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TWO MEASURES OF MECHANICAL MOTHER MOVEMENT FORM
Second edition
Redesigned and supplemented
Ph.D., Yudin Sergey Yurievich
In the modern course "Theoretical mechanics" two measures of mechanical motion are used: momentum (momentum), expressed by the dependence mv and kinetic energy, expressed by the dependence mv 2/2, which introduces uncertainty into the nature of this motion. Although formally in the XIX century, Engels in his work "Dialectics of Nature" legalized both measures of motion, but there remained a lot of questions. Therefore, I decided to publish this article on the Internet, which was written back in 1989 and is now one of the sections of my book "Modeling Systems and Optimizing Their Parameters", so the numbering of formulas and figures is given in the numbering adopted in the book.
Why these measures are two, for which they are needed and what requirements should be met by the measure as such. In studying the motion of mechanical systems in dynamics, they are solved as a direct problem, when, given the forces and initial coordinates and velocities, it is necessary to find out how the system will move in the future, and the inverse, when, according to a given law of motion, it is necessary to find a function by which the forces must change The system moved according to this law (they are usually called, respectively, the second and first tasks of dynamics). In solving inverse problems, there are usually no complexities, but when solving a direct problem, it turns out that if we describe the motion of a real mechanical system, rather than an idealized one, i.e. Educational, then it is almost never possible to solve such differential equations analytically [7] , since They will almost always be nonlinear and most often nonlinearity will be caused by the forces of dry friction, which in one way or another are present in all mechanical systems. But if we make some assumptions that allow us to linearize the differential equations, i.e. Describe the motion of a similar system, then such equations can be solved analytically. However, in this case the solution can be very difficult. But if we do not need to describe the entire process of motion and we are only interested in the position of the system before and after what happens, for example, before and after, then it is possible to solve such problems without using Newton's laws to describe the motion of this system, and using conservation laws Mechanical motion, in which motion is precisely measured by these two measures. True, some authors point out that it is necessary to use these measures very carefully. For example, [3] writes that conservation laws never give an unambiguous answer about what will happen, and [2] writes that when it is possible to solve a problem through the conservation law, it is necessary to check whether in this case the second Newton law .
Now let's consider what requirements should be met by the measure. Under the measure of mechanical motion, I understand a quantity that objectively, comprehensively and conveniently reflects the essence of this movement. The last requirement for convenience follows from the requirement for methods of calculating mechanical systems that should be convenient, otherwise they will not survive in the social form of the motion of matter. We are now not using the Al Khorezmi method or using the compass and ruler to find the roots of the quadratic equation, although they also give the correct answer. Yes, and the main incentive of Copernicus, as he himself said, was precisely the simplification of Ptolemy's technique, which became not convenient for the rapid development of navigation, although it gave a very precise answer. Based on these requirements and consider these two measures of mechanical motion.
About which of these measures is true, i.e. Objectively and comprehensively reflects the essence of this movement, the dispute has lasted for more than three centuries. But in modern textbooks this issue is not given almost no attention. But if these two measures are equal, then the conclusions resulting from the properties of these measures can be extrapolated to the whole of Nature, and the conclusions for different measures can be different and critics of modern physics very often use the presence of two measures as an argument in their rightness and some even Manage to create a new physics, where there is no such measure as energy (I do not give the link not to propagandize sorcerers from science).
Contemporary mathematical physics, unlike the physicists and mathematicians of the past, who were always a little philosophers, and physics itself was called "natural philosophy", did not pay attention to such trifles as two measures of mechanical motion in their textbooks. Yes, there are some measures, some authors of textbooks, for example, Landau and Lifshitz, do not pay any attention to the physical meaning of the basic axioms of mechanics, and on the 10th page of the textbook [6] they introduce the principle of least action and from it, as from the cornucopia, All the laws of mechanics fall asleep mathematically. Incidentally, one of them says that if the mechanical system is not described by the Lagrange function, then this is not mechanics (although the authors use the term not classical mechanics). And here the authors just liken themselves to Mitrofanushka from Fonvizin's comedy "The Minor" , who on the question "Is the door is substantive or adjective?" Answered in a similar vein - "the one that lies in the closet, is substantive, and the one that is adjoined to the loops is adjective." But the system will not be described by the Lagrange function, for example, in cases where dry friction is present, i.e. Practically never, but the authors continue in the same spirit and write that if the friction in the system turns out to be weak and neglected and the masses of the elements connecting the system into a single mechanism, then in this case we can use the Lagrange function and, therefore, the system becomes immediately Mechanical.
