About friction

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What is Aristotle’s mistake?

Aristotle is usually remembered for criticism. And this is wrong with him, and this. He especially gets it in the mechanics course. After all, it is well known that force is proportional to the acceleration of a body, and a body presented to itself moves at a constant speed. But Aristotle argued something else: the speed of a body is proportional to the force acting on it. And if you don’t apply force, the body will remain. True, our everyday experience rather confirms the statement of Aristotle than of Newton, whose two laws we formulated above. Indeed, in order for a body to move, it only needs to be pulled, that is, a force must be applied to it, and as soon as the applied force disappears, the body is abandoned. Aristotle’s statement, at first glance, agrees better with observed phenomena than Newton’s laws. That is why it was so difficult to appeal to the authority of Aristotle.

The agreement between theory (Newton’s laws) and experiment (observations of moving objects) is revealed if the friction force is taken into account. When, as Aristotle thought, nothing acts on the body and therefore (!) it is left, a frictional force actually acts on it, which slows down the movement.

Trene and atoms

What is friction force? What is its nature? Most often we remember it when studying the operation of mechanisms, thinking about the sliding of a sled on a snowy bank or a boat on a water surface. In general terms, it is possible to understand what happens when two bodies come into contact. Their atoms (slide and snow or bearing and frame) interact with each other *). When moving, the atoms of one body try to entrain the atoms of another. If they succeed, then one or both of them are destroyed. We usually say - they are afraid. Let us, in order not to complicate the consideration, neglect the wandering. At not very high speeds and fairly hard parts, this is quite acceptable. So, the atoms of one body fail to entrain the atoms of another body. What’s going on anyway? Atoms of a solid body (crystal) are located in certain positions in equilibrium positions. True, the term “located” has several conditions, since atoms are always in motion: they perform small oscillations around equilibrium positions. The intensity of these oscillations is determined by temperature. The higher the temperature, the greater the energy and amplitude of atomic vibrations. The atoms of the crystal interact with each other. Changing the movement of atoms

*) The somewhat vague word “interact” means that atoms either stick to each other or repel each other. In reality, they both press in and push away. Far away, they hug, and close, they push away. Just remember that scales in the world of atoms are angstroms (1A°=10⁻⁸cm). “Far” means a distance much greater than 1A, close - about 1A. When the atoms of different bodies come into contact, they are located at distances from each other that significantly exceed 1A.

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Fig. 1.

ma or group of atoms is transferred to neighboring…

When one body moves relative to another, touching it, the atoms of these bodies “cling” and swing each other. If you are not afraid of vulgarization, then this process can be imagined like this: a person runs away from the pendants hanging nearby and swings them. Or better yet: the pendants are interconnected. By swinging one or more, all the others are forced to move (Fig. 1).

Vibrations of atoms located on the surface of the body are transmitted to atoms located in the depths. The transfer of energy from the surface to the depths of the body is a complex process that has not yet been fully traced (although its general features are understood). The path of energy transfer depends significantly on what the solid is. For example, in a metal, electrons can be carried away. But, whatever the path of energy removal from the surface, ultimately

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The interaction of one body with another leads to an increase in temperature. In addition, during the friction process, a change in the structure of the rubbing surfaces occurs. They must be taken into account if you want to understand all the phenomena accompanying trenna *). However, as long as we are not trying to construct a quantitative theory, it is not very important for us to know exactly what the process is in a solid. It is important to understand: the energy of a moving body is partially spent on exciting the movement of atomic particles that make up solids.

We can say that corpses “work”, only this work is done in vain - energy is wasted on heating the body, which most often comes to be artificially cooled (a Stone Age man, accustomed to adding fire to the morning, would not agree with us).

The greater the mutual speed of the bodies, the more energy is spent on rocking the atoms, which means the more work the bodies produce. On the other hand, it is known that work per unit time (power) is equal to force multiplied by speed. This force is the force of trenya.

We spring formally “introduced” the power of trenya. True, they tried to explain its nature.

You can do it a little differently. Find out how much momentum (amount of motion) is expended by a moving body per unit time on the excitation of atomic vibrations. According to Newton’s law, the change in the momentum of a body per unit time is equal to the force acting on it. This gives us the opportunity to determine the strength of trenya. Of course, both definitions lead to the same meaning of force.

The force of trenya characterizes the interaction between macrosoppe bodies, and although trenna is the result*

*) Teams of scientists are occupied with the problem of trenya, using the most modern research methods and the latest ideas about the movement of atomic particles in solids.

