
Let us select a certain volume in a liquid in equilibrium, then cut it into two parts by an arbitrary plane AB and mentally discard one of these parts, for example the upper one. In this case, we must apply forces to the plane AB, the action of which will be equivalent to the action of the discarded upper part of the volume on the remaining lower part of it.
Let us consider in the section plane AB a closed contour of area ΔF, which includes some arbitrary point a. Let a force ΔP act on this area.
Then the hydrostatic pressure whose formula looks like
Рср = ΔP / ΔF
represents the force acting per unit area, will be called the average hydrostatic pressure or the average hydrostatic pressure stress over the area ΔF.
The true pressure at different points of this area may be different: at some points it may be greater, at others it may be less than the average hydrostatic pressure. It is obvious that in the general case, the average pressure Рср will differ less from the true pressure at point a, the smaller the area ΔF, and in the limit the average pressure will coincide with the true pressure at point a.
For fluids in equilibrium, the hydrostatic pressure of the fluid is similar to the compressive stress in solids.
The SI unit of pressure is newton per square meter (N/m2) - it is called pascal (Pa). Since the value of the pascal is very small, enlarged units are often used:
kilonewton per square meter – 1 kN/m 2 = 1*10 3 N/m 2
meganewton per square meter – 1MN/m2 = 1*10 6 N/m2
A pressure equal to 1*10 5 N/m 2 is called a bar (bar).
In the physical system, the unit of pressure intention is dyne per square centimeter (dyne/m2), in the technical system it is kilogram-force per square meter (kgf/m2). In practice, liquid pressure is usually measured in kgf/cm2, and a pressure equal to 1 kgf/cm2 is called technical atmosphere (at).
Between all these units there is the following relationship:
1at = 1 kgf/cm2 = 0.98 bar = 0.98 * 10 5 Pa = 0.98 * 10 6 dyne = 10 4 kgf/m2
It should be remembered that there is a difference between the technical atmosphere (at) and the physical atmosphere (At). 1 At = 1.033 kgf/cm 2 and represents normal pressure at sea level. Atmospheric pressure depends on the altitude of a place above sea level.
Hydrostatic pressure measurement
In practice, various methods are used to take into account the magnitude of hydrostatic pressure. If, when determining hydrostatic pressure, the atmospheric pressure acting on the free surface of the liquid is also taken into account, it is called total or absolute. In this case, the pressure value is usually measured in technical atmospheres, called absolute (ata).
Often, when taking pressure into account, atmospheric pressure on the free surface is not taken into account, determining the so-called excess hydrostatic pressure, or gauge pressure, i.e. pressure above atmospheric.
Gauge pressure is defined as the difference between the absolute pressure in a liquid and atmospheric pressure.
Rman = Rabs – Ratm
and are also measured in technical atmospheres, called in this case excess.
It happens that the hydrostatic pressure in a liquid is less than atmospheric. In this case, the liquid is said to have a vacuum. The magnitude of the vacuum is equal to the difference between atmospheric and absolute pressure in the liquid
Rvak = Ratm – Rabs
and is measured from zero to the atmosphere.
Hydrostatic water pressure has two main properties:
It is directed along the internal normal to the area on which it acts;
The amount of pressure at a given point does not depend on the direction (i.e., on the orientation in space of the site on which the point is located).
The first property is a simple consequence of the fact that in a fluid at rest there are no tangential and tensile forces.
Let us assume that the hydrostatic pressure is not directed along the normal, i.e. not perpendicular, but at some angle to the site. Then it can be decomposed into two components - normal and tangent. The presence of a tangential component, due to the absence of forces of resistance to shearing forces in a fluid at rest, would inevitably lead to the movement of the fluid along the platform, i.e. would upset her balance.
Therefore, the only possible direction of hydrostatic pressure is its direction normal to the site.
If we assume that the hydrostatic pressure is directed not along the internal, but along the external normal, i.e. not inside the object under consideration, but outside from it, then due to the fact that the liquid does not resist tensile forces, the particles of the liquid would begin to move and its equilibrium would be disrupted.
Consequently, the hydrostatic pressure of water is always directed along the internal normal and represents compressive pressure.
From this same rule it follows that if the pressure at some point changes, then the pressure at any other point in this liquid will change by the same amount. This is Pascal's law, which is formulated as follows: The pressure exerted on a liquid is transmitted inside the liquid in all directions with equal force.
The operation of machines operating under hydrostatic pressure is based on the application of this law.
Another factor influencing the pressure value is the viscosity of the liquid, which until recently was usually neglected. With the advent of units operating at high pressure, viscosity also had to be taken into account. It turned out that when the pressure changes, the viscosity of some liquids, such as oils, can change several times. And this already determines the possibility of using such liquids as a working medium.
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 liquid press on the middle ones, those on the lower ones, and the latter ones on the bottom.
The pressure exerted by a fluid at rest is called hydrostatic.
Let us 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 column of liquid, i.e. the product of the mass of the liquid (which can be found by the formula m = ρV, where volume V = Sh) and the acceleration of gravity g:
F = mg = ρShg.
Let us equate both expressions for the pressure force:
pS = ρShg.
Dividing both sides of this equality by area S, we find the fluid pressure at depth h:
p = ρgh. (37.1)
We got hydrostatic pressure formula. Hydrostatic pressure at any depth inside a liquid does not depend on the shape of the container in which the liquid is located and is equal to the product of the density of the liquid, the acceleration of gravity 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 very high pressure. In 1648, this was very convincingly demonstrated by B. Pascal. 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 withstand it, and it cracked (Fig. 99).