To be fair, it should be noted that the textbook of the 1965 edition, when Ahnezer was among the co-authors, was similar to most textbooks on Theoretical Mechanics, but four years later the authors changed their views sharply. And this is especially strange, considering that at the conference in Kiev in 1959, L.D. Landau stated that the Lagrangian is dead and should be buried with all proper honors. By the way, it is this textbook that is so zealously advocated by the powers that be in science that it even received the popular name "Landivshits textbook" (after Lifshitz's death and beginning from the 4th edition only Pitaevsky LP is edited ). In more detail we will return to the possibilities of Lagrange's equations in Section 2.1.2 , and now let us return to the question of two measures, since A misunderstanding of this question leads us, for example, to the conclusion that the movement proceeding according to the laws of classical mechanics is irreversible when considering the oblique impact, and in all textbooks, for example, [2, 6] it is written that it must be reversible. However, when considering two balls of the same mass with an oblique impact, when one is at rest, it is directly stated [2, 6, 9] that they will fly at an angle of 90 degrees . But if we now make a return stroke, the balls will simply exchange speeds and one of them will no longer be at rest.
The first to think about the extent of the movement was Leonardo da Vinci and Galileo . Then, in the XVII century, a dispute arose between Descartes ' supporters, who claimed that mv and supporters of Huygens and Leibniz , who claimed that mv 2 . Galileo only suggests that the motion of a body (he called it a momentum or a moment) is proportional to both the mass and speed of this body. Descartes, in general, takes the product of the mass of a moving body by its velocity as the only measure of its motion. From this moment, a dispute begins as to the measure of mechanical motion, since Huygens discovered that in the case of an elastic impact, the sum of the products of the masses of two bodies per square of their velocities, and remains unchanged before and after the impact.
But, two different formulas can not reflect the same value, therefore Leibniz at the end of the seventeenth century takes as the measure of the actual motion mv 2 and calls this value "living force" . He called the term "dead forces" the pressures or thrust of resting bodies, which are measured by the product of the mass at the speed with which the body would move, if it moved from a state of rest to a state of motion or vice versa, as in the case of an absolutely inelastic impact. This view of things did not suit the supporters of Descartes and therefore a fierce verbal dispute about the two measures of the movement continued.
In the middle of the eighteenth century, d'Alembert managed to shift this dispute into a more realistic plane. He argued that under the power of moving bodies one should understand only their ability to overcome obstacles or resist them, and therefore the force should not be measured either through mv or through mv 2 . Further, D'Alembert tries to link both measures of motion, arguing as follows. Mass 1 , having a speed of 1 , compresses the 1-y spring into unit of time. Mass 1 , having a speed of 2 , compresses the 4th springs, but uses 2 units of time for this, i.e. Compresses in a time unit only two springs. So, if we divide the action into the time it takes for us, then we will return from mv 2 back to mv . From the above arguments, the conclusion suggests that under mv 2 . D'Alembert understands the kinetic energy or work that the body can do with this energy, and under mv - the work per unit of time, i.e. power. And so, the logic of Dahlambert's reasoning is correct, but the mathematical expressions used for kinetic energy and power do not correspond to modern expressions for these quantities, but his principle of deriving differential equations of motion of bodies, formulated by d'Alembert in the following form, "the momentum lost for an element of time The system of vectors balancing through the connections of the system " now could be called the power equation. And once under the amount of motion lost for the time element dt , d'Alembert understands the value of d mv , then at that time it meant that the final verdict of D'Alembert was in favor of mv .
Here it should be noted that at the time when these events developed, there were no generally accepted concepts such as work, power, strength, because They were just formed and all the unknown phenomena were explained by the action of their strength (digestive force, friction force, the power of love, the power of the mind, etc.), and the fact that "living force" and kinetic energy are one and the same will only become known In the second half of the nineteenth century, and the "living force" will not only be called kinetic energy, but will also be measured not as mv 2 , but as mv 2/2. Therefore, when considering this issue, each of the scientists operated on parameters in its individual coding, which did not coincide with the coding of other scientists, and hence the continuation of the dispute, since One does not quite understand what the other says. Those. We again encounter the problem of encoding and scaling parameters that we considered earlier, which at one time did not allow Archimedes even to describe the work of the invented screw.
Engels, as a philosopher, in contrast to modern mathematician-physicists, understands that "The task is to find out why the movement has a two-fold measure, which is unacceptable in science, as in trade . " Therefore, he, already familiar with the law of conservation of energy and terms such as work and power, considering the then known cases from the practice in which the motion is transmitted according to the formulas mv and mv 2 , makes the following conclusion. "Thus, we find that a mechanical movement does have a double measure, but we are convinced that each of these measures is valid for a very specific limited range of phenomena. If the mechanical movement is already present, is transferred in such a way that it is retained as a mechanical motion, then it is transmitted according to the formula for the product of mass per speed. If it is transmitted in such a way that it disappears as a mechanical movement, resurrecting again in the form of potential energy, heat, electricity, etc., if in a word, it turns into some other form of motion, then the amount of this new form Movement is proportional to the product of the initially moving mass per square of the velocity . "
As we see the question is really very complicated, and even such an enlightened philosopher as Engels could not solve it, because when the elastic bodies are struck, the mechanical motion is first transformed into the potential energy of the two springs (elastic spheres), and then again into the kinetic, ie, The measure mv is valid also in those cases when one form of motion passes into another. And three pages earlier, he himself proves that in all known cases, when the expression mv (this is the lever and impact of two elastic bodies) is true, the expression mv 2 is also true. But in the derivation, for some reason, he concludes that when the mechanical motion is transformed into a mechanical one, only the formula mv is valid. By the way, the example with levers is simply incorrect, because No transfer of motion is carried out and the lever with two weights at the ends must not be regarded as a separate movement of one cargo transferring traffic to another cargo but as a closed unified system moving uniformly according to Newton's 1st law with respect to rotational motion, which we will talk about further .