Because of the interaction of individual atomic particles, the frictional force is strikingly different from the forces that act between atoms and molecules. Indeed, the forces acting between atoms and molecules depend only on the distance between them, and the force of friction depends on the relative speed of the body. However, this is not the only reason why the frictional force differs from the forces of interaction between microsoptic bodies. It has properties called by different words, the meaning of which is the same:

Irreversibility, dissipation, loss

Under the influence of the force of training, the body always loses energy and is always thermalized. This is due to the fact that the energy of movement of bodies during friction is spent on exciting the movement of atomic particles. The atoms are unimaginable (10^23 particles per 1 cm³), they move randomly. The energy transferred to it is simply dissipated (dispersed, from the French word “dissiparer” - to dissipate) between countless atoms. The new owners of energy continuously exchange it with each other and are not at all inclined to repay the debt to the macrosoptic body. The described process is, of course, a transition from integral energy to warm energy. We see that this process is irreversible.

A fruitful method of generalization is that one of the main properties inherent in a given concept is taken as its new definition. Let us use this method to generalize the concept of trenya. The most important property of trenya, as we have seen, is irreversibility associated with the transition of the energy of ordered motion into heat. The acceptance and rejection of this property will be called tren any irreversible process that accompanies the transition of the energy of ordered motion into warm energy. Terrier, we can talk about the friction of two counter flows of gas - they are mutually thermalized and heated, about

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friction of a solid body against a gas, etc.

What about the friction force? It is still accompanied by a friction process. In fact, during the process of friction, the energy of ordered motion decreases, which means its speed also decreases. In other words, acceleration is directed in the direction opposite to the speed of the body.

Let’s remember Newton’s second law. Where there is acceleration, there is force. This force is the friction force. It is directed in the direction opposite to the speed, it always slows down…

Simple case

There are cases when the calculation of siliprene is relatively simple and does not require a deep penetration into the theory of solids.

Let, for example, a solid body move through a fairly rarefied gas. Then silipren is caused by collisions of gas atoms with the surface of the body.

Before proceeding to the calculation of silipren, we will consider a separate act of collision. A particle with momentum $P$ falls on the surface of the body. The collision of an atom with a surface is a complex event. Let’s think about what could happen. Atom can remain on the surface (float), can penetrate deeply into the body, or can be reflected from the surface of the body. With reflection (we will be interested in this particular silipran, we will explain how much later) various situations are also possible. For example, an Atom can innise or be knocked out of an atom from a surface, a molecule can disintegrate into its constituent atoms, and so on. In short, a lot will appear, but we will assume that atom was simply reflected, like a ball from a wall. Not for simplification, although this situation is indeed the simplest, but because in the overwhelming majority of silipevs, when a body moves through a gas not too quickly, this is exactly what happens - the atoms are simply reflected (bounced off) from the surface of the body ). Large complex situations (the introduction of atoms into a solid body, nucleation, etc.), of course, are also observed and studied in detail. Each of these processes occurs with most probability at a certain particle speed. For example, the higher the particle speed, the greater the likelihood that an atom will penetrate to a relatively greater depth into a solid body. To imagine what speeds we are talking about, we note: an oxygen atom incident on the surface of a solid body at a speed of $10^7$ cm/sec*) penetrates approximately 10-12 atomic distances into the depth of the solid body. The average speed of $v$ atoms of a gas with a mass $m$, as shown by the molecular-kinetic theory of gases, is equal to $\sqrt{\frac{kT}{m}}$, where $T$ is the temperature in absolute units, and $k$ is the so-called Boltzmann constant, equal to $1,4 \cdot 10^{-23}$ J/sec. At $T = 300^\circ$ K ($=27^\circ$ C) the average speed of an oxygen atom is less than $10^5$ cm/sec. If a body accumulates in a gas, then the average speed of gas molecules with the surface of the body approximately coincides with the average body speed; if the body moves, it differs from it. However, in the case when the speed of a moving body is small compared to the average solid speed (you should never go through the minimum partial silipevs, in which the question under study looks simpler), the speed of the moving body is almost no different from the solid body, and the energy of the atom is simply not enough to penetrate into the depths of the solid body.

Although we called the essence of reflection of atoms from the surface of a solid body the simplest, it also needs to appear…

*) Fortunately, very rarely water molecules from the air float to the flying person. Otherwise, the description of glaciation would have increased many times.
**) The kinetic energy of oxygen atom is equal to 10 keV (see below).

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Fig. 3.
Now we have really moved away from the subject that interests us.

What is the friction force?

