PRESSURE

The physical meaning of this paradox is that the weight of the liquid in the container is different from the pressure force on the bottom (for the middle and right containers).
How to "reconcile" our assumption about the mass (weight) and the result of the experiment?
The pressure at the bottom of each vessel can be calculated using the formula p = ρ g h. If the same liquid is poured into containers to the same level, the pressures will be equal, because the heights of the liquid columns are equal.
The force with which the liquid presses the bottom of each vessel can be calculated from the formula F = pS. Inserting the expression for pressure, we get: F = ρ g h S. All values on the right-hand side of this equality are the same for all three vessels. It follows that the forces acting on the bottom of all three containers will also be equal. Namely, this formula contains neither the mass nor the weight of the liquid, so we can conclude that the pressure forces on the bottom of the container do not depend on these values.

The hydrostatic paradox can be clearly demonstrated by communicating vessels. These are containers interconnected by liquid passages and have a common bottom. In them, the surfaces of a homogeneous liquid at rest are at the same level, regardless of the shape and size of the individual container. Because the pressure on the vessel walls is equal at any horizontal level.

Liquids act by pressure both on the bottom and on the walls of the container. And that pressure depends on the depth, which can be easily demonstrated by the outflow of the jets from the holes on the vessel and by comparing their ranges on the ground, which should be sufficiently lower than the lowest opening of the vessel itself.

The water tower is a facility in the water supply network of lowland settlements. It is a tank raised high above the ground to store drinking or industrial water. With an elevated tank, not only a temporarily sufficient amount of water is achieved, but also sufficient and uniform pressure in the water supply network. Pressure oscillations on the inlet side (filling) and water consumption fluctuations on the outlet side (discharge) are compensated by the height and water supply. The result is a lower load on the filling pump and the creation of pressure in the supply network.
For sufficient pressure, all consumers must be lower than the tank of the water tower (principle of communicating vessels). Discharge points that are higher than the water tower level, such as tall buildings, require their own booster pump system

In addition to liquids, pressure also exists in others fluids, as well as in air. The first proof of atmospheric pressure was performed by the Italian physicist Torricelli 🔎Torricelli, Italian physicist, mathematician and good friend of Galileo Galilei. His invention of the mercury barometer (1643 or 1644) is notable, with which he rejected the common prejudice about the fear of emptiness (lat. horror vacui). Torricelli filled a sealed glass tube with mercury and immersed the opening of the tube in a vessel filled with mercury. The mercury only partially flowed out of the tube into the vessel, and an empty space, a vacuum, was created above the column of mercury in the tube. Torricelli interpreted this phenomenon as atmospheric pressure acting on the surface of the mercury in the vessel and maintaining the balance of the column of mercury in the tube. This experiment proved the existence of atmospheric pressure and showed how it can be measured based on the movement of the mercury column.. He filled the tube (actually a long test tube) to the top with mercury, closed the opening with his finger and turned the tube with the opening downwards. Keeping his finger on the opening, he dipped that end of the tube into the container of mercury. When the opening was immersed, he removed his finger and let the mercury flow out. Soon the mercury stopped flowing out at a column height of 760 mm.
The air pressure on the surface of the mercury in the container equalized with the pressure of the column in the tube. Since then, the height of the mercury column has served as a barometer. Atmospheric pressure acts on the height of the column and without this pressure the mercury in the tube would completely leak out. However, when Torricelli removed his finger from the opening of the tube after the mercury immersion, he did NOT let go of the tube from the hand he was holding. So should we then say that the column of mercury holds the atmospheric pressure or that Torricelli held it with his hand? The answer is in the article: Torricelli and 1st Newton's Law.

Upper - Systolic blood pressure
The phase in which the heart muscle contracts and pushes blood into the arteries is called systole. The pressure at which the artery wall briefly expands is known as the upper or systolic blood pressure. We can also feel this pressure wave as either (pulsation) by feeling a vein on the neck or wrist.

Lower - Diastolic blood pressure
During diastole, the heart muscle relaxes again and blood can flow back to the heart through the veins. During this brief phase of relaxation, the pressure in the vessels is understandably reduced. The diastolic blood pressure value is always lower than the systolic value and is also called the lower measured value.

Even today, blood pressure is expressed in units of mmHg - millimeters of mercury - since it was originally determined using a mercury column. One millimeter of mercury column equals 133 Pa. The first given value is always the systolic measured value (upper pressure), the second value describes the diastolic (lower) blood pressure.

The optimal blood pressure value measured at rest is 120/80 mmHg. But slightly higher values are still considered normal. Only above 140/90 mmHg do doctors speak of elevated blood pressure, which is professionally called arterial hypertension 🔎With regard to the measured values of arterial pressure in the doctor's office, arterial pressure is classified as
• optimal (<120 / <80 mmHg )
• normal (120-129 / 80-84 mmHg )
• normaly high (130-139 and/or 85-89 mmHg)
•1st degree hypertension (140-159 and/or 90-99 mmHg)
•2nd degree hypertension (160-179 and/or 100-109 mmHg)
•3rd degree hypertension (≥180 and/or ≥110 mmHg)
• isolated systolic hypertension (≥140 / <90 mmHg). .