Let's try to deal with this issue ourselves and turn to formulas and experiments. Let the constant force F act along an axis, for example, x , in an inertial frame of reference on an absolutely rigid body of mass m . Then, according to the law of conservation of energy, the kinetic energy acquired by the body will be equal to the work of the perfect force F on the path x and will be expressed by the dependence
We differentiate both sides of this equation in time t and obtain the power equation for mechanical motion
Since our body is absolutely rigid, the velocity of moving the center of mass of the body vm is equal to the speed of application of the force vf, and both parts of the equation can be divided by the speed. Thus, from the power equation we came to the second Newton law in Euler's formulation.
If now expressing the acceleration a as dv / dt in this equation, and then multiplying both sides of the equation by dt , integrating the left-hand side with respect to v and the right-hand side with respect to t , then we arrive at the equation where the measure mv is used and another interesting quantity F * T , i.e. "Force pulse" for a time interval from zero to t .
Where v0 is the velocity of the body at the initial time t = 0 ;
It is reasonable to raise the question if all the expressions (2.1.1), (2.1.2), (2.1.3) and (2.1.4) are mathematically dependent, then why only the first and last formulas pretend to be a measure of motion. As for the measure mv 2 , the law of conservation of energy so many times proved its validity, that simply can not be subjected to any doubts. And that's why the measure mv pretends to the same role of the measure of mechanical motion? Indeed, with just as good a success, we can declare that formulas (2.1.2) and (2.1.3) are also laws of conservation, respectively, of power and forces, and even in vector form. There will be several explanations here: firstly, this is a purely historical aspect, and secondly, the impossibility of organically combining it with the rest of the formulas due to "inconvenient" dimension, thirdly - the existence of a "theory" of impact, where the expression mv is not only used for calculations, but And the formulas have a certain physical meaning, and in the fourth, the value of mv is also preserved by inertia, like mv 2, in contrast to the forces and powers that should be zero in this case (although a particular case of inertial motion is possible, when The force vector is constantly perpendicular to the velocity vector - the motion along the circumference). As for the historical aspect, it will not be eliminated, but the rest we will try to correct, so that the expression mv does not claim the role of the same measure of motion as mv 2 , and at the same time the expression (2.1.4) found an organic connection with existing physical Concepts that characterize the movement of the body.
At first glance, the expressions (2.1.1), (2.1.2), (2.1.3), (2.1.4) since all of them are mathematically related to each other describe the motion of the body equally objectively. However, in expression (2.1.4) there is one peculiarity: the force must be a function of time or be constant, because Only in this case we can take the integral of F * dt . This requirement is contrary to the laws of Nature, t. All the forces in Nature acting on the bodies (not counting the inertia forces) either depend on the coordinates (the forces of gravitational or Coulomb attraction and the force of elasticity) or on the speed (Lorentz force, Coriolis force, reactive and fluid friction) or on the velocity gradient (the force of dry friction) . This fact is indicated by many authors of textbooks [2, 3] , and in classical mechanics, as they say [4] , the forces in general depend only on the coordinates. But for some reason, again, in many textbooks, for example, [1] it goes without saying that the force can depend on the time, and as an example, the centrifugal force is shown when the unbalanced rotor in the motor rotates (this example was discussed in Section 1.1 ) .
But in this example, the law of changing centrifugal force from time only approximately for the solution of practical problems can be considered admissible (I simply did not see other examples). After all, even if we assume that the load in the electric network from our electric motor will be constant and, consequently, the speed of rotation of the turbine rotor at the power plant and thus the frequency of rotation of the magnetic field in the stator of our electric motor will be stable, then its characteristic with a change in load from the imbalance is still not Will allow the rotor to rotate evenly. And let the law of variation of this force be slightly different from the strict sinusoidal dependence in the function of time, this does not allow us to assume that the forces in Nature can depend explicitly on time. Although in various control systems, we can of course programmatically specify any kind of force dependence on time.
On this issue, I only mentioned in the textbook [2] a slight mention of the fact that the force appears to have two characteristics, but its temporal characteristic is placed at the forefront (although it is also written here that forces can only be a function of coordinates and Speeds). The author writes: "Along with the time characteristic of the action of force - its impulse - in mechanics, the spatial characteristic of the action of force, called mechanical work, also plays an important role." That's just not saying that it will be already completely different forces, which we'll talk about in more detail on a specific example after. And now just recall that in mechanics, along with other postulates, it is accepted that the work (energy) is a product of force on the path of its action and if the force is not constant, then for calculation it is necessary to take the integral of F (x) * dx or take the product of medium strength Along the way (not by time) and multiply by the path. And if we take into account that force can only be a function of coordinates or velocities, then this postulate acquires a theoretical justification.