The results we obtained are valid not only for liquids, but also for gases. Their layers also press on each other, and therefore hydrostatic pressure also exists in them. 
1. What pressure is called hydrostatic? 2. What values does this pressure depend on? 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. Cover the holes with plasticine and fill the vessel with water. Open the holes and watch the streams of water flowing out (Fig. 100). Why does water leak out of the holes? What does it mean that water pressure increases with depth?
Pressure is a physical quantity that plays a special role in nature and human life. This phenomenon, invisible 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 it exists and how to find pressure (formula) in different environments.
What is pressure in physics and chemistry?
This term refers to an important thermodynamic quantity, which is expressed in the ratio of the pressure force exerted perpendicularly 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 of the system.
In physics and chemistry it is denoted by the letter “P”, which is an abbreviation of the Latin name of the term - pressūra.
When talking about the osmotic pressure of a fluid (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 in question is measured in pascals (Cyrillic - Pa, Latin - Ra).
Based on the pressure formula, it turns out that one Pa is equal to one N (newton - divided by one square meter (unit of area).
However, in practice it is quite difficult to use pascals, since this unit is very small. In this regard, in addition to SI standards, this quantity can be measured differently.
Below are its most famous analogues. Most of them are widely used in the former USSR.
- Bars. One bar is equal to 105 Pa.
- Torrs, 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 atm.
- Kilogram-force per square centimeter. Ton-force and gram-force are also distinguished. In addition, there is an analogue to pound-force per square inch.
General formula for pressure (7th grade physics)
From the definition of a given physical quantity, one can determine the method for finding it. It looks like in 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 turns out to be related to other thermodynamic variables: volume and mass.
For pressure, the following principle applies: the smaller the space that is affected by the force, the greater the amount of pressing force that falls on it. If the area increases (with the same force), the desired value decreases.
Hydrostatic Pressure Formula
Different states of aggregation of substances provide for the presence of properties that differ 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. However, it cannot be used to calculate atmospheric pressure due to the difference in altitude and air density.
In this formula, p is the density, g is the acceleration due to gravity, and h is the height. Based on this, the deeper an object or object is immersed, the higher the pressure exerted on it inside the liquid (gas).
The option under consideration is an adaptation of the classic example P = F / S.
If we remember that the force is equal to the derivative of mass by the speed of free fall (F = mg), and the mass of the liquid is the derivative of volume by density (m = pV), then the formula pressure can be written as P = pVg / S. In this case, volume is area multiplied by height (V = Sh).
If we insert this data, it turns out that the area in the numerator and denominator can be reduced at the output - the above formula: P = pgh.
When considering pressure in liquids, it is worth remembering that, unlike solids, 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 = P 0 + 2QH. In this case, P 0 is the pressure of the non-curved layer, and Q is the tension surface of the liquid. H is the average curvature of the surface, which is determined according to Laplace's Law: H = 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 for 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 often found, 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.
It's quite easy to define. It is equal to the sum of the pressure of each component of the mixture under consideration (ideal gas).
It follows from this that the partial pressure formula looks like this: P = 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 air pressure. However, some people 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. This means that we should take the above-mentioned P = 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 in question using the above-mentioned 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, over several centuries a number of devices have been developed that do this instead of people.
In fact, almost all devices of this kind are a type of pressure gauge (helps determine pressure in gases and liquids). However, they differ in design, accuracy and scope of application.
- Atmospheric pressure is measured using a pressure gauge called a barometer. If it is necessary to determine the vacuum (that is, pressure below atmospheric), another type of it is used, a vacuum gauge.
- In order to find out a person's blood pressure, a sphygmomanometer is used. It is better known to most people as a non-invasive blood pressure monitor. 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 location of measurement.
- Pressure drops in the environment (in English - pressure drop) are determined using differential pressure meters (not to be confused with dynamometers).
Types of pressure
Considering 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
- Excessive.
- Vacuum metric.
- 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 using the formula P = P 2 + P 3 or P = P 2 - P 4.
The starting point for absolute pressure under the conditions of planet Earth is the pressure inside the container from which air has been removed (that is, a classic vacuum).
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 as atmospheric.
It is classified as one and its value varies depending on the place and time of measurement, as well as weather conditions and location above/below sea level.
The magnitude of barometric pressure is equal to the modulus of the atmospheric force over an area of one unit 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 normal 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, water in kettles boils much faster in the mountains than on the stove at home. The fact is that pressure affects the boiling point: as it decreases, the latter decreases. And vice versa. The operation of such kitchen appliances as a pressure cooker and autoclave is based on this property. The increase in pressure inside them contributes to the formation of higher temperatures in the vessels than in ordinary pans on the stove.
The barometric altitude formula is used to calculate atmospheric pressure. It looks like in the photo below.
P is the desired value at altitude, P 0 is the air density 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 Eichler number equal to 2.71828.
Often in the above formula for atmospheric pressure, K - Boltzmann's constant is used instead of R. The universal gas constant is often expressed through its product by Avogadro's number. It is more convenient for calculations when the number of particles is given in moles.
When making calculations, you should always take into account the possibility of changes in air temperature due to a change in meteorological situation or when gaining altitude above sea level, as well as geographic latitude.
Gauge and vacuum
The difference between atmospheric and measured ambient pressure is called excess pressure. Depending on the result, the name of the quantity changes.
If it is positive, it is called gauge pressure.
If the result obtained has a minus sign, it is called vacuummetric. It is worth remembering that it cannot be greater than barometric.
Differential
This value is the difference in pressure at different measurement 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 what formulas it is found, we can conclude that this phenomenon is very important, and therefore knowledge about it will never be superfluous.