So, let’s return to the calculation of siliprene acting on a body moving at a relatively low speed $u$ through a rarefied gas, the temperature of which is $T$. It is convenient to consider the body to be at rest and the gas to be moving at speed $-u$. Let us select an infinitesimal area $^*$ $\Delta S$ (Fig. 3) on the surface of the body. The momentum of a gas molecule, if it collides elastically with a body at a point on the visible area, will change to $2P_n$, where $P_n$ is the projection of the molecule’s momentum onto the normal to the surface of the body. This means that a gas molecule, when colliding with a body, gives it an impulse equal to $2P_n$. Now our task is to calculate the total impulse transmitted by the gas to the body per unit time. The change in momentum per unit time is the force (in this case, acting from the gas on the body). To calculate the total transmitted momentum, it is necessary, firstly, to take into account all the molecules falling per unit time onto a given area, and secondly, to sum over all areas.

*) See regarding the “infinitesimal platform” the article by N. Ya. Vilenkin “Secrets of Infinity” («Kvant» No. 3, 1970).

A similar operation was carried out in the article by Ya. A. Smorodinsky “Ideal Gas” («Kvant» No. 10). Here we will use ready-made results. The gas pressure on a wall at rest is equal to $1/3$ nmv², where $n$ is the number of gas molecules per unit volume, $m$ is the mass of the molecule and $v^2$ is the arithmetic mean of the squares of the velocities of the molecules. If the body ends ($u = 0$), the number of blows to the right and left is on average the same, and therefore no impulse is transferred to the body. At $u \neq 0$ this is not the case. Let the body move. In this case, the speed of molecules hitting the front is $u$ greater, and the speed of molecules hitting the rear is $u$ less. Therefore, the force acts on section $\Delta S$ of the moving plane

$$F = (P_{\text{спереди}} - P_{\text{задди}}) \Delta S = \frac{1}{3} nm \Delta S \left[ (v + u)^2 - (v - u)^2 \right] = \frac{2}{3} nmvu \Delta S.$$

Recalling that $v = \sqrt{\frac{3kT}{m}}$ (see the article “Ideal Gas”), we obtain

$$F = \frac{2}{3} n \sqrt{3kTm} u \Delta S.$$

For a body of arbitrary shape we must obtain the formula

$$F = \beta Sn \sqrt{mkT} u.$$

Here $S$ is the surface area of ​​the body, and $\beta$ is a numerical coefficient, depending on the shape of the body. It takes into account that molecules enter one area perpendicular to the surface, and others at an angle.

If we use Clapeyron’s law $PV = NkT$ ($N = nV$ is the total number of the gas part in the volume $V$), then we can express the friction force through the pressure $P$ and temperature $T$ of the gas:

$$F_{\text{Tp}} = \beta SP \sqrt{\frac{m}{kT}} u.$$

The result was natural: at fixed temperatures and

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speed of the body, the greater the friction force, the greater the gas pressure.

Gas viscosity

We have described in some detail why, when a body moves through a gas, a force acts on it - the force of friction - the result of the impacts of individual molecules on the surface of the moving body. At the same time, we were not at all interested in what happens to the gas, to its molecules after the collision. Let us think for a moment about the fate of the molecules that come into contact with the body. It is clear that if we had the opportunity to change the speed of all molecules interacting with a body, we would be convinced that their average speed is not equal to zero. Molecures, albeit slightly, are carried away by a moving body. And if you change the speed of molecules far from the body, then (in the absence of finite wind, their average speed is zero). In slightly different words: near the surface of a moving body there is an ordered movement of gas, but far from the body there is none. There is a certain impulse associated with ordered movement. A layer of gas in the surface layer collides more often with a moving body than other gas molecules. Because of this, their average momentum would have to increase if there were no process of momentum removal from the boundary layer.

This process is easy to understand. Gas molecules that escape from the boundary layer simply carry away excess momentum with them. Far from the body, they “disperse” it in multiple collisions with other molecules or in collisions with bodies at rest.

The ability of a gas to transfer impulse from one place to another and, most importantly, to waste it irreversibly is called viscousness. Viscosity and friction are the same. Viscosity is an interesting phenomenon in which collisions between molecules play a major role. We neglected when calculating the friction force acting on the moving body from the gas side.

We were wrong about this. But how? Did you exaggerate or underestimate the friction force? It’s easy to understand that they exaggerated. Indeed, we believed that a molecule, once in contact with a body, would never return to it and, therefore, would not be able to “return” to the body the impulse it received. In fact, if we carefully monitor many molecules colliding with the body, we will be convinced that some of them return to the body (after colliding with any of the gas molecules), others give the excess implus acquired from the body to their collide partners, and they, in turn, can collide with the body. Strictly speaking, as a result of such new molecules interacting with each other, a partial return of the body’s impulse occurs, which means that the body loses less impulse than we thought.

The formula derived here is applicable only to very rarefied gases, in which the average stretch between stolnov molecules is much greater than the size of the body. If we calculate the friction force acting on dust particles with dimensions not exceeding several microns, our air may also fit into the category of rarefied gases.