Thus, the dependence (2.1.4) is partially artificial, i.e. Completely suitable only for solving real problems and, therefore, the measure of mv to claim the same universality in Nature as a measure of mv 2 can not. But, at the same time, with the help of this expression, many simple real problems can be solved even when one form of motion passes into another. For example, the motion of the body along an inclined plane, when the potential energy passes into kinetic, and kinetic due to friction into thermal, because The frictional force on the surface and the tangential component of the attractive force do not depend on either the displacement or the velocity, i.e. Are constant.
Where: a - angle of inclination of the plane to the horizon; F is the coefficient of friction of the body on the plane; G - acceleration of gravity.
We now divide both sides of expression (2.1.4) by m and transfer v0 to the right-hand side and we get v = v0 + F * t / m , and since At constant force F / m this is a constant acceleration a , then we can write v = v0 + a * t . Thus, the expression (2.1.4) is equivalent to the formula for determining the velocity for accelerated motion and, in doing so, has a completely defined meaning and retains all its properties in this record, which is valid not only when F = const, but also in those cases when It is a function of time. For example, if F = F0 * sin (w * t) , then v = v0 - F0 * (cos (w * t) - 1) / m / w . And the only requirement to apply the measure mv is the presence of forces that depend on time or constants. Thus, the measure mv corresponds only to one requirement imposed on the measure of motion-objectivity, and then in a particular case, and therefore it can be that only for the solution of educational problems, except for the simplest case when all the forces acting on the system are constant.
True, some opponents may object to me that the value of mv is a vector value (although originally in Descartes the expression mv , as a measure of motion, was a scalar quantity), and mv 2 is scalar and therefore mv 2 can not in any way replace mv , for example , In determining the velocities of ball expansion after impact. To this I can answer that, as will be shown below, no vector merits of this measure will help her to determine either the direction of the balloon's expansion or the magnitude of their velocities. And its only vector advantage should be recognized, the fact that the direction and magnitude of the center-of-mass velocity of the entire closed system will remain unchanged for any displacements of individual parts of the system among themselves due to the internal energy of the system. And this is how you understand the formulation of Newton's 1st law for a closed system, i.e. For the motion of its center of mass. And the fact that the law of conservation of momentum is equivalent to 1- Newton's law is directly pointed out by the authors [5, 8] . By the way, as many authors write, many problems using the measure mv are much more convenient to solve in the so-called C-system , i.e. The system of reference of a system connected with the center of mass, which is translationally moving relative to some inertial frame of reference. But if someone really likes to call the first law of Newton - the law of conservation of momentum, then in principle I have no objections. That's just not clear why everything is complicated and introduce to mechanics one more measure of motion. On this fan to invent a bicycle, I would like to recall the idea of Lomonosov about the simplicity of Nature and the need to abandon the complications in its description, if it is possible to describe it simply. And, using the measure mv 2 , you can successfully apply not only the formula (2.1.1) , but also the formulas (2.1.2) and (2.1.3) , where the force u is a displacement function and, therefore, in these formulas is used Measure mv 2 . And, what is very important for processing experimental data from oscillograms that are always written with a time sweep, as will be shown later, you can use instantaneous power values not only in the displacement function, but also as a function of time. And although in the ergodic processes the deviation from the requirement to calculate the average values of the parameters along the path and not the time leads to an error of only about one percent, this is not an experimental error, but a systematic error, which for theoretical conclusions is not permissible, and in nonergodic processes the error can reach And tens of percent.
But maybe there are problems that we can not solve without resorting to mv or mv-based theory gives better results than mv 2-based theories. In this case, naturally, the measure of the motion mv will have the right to exist, even if only for this particular case of motion. The only dynamic region (not counting quantum mechanics, where truth is no dynamic) based on the measure of motion mv is the "theory" of the impact. What are the possibilities of this "theory"? It is believed that we can determine the magnitude and direction of the velocities of the bodies after the impact, and the magnitude of the maximum impact force. But, this "theory" has significant drawbacks. We can not determine neither the magnitude nor the direction of the velocities of the bodies moving and the forces acting on them during the impact, but also the time of interaction of two bodies. In addition, with an imperfect impact, we are forced to use the rate recovery factor, which must be obtained experimentally, and which, in addition, is not invariant to the magnitude of the relative velocity of colliding bodies and their shape. To determine the maximum force at impact, the approximate dependence of the change in force upon impact is used, and the experimentally obtained value of the impact time, since Neither this nor the other does not follow from the "theory" . Thus, from the so-called "theory", practically nothing remains and we see simply a set of empirical dependencies. But, if we do not yet have other means to describe the motion of bodies upon impact, then this semi-theory and measure mv still have the right (at least temporary) for existence.