Both upper and lower blood pressure are constantly adjusting to the demands of our body. During physical exertion, the heart pushes more blood into the body, which increases the pressure. Blood pressure also increases with stress and excitement. By slower or faster heartbeats and narrowing or widening of blood vessels, blood pressure optimally adapts to different situations in our life. This regulation of blood pressure is vital for our body.

Wind is the movement of air over the Earth's surface, caused by unequal heating of the Earth's surface, which leads to differences in the pressures of different areas. Wind strength can vary from a light breeze to hurricane force and is measured on the Beaufort wind scale.

Wind pressure or wind power is an important quantity when calculating the effect of wind on an object, for example large surfaces, buildings or bridges. The dynamic pressure is the product of the aerodynamic coefficient c (depending on the shape of the object) and the wind stagnation pressure. The force is obtained by multiplying the pressure and the surface area of the object.

Numerical simulation:
A crosswind (100 km/h at a height of 10 m) hits a truck while driving. The truck is 4 m high and 8.5 m long and has a closed cargo area. Overpressure is created on the windward side (left side) due to air compression, and negative pressure on the leeward side. This can cause the truck to overturn. With a wind speed of 100 km/h (27.7 m/s) and an aerodynamic drag coefficient c = 1.05 for the shape of a cuboid, the lateral force on the truck is about 16800 N.

Wind power, in some climates, can reach hurricane proportions and leave dramatic destructive effects. Such natural disasters can wipe out entire settlements from the face of the earth, especially in poor areas where houses are built mainly from cheap, light materials.

At the time of the emergence of experimental science, in the 16th century, many scientists debated the question of whether there really is an empty space. Even the mayor of Magdeburg (and a physicist) Otto von Guericke 🔎Otto von Guericke (Magdeburg, 1602. – 1686.), German physicist and engineer. He invented the air pump in 1650 and founded vacuum physics. His experiment from 1654 in Magdeburg is well-known, when 50 people could not lift the lid of the kettle from which air was sucked with his suction pump. Guericke was engaged in research on the connection between the state of the barometer and atmospheric conditions, the astronomy of comets and the use of electricity in obtaining light. He is also known for the fact that in 1672 he was the first to observe the phenomenon of electric repulsion of the electric charge of the same name, and in 1663 he constructed the first electrostatic friction machine (a type of electroscope). dealt with this question.

 „Weil die Gelehrten nun schon seit langem über das Leere, ob es vorhanden sei, ob nicht, oder was es sei, gar heftig untereinander stritten (...) konnte ich mein brennendes Verlangen, die Wahrheit dieses fragwürdigen Etwas zu ergründen, nicht mehr eindämmen (...)"[1]

 „While scholars, for quite a long time now, have been vigorously arguing with each other about the void, whether it exists, or does not exist, or what it is at all (...) I could no longer hold back my burning desire to grasp the truth of that doubt to some extent (...)"

[1] Experimenta nova (ut vocantur)Magdeburgica De Vacuo Spatio

The original hemispheres used in the Magdeburg experiment are kept in the Technical Museum in Munich.

Model of Magdeburg spheres for school experiments in physics, their diameter is about 10 cm, so they can be disassembled with a force corresponding to a weight of about 75 kg.

The hemispheres are pressed together with a seal at the joint. After the air is sucked out of them with a vacuum pump (the empty space in them is a relatively rough vacuum), the external pressure creates a force that prevents them from being disassembled.

hemispheres filled with air    and         after vacuuming

Guericke's spectacular vacuum experiments began in 1650 and made him a famous physicist and engineer. The culmination of his distinguished research series was an experiment on the effect of air pressure: the Magdeburg hemisphere.

The demonstration of his famous experiment was performed for the first time in 1657 in the city of Magdeburg:

For this purpose, Guericke used a vacuum pump to exhaust the air from two hollow copper hemispheres, which were sealed at the joint with a leather seal soaked in oil and wax, and had a diameter of 42 cm. The air acting on the hemispheres from the outside pressed the halves together. To show how much pressure the atmosphere causes, Guericke harnessed eight horses to each hemisphere, and they were unable to separate the hemispheres.

What is the horse's pulling force?

Depending on the friction under the hooves, the horse can pull with a force always less than its own weight. The force of friction is generally the product of the pressure on the surface (i.e. the weight of the horse) and the coefficient of friction which depends on the surface (grass, asphalt, ice, gravel) and ranges between 0.2 and 0.4. So a 700 kg shod horse can pull with about 1500 N on dirt, but it cannot pull at all on ice.

