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 form 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 height of the column they 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.
Liquids and gases transmit pressure applied to them in all directions. This is stated by Pascal's law and practical experience.
But there is also its own weight, which should also affect the pressure existing in liquids and gases. Weight of own parts or layers. The upper layers of liquid press on the middle ones, the middle ones on the lower ones, and the last ones on the bottom. That is, we we can talk about the existence of pressure from a column of resting liquid on the bottom.
Liquid column pressure formula
The formula for calculating the pressure of a liquid column of height h is as follows:
where ρ is the density of the liquid,
g - free fall acceleration,
h is the height of the liquid column.
This is the formula for the so-called hydrostatic pressure of a fluid.
Liquid and gas column pressure
Hydrostatic pressure, that is, the pressure exerted by a liquid at rest, at any depth does not depend on the shape of the vessel in which the liquid is located. The same amount of water, being in different vessels, will exert different pressure on the bottom. Thanks to this, you can create enormous pressure even with a small amount of water.
This was demonstrated very convincingly by Pascal in the seventeenth century. He inserted a very long narrow tube into a closed barrel full of water. Having risen to the second floor, he poured only one mug of water into this tube. The barrel burst. The water in the tube, due to its small thickness, rose to very high altitude, and the pressure grew to such levels that the barrel could not stand it. The same is true for gases. However, the mass of gases is usually much less than the mass of liquids, so the pressure in gases due to their own weight can often be ignored in practice. But in some cases you have to take this into account. For example, Atmosphere pressure, which presses on all objects on Earth, has great importance in some manufacturing processes.
Thanks to the hydrostatic pressure of water, ships that often weigh not hundreds, but thousands of kilograms can float and not sink, since the water presses on them, as if pushing them out. But precisely because of the same hydrostatic pressure on great depth our ears are blocked, and we can’t go down to very great depths without special devices- diving suit or bathyscaphe. Only a few sea and ocean inhabitants have adapted to live in conditions of strong pressure at great depths, but for the same reason they cannot exist in upper layers water and can die if they fall into shallow depths.
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 the liquid produced external forces, is transmitted in 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 – fluid 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 closed loop area ΔF, including 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 the newton per square meter(N/m 2) - 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 a physical system, the unit of pressure intention is the dyne per square centimeter(dyne/m2), in technical system– 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 they use various ways taking 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 changes at some point, then the pressure at any other point in this liquid changes 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 on high blood 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 pillar 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 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?