For those who are not familiar with my book, it is necessary to report that when modeling mechanical systems to compose the equations of their motion, I, as a general principle, use the power equation, i.e. I apply the energy method and, therefore, recognize only the measure of motion as mv 2 . In this case, I do not have active forces and reaction forces in the equations, and all forces are equivalent and I write the equations not in vector form, but in the axes of the Cartesian coordinate system, which I consider more natural for Nature. At the same time, we will not have any connection equations, since There will be no links themselves, and I take the signs of the magnitude of the forces and velocities in accordance with the direction of the axes and if the power is obtained with the plus sign, it increases the kinetic energy of the system, and if it decreases, it decreases accordingly. In those cases where the bodies have moments of inertia and rotate, I additionally write three more equations of rotation of the bodies around the axes of the Cartesian coordinate system in accordance with the equation of powers for rotational motion. In this case, three Newton's laws for rotational motion will look as follows
Where: w - angular velocity, J - moment of inertia, M - torque.
In the vast majority of cases, the power equation, i.e. The principle of d'Alembert in my modern interpretation is simplified to its classical interpretation given in textbooks, i.e. Up to the principle of kinetostatics, but this does not mean that we must always act this way. So when modeling processes, where we have kinematic slippage (both elastic and inelastic), i.e. There is no rigid relationship between the translational and angular velocities of parts of the system, for example, in V-belt transmissions or when driving a car wheel, the use of the power equation leads to fundamentally different solutions. In particular, it helped me to significantly improve the modern theory of rolling the wheel and get rid of the mythic resistance to the movement of the wheel present in it when it moves over an unshielded base, replacing it with the moment of resistance from elastic slipping (not to be confused with slipping) of the individual parts of the tire in the contact spot relative to each other Friend. Thus, another "magic force" from the past has become less and who is interested in this issue can find an answer to it at ser.tk.ru.
Let's now, using the principles of modeling mechanical systems outlined above, let's see if we can get the same results when struck, and if possible and better, without resorting to the help of the mv measure, but simply simulating the impact. Let us describe the motion of an absolutely elastic body (ball), whose entire mass is concentrated at its center of mass, when it hits an absolutely rigid wall, using equation (2.1.2) and In our case vm = vf we arrive at the particular case, that is, The equation (2.1.3)
Where: c - the rigidity of the body in the longitudinal direction.
The solution of this differential equation is known
Where: k1, k2 are constants determined from the initial conditions; W is the angular frequency of the oscillations.
For body parameters m = 1 kg , c = 10000 n / m and initial conditions v0 = 10 m / s , x0 = 0 for t = 0, the solution of equation (2.1.10) takes the form
The change in speed at impact and the impact force, which is defined as c * x , in the time function t and in the displacement function x, we will graphically depict in Fig . 2.1.1 .
Fig.2.1.1. Dependence of the impact force F and the velocity of the ball V as a function of time t and its deformation x.
Thus, we can find not only the value of the speed of the ball after the rebound, but also obtained dependencies for finding the force and velocity during the impact, which can not be obtained from the measure of the motion mv . Here it should be noted that the calculation of the average force, as well as the maximum, according to the formula based on mv, is not very unambiguous, even if we know all the other parameters of the impact, if we do not take into account what this force is. For example, using the data obtained by us in the example considered, and substituting them into the formula recommended by the "theory" of impact to calculate the average force, we get
But only the authors of textbooks nowhere write that this is not the force to which we are accustomed, i.e. On the displacement, and the average force over time, because the average force for displacement will be 500 n . The maximum force for impact, according to the "theory" of the impact based on mv , for example, in the textbook [1] is recommended to be defined as the doubled value of the average force, because This is the only reasonable recommendation for the unknown dependence of the change in force, which (the dependence) "theory" based on mv simply can not give. But we have such a dependence and, according to Fig. 2.1.1, the maximum force at impact was 1000 n , which is much less than 1274 N , which can give the "theory" of impact based on mv . But as we shall now see, formula (2.1.13) is in principle valid, but for another force, i.e. Not spatial, but temporary power. In this case, the work done by the force with an impact Ax is equal to the product of the average value of the force for the displacement Fxcr multiplied by this displacement x , but not At equal to the average value of the force in time Ftcp multiplied by the displacement calculated from the mean velocity in time x = Vtcp * t or The average value of the power obtained from the average values of the force Ftcp and the velocity Vtcr in time multiplied by the action time, which is easy to verify by comparing this work with the kinetic energy Ak lost by the ball before it stops at maximum strain.
But if we take from instant power
In the interval from zero to 0.0157 seconds of the integral we get again the correct answer
Thus, from the mathematical point of view, the forces Fx and Ftcp are quite acceptable for calculations, and the question is only for which calculations and how to apply them, so as not to lose the physical meaning of these calculations both in the simplest example of an elastic impact considered now, and in other more complicated cases , As, for example, in thermodynamics, when determining the gas pressure on the vessel walls.