 Guericke wrote in his main work "Experimenta nova (ut vocantur) Magdeburgica De Vacuo Spatio":

[1] translation:
[ „Only with a leather ring as an insert, those hemispheres are pressed against each other, and then the air is...suddenly depleted. I saw then, with how much force both hollow halves pressed to insert the ring! And under the action of air pressure, they remained so firmly attached to each other, that even 16 horses could not separate them, or separated them only with great difficulty. If sometimes, but with the greatest effort, they manage to separate them, then a shot rings out as if from a flintlock rifle". ]

With his experiments, he disproved the then-prevailing belief about "Horror vacua" (lat. Fear, horror of emptiness), the so-called nature's fear of emptiness, which people at that time used to explain, among other things, the principle of pumping with suction pumps. Guericke showed that empty space really exists and how man can realize it. At the same time, it was clarified that air, like any other object, has weight. And according to this, the Magdeburg hemispheres are not "attracted to each other by the internal vacuum", but due to the high air pressure from the outside, they are pressed against each other.

Calculation of the force required to separate the hemispheres

CAUTION! HIGH TEMPERATURE! In this experiment, heated objects are used to high temperature! Everyone has probably already crumpled a drunk can of some refreshing drink with their hand. Of course, it wasn't difficult because the can is empty and is made of thin aluminum sheet, so it easily bends under the pressure of the hand. However, an interesting and simple experiment will show us that a can can be crumpled in a completely unexpected way due to atmospheric pressure. Pour water into the empty can, but no more than one finger (about 0.3 dl). Then put the can on a gas or electric stove and let the water boil. If the water in the can is boiling, it will be seen by the cloud of condensed water vapor above the opening. Let it boil for 20 to 30 seconds, so that the inside of the can is completely filled with steam. While we wait for the water to boil, prepare a shallow bowl with cold water. The depth of the water in that bowl should be 2 - 3 cm. Using a cloth or a glove, so as not to burn yourself, take the can off the stove and quickly immerse it, turned upside down, in cold water, so that the water covers the entire opening (see the animation on the right). At the same time when the can comes into contact with cold water, some invisible force will crumple it so quickly that it will almost fall out of our hands. Why did this happen?   Why is boiling water needed?     Would the experiment work with only air in the can? In an open can, the pressure is equal to the external pressure. For a successful experiment, it is crucial that the can be filled with water vapor. Upon sudden cooling, in contact with water, the steam condenses and immediately turns into a liquid state. At the same time, its volume decreases about 1600 times !!! The can is then almost empty, as if vacuumed. We might expect the water from the container to fill the can. A little water may enter, but the pressure in it drops so quickly that the water, due to its viscosity, cannot enter through the relatively narrow opening. That's why the pressure difference tends to crush the can. Let's estimate how much force was acting. Assume that the temperature of the hot gas in the can is T1 ≈ 370 K (97 ℃), and that the atmospheric pressure inside is p1 ≈ 105 Well. By sudden cooling, let the temperature drop to T2 ≈ 300 K (27 ℃). This means that with unchanged volume (isochoric change), the new pressure p2 will be obtained from the constancy of the ratio of pressure and temperature: p1 / T1 = p2 / T2   ⟹  p2 = p1 T2 / T1 If we include the assumed values, we will get the approximate amount of pressure after cooling: p2 = 105 300 K / 370 K Pa ≈ 81000 Pa that is, the difference between the pressure outside and the pressure in the can is Δp ≈ 19000 Pa From the diameter and height of the can, which has the shape of a cylinder, it is easy to calculate its surface area, which is about 3 dm². Since force is the product of pressure and surface area, we arrive at the amount of: F = Δp·S = 19.000 Pa ·3·10-2 m2 = 570 N A force of 570 N corresponds to a body weight of about 60 kg! The above calculation is valid only for gases that would remain gaseous after cooling, eg air. However, the water vapor turns into a liquid condensate upon cooling, and this condensation occurs quickly. The air would cool slowly and would not reduce its volume 1700 times but only by 1/5. Water vapor volume 330 cm³ by changing to a liquid state, it reduces the volume to only 0.2 cm. Assuming that all the vapor in the can condenses, which is probably not the case. For our can, this means a pressure change of about 10⁵ Well, what by multiplying by an area of 3 dm² gives a force of 3,000 N. This force corresponds to the weight of a body with a mass of approximately 300 kg! Remark: Where did the 1700 times volume reduction come from? A mole is a measure of the amount of a substance. We will have one mole of any substance if we take as many grams of that substance as its relative atomic or molar mass. For water, this means that 1 mole of water has 18 grams, (H2O = 2 hydrogen atoms (2 × 1 g/mol) and 1 oxygen atom (16 g/mol)). Those 18 g of water have a volume of 18 ml (density of water is 1 g/cm³). When liquid water turns into steam (gas), then a mole of gas at a temperature of 373 K or 100 ℃ occupies a volume of 30.6 liters (at a pressure of 105 Pa). And 30.6 liters is 30600 ml. That is 1700 times more than 18 ml of liquid. If we include the following values in the gas equation p·V = n·R·T: number of moles n = 1;  pressure p = 105 Pa;   temperature T = 373 K;   gas constant R = 8.314 J/mol·K we can calculate the volume of one mole of steam: V = n·R·T / p = 1 ·8,314 J/mol·K ·373 K / 1,01325·105 Pa = 0,03061 m³ = 30,61 L