Similarly, by simulating the whole process, one can also obtain dependencies along which the velocities are determined for an absolutely elastic or absolutely inelastic frontal impact of two balls. For example, the motion in the frontal collision of two absolutely inelastic balls moving along the x axis is described by a system of differential equations (2.1.14) , solving it by finding the velocities of both balls, which are expressed by the dependences (2.1.15) and (2.1.16)
Where: v1 , v2 - velocities of balls before impact; 
- velocity of the point of contact; D1 , d2 - dissipative parameters of the balls; A1 , a2 are constant coefficients, determined from the dependences
It follows from formulas (2.1.15) and (2.1.16) that for the time tending to infinity, the first terms of these expressions tend to zero and, consequently, at the end of the impact, the velocities of both balls are expressed by the second part of equations (2.1.15) and 2.1.16) , which are identical. Consequently, the velocities after the impact will be equal and after substituting the values of a1 and a2 in these expressions is defined as u
The motion in the frontal impact of two absolutely elastic spheres is described by the system of equations (2.1.20) , and the solutions of these equations for the velocity of the balls are given by expressions (2.1.21) and (2.1.22)
Where: v1 , v2 - velocities of balls before impact; Ce = c1 * c2 / (c1 + c2) is the total rigidity of the two balls; B1 , b2 are constant coefficients (b1 = ce / m1, b2 = ce / m2)
After a while 
Balls will come out of contact and fly at constant speeds. Substituting this value of time and the values of the coefficients b1 and b2 in dependence (2.1.21) and (2.1.22) , after the transformations we obtain
Where: u is the velocity of the balls after an absolutely inelastic impact, i.e. When the system completely consumes its internal energy, determined by formula (2.1.19)
In an analytical solution of these two problems, which I received for the first time (at least in the open press, I have not seen any of these calculations anywhere), it has been assumed that when struck in the contact spot, there will be no dry friction, which, however, Impact of two balls of the same diameter and the same rigidity, and assumed that the friction inside the material of the balls for an absolutely inelastic impact will be liquid, i.e. The frictional force will be proportional to the rate of deformation of the ball. These assumptions required that the differential equations describing the motion of the balls be linear, since Otherwise we would not be able to solve them analytically. But, with the help of differential equations, similar to the above, it is possible to describe the impact of any bodies at any angle, and with dry friction, and with variable rigidity, and of various shapes. Only, to solve such equations it is necessary to use numerical methods of integration, for example, the Runge-Kutta method, which I realized in the Udar23 program, which everyone can download from my home page ser.tk.ru.
It follows from the particular solutions (2.1.19) , (2.1.25) , (2.1.26) that the solutions coincide with the equations given in the "theory" of the impact based on mv at the moment when the blow is already completed, and the general solutions of the systems ( 2.1.14) and (2.1.20) allow us, in addition, to describe in detail the process of impact itself. Thus, here again the expression mv can not claim the role of another measure of motion, since Not only does not give us anything new, but the solutions obtained with the help of this measure are only a partial solution of the general solution obtained without this measure and, therefore, see Lomonosov's statement given above.
Even more obvious is the inconsistency of the measure mv in solving problems, when two balls hit at an angle to each other, i.e. With oblique impact. Here, even a particular solution, i.e. When the balls came out of contact with each other, does not correspond to reality. For example, if you hit absolutely elastic and absolutely smooth two balls of the same mass m1 = m2 = 1 kg and the same stiffness c1 = c2 = 400 n / m at an angle of 90 degrees Fig.2.1.2. It turns out that if their velocities before impact were v10 = 1 m / s and v20 = 10 m / s , then after impact according to the existing "theory" of impact they will be v1 = v2 = 7.1 m / s . In fact, the balls will scatter as shown in Fig. 2.1.2 with velocities v1 = 2.6 m / s and v2 = 9.7 m / s , which were obtained by solving this problem numerically using the Udar23 program ).
Fig.2.1.2. The velocities of the two balls before and after the oblique impact, calculated from the equations of motion by the energy method obtained with the indices mv ^ 2 and using the classical "theory" of impact with the indices mv in the left figure and the dependence of the velocities of the balls after the impact on the rigidity of the balls in the right figure.
What explains this discrepancy between the solutions obtained. And the fact that to solve this problem analytically, we need to have four equations for finding two speeds with respect to two coordinates x and y , and we have only three of them. If we use the first law of Newton , as the law of conservation of momentum, ie. As a measure of mv , we can derive from it two equations. Another equation gives us the law of conservation of energy. Balls are absolutely elastic and absolutely smooth (ie absent both friction inside the balls and in the contact spot), but we do not have the fourth equation and take it from nowhere. But in order to somehow solve this problem, some assumptions are used, as, for example, it was done in [9] and it is assumed that Both balls are absolutely smooth, therefore, there will be no tangential forces in the contact spot and, consequently, the tangential velocities of the balls will not change, but only normal forces from the action of only normal forces in the contact spot will change; As in a frontal impact and from here we get another equation.
But, introducing such an assumption, the authors of textbooks forget to say that for this purpose the time of impact must be zero, i.e. The system must be three times ideal, which is not possible even theoretically. On the one hand, the bodies must be absolutely elastic and smooth, so that there is no dissipation of energy, and on the other hand, absolutely rigid so that the impact time is zero, or else during the time of impact of the vector of normal forces in the contact spot they will turn around at some angle. For example, in the example considered by us during the impact time, the vector of normal force in the contact spot manages to rotate through an angle of about 60 degrees, which is clearly visible on the graphs during the Udar23 program . But if we gradually increase the rigidity of the balls, i.e. To decrease the contact time on impact, we see that the velocities of ball expansion will gradually approach the solution at the time of impact equal to zero, i.e. With the rigidity of the balls equal to infinity, as shown in the right figure. 2.1.2 . But if we take an absolutely rigid body so that the impact time is zero and the vector does not have time to turn around, we will get the force at the impact equal to infinity and, therefore, we will not scatter balls in different directions, but their fragments.
And now let's return to the problem of reversibility of processes in mechanics, which I mentioned earlier, which occurs when using the measure mv . If we are in the example of an oblique impact of absolutely elastic and smooth balls, that is, An ideal system consisting of two balls, after the balls are scattered, stop the time, and then pressing the "Reverse" button in the Udar23 program, we will make the balls move with their velocities, but strictly in the opposite direction, then we simulate the time machine. And since we have an ideal system (in other words, without friction), then we must observe the reversibility of the process in time, i.e. The effect should be as if we were rolling the film with the impact of the balls in the opposite direction, and in this case we can not determine in any way whether the film is moving in the forward or reverse direction, i.e. After the return stroke, the velocities of the balls should become the same as before the impact, i.e. 1 m / s and 10 m / s , which we observe in the program, where the measure mv 2 is used . And with the use of the measure mv after the first impact, we obtained identical velocities of the balls v1 = v2 = 7.1 m / s and the balls thus fly at an angle of 90 degrees . If now to make a return stroke, then again we will get the same speeds of 7.1 m / s , tk. Balls simply exchange speeds. And now, however much we force the balls to make a direct and reverse stroke, the speed will always be 7.1 m / s , i.е. Will never become 1 m / s and 10 m / s and, consequently, there will be no reversibility of the process.
Thus, we can state the complete inconsistency of Engels' conclusions that the mechanical motion has a double measure of motion and that in some cases, it is measured as mv , and in others, as mv 2 . The same rare cases when with the help of mv it is possible to obtain some results should be attributed not to the law of conservation of momentum, as yet another measure of mechanical motion, but to the natural conclusions resulting from the mathematical interpretation of Newton's 1st law for the motion of its center Mass. But in order to finally finish with this question, it is necessary, at least in brief, to dwell on the magnitude of the derivative of mv , namely the "moment of momentum" , i.e. For a material point as it moves along a circle of radius R, this will be mv * R = m * w * R * R = J * w . In the course "Theoretical mechanics" of this measure of motion, the head is necessarily dedicated and with the same rights as mv , only with rotary motion. True, the tasks that are solved with this measure of motion and, as in the case of mv , do not correspond to the areas of application of the two measures of motion indicated by Engels, and, in addition, include not only tasks for rotational motion of systems, but also tasks for Movement along the circumference of individual bodies during their movement only under the influence of central forces. A particular example of such problems is the motion of the planets of the solar system, which obeys Kepler's second law, which is the same as the theorem on the conservation of the "angular momentum" , where the kinetic energy of the planets is then destroyed, passing into a potential one, again arising from the potential one. As we can see the conservation theorem for the "angular momentum" (2.1.27) , this again, just as in the case of translational motion, is simply the mathematical formulation of Newton's 1st law only for rotational motion with its specificity, and here we do not have the mass m , And the moment of inertia J and not the linear velocity v , but the angular velocity w .
Let us consider an example. A ball of mass m is attached to a thread passed into the hole of the horizontal surface and moves around the circumference around this hole with an angular velocity w1 , and the length of the filament is equal to R1 . Those. The ball moves in inertia along the circumference only under the influence of the tension of the thread, i.e. The central force (the friction of the ball on the surface is neglected). Then the thread is slowly pulled into the hole until the ball starts to rotate around the circle of radius R2 . It is necessary to determine the angular velocity w2 . This is a very typical problem in the course of theoretical mechanics and it is solved on the basis of the theorem on the conservation of the "angular momentum", because Here the mechanical motion has become mechanical and thus, in the form of a product of the vector mv, the radius vector must be preserved and, consequently, the so-called kinetic moment must be preserved.
J1 * w 1 = J2 * w 2 (2.1.27)
Where: J1 , J2 are the moments of inertia that, for a ball moving along a circle under the action of some kind of central force that does not allow it to fly away from the action of inertia forces with an instantaneous rectilinear motion, are equal
Although we can assert that equation (2.1.27) is not a theorem about the conservation of the angular momentum , but simply the ball moves in inertia according to Newton's 1st law, and therefore the product of the moment of inertia by the angular velocity should not change , And hence, knowing what moment of inertia the ball has when it moves along the radius R2 , find w2 . To solve this problem, we can completely simulate the whole process of ball motion and find everything that interests us, as it was with a stroke, and even much more than what interests us at the moment. For example, we can observe a very interesting phenomenon when the ball initially flew freely, and then the thread was stalled and the ball moved under the action of the central force along the ellipse with a slight rotation of the ellipse. Such a movement of Mercury is explained by the theory of relativity of Einstein (the truth is with a big mistake) and is considered one of its experimental proofs, since it is believed that such a movement can not be explained by the laws of conventional mechanics. And if you use the program Konma2 , which you can download from my home page, then you too can do this experiment (part of this experiment is given in the screenshot in the Programs section). But if you do not want to consider the process of ball motion in all the details and you are only interested in the initial and final states of the system, then you can learn this in the same way and without resorting to Newton's 1st law, and using the measure of motion mv 2 as the law of conservation of energy . After all, if we look at what the kinetic energy of the ball became, after we dragged it, we will find out that it has become larger. Consequently, here not only the energy of mechanical motion along one circle has passed into the energy of mechanical motion along another circle, but some other energy has been transformed into kinetic. Naturally, this will be a work done by dragging the ball from R1 to R2 . The difference of the kinetic energies of the ball at the points R2 and R1 is equal to the work of dragging the ball and, consequently,
Where: w is the angular velocity, depending on the variable R
In order to find the functional dependence w = f (R) , we replace R2 by the variable t and w2 by w (t) . Now we differentiate the expression (2.1.29) with respect to the variable t . As a result, after the transformations, we obtain the differential equation (2.1.30) , whose solution is (2.1.31)
If we now multiply the right and left sides of this equation by m , we obtain
those. Equation (2.1.27) . Thus, since (2.1.32) was derived from the general law of conservation of energy, we can rightly say that it is a special case of the law of conservation of energy when the body moves around the circumference under the action of central forces and the measure of motion "the angular momentum" does not have To this law ( 1- Newton's law for rotational motion) is irrelevant.
Very often, the theorem on the conservation of the "angular momentum" of the system with respect to the fixed axis is used, if the sum of the moments of the external forces is zero. For example, a person standing on a bench Zhukovsky, can turn rotating over his head an umbrella. Using the theorem on the conservation of the "angular momentum" of the entire system, one can find the speed of rotation of a person, knowing formula (2.1.27) , knowing the moment of inertia of the umbrella and the person, and the speed of rotation of the umbrella. But this problem can be solved with the same success without the "momentum of the momentum" and the law on its conservation, and using Newton's first law for the rotational motion of a closed system or equation (2.1.34) for the uniformly accelerated motion along the circumference similar to equation With rectilinear motion of the body
And since the angular acceleration e , according to Newton's 2nd law, is M / J , then for w0 = 0 for man and umbrella we have
And since according to Newton's 3rd law the turning moment applied to the umbrella is equal to the moment applied to the person, then M1 = M2 and hence the dependence (2.1.27) is obtained in a natural way , which can be understood and logically explained.
Thus, if we state that we can give this product of mass to the speed, i.e. Mv, if, in addition to the mathematical interpretation of Newton's 1st law for the motion of the center of mass of a closed system, we add to it some other meaning as another measure of mechanical motion, then we have to state that nothing but a headache. So why reinvent one more measure of the mechanical form of the motion of matter? Is it only so that students can practice in mathematics on the simplest teaching tasks? From all that has been said above, it follows from the practical conclusion that the presence of two measures of the mechanical form of the motion of matter in textbooks on "Theoretical Mechanics" is not only not justified, but also harmful. The exclusion of sections related to mv , as with another measure of movement, will result in not only simplifying the presentation of the material, but also significantly improve the quality of students' knowledge. This is all the more justified by the fact that with the development of computer technology, it is possible to solve any system of differential equations by numerical methods and, in doing so, to practically automate not only the process of writing equations using the power equation, but also their solving by numerical methods. At the same time, the bulk of engineers do not have to memorize particular solutions to specific problems for the successful development of new models of equipment. And experts involved in solving fundamental problems do not need to solve a specific problem, but a similar one. After all, in Nature there are no such phenomena that would be described by linear differential equations, for which one can find an analytical solution and we have to replace the real problem of a similar one, i.e. Linearize the differential equations. And, consequently, the main task of the course of "Theoretical Mechanics" is to use the power equation (2.1.2) , where there is always an understandable physical meaning of the laws of Nature, to teach the student to make systems of differential equations, i.e. Model the world around us and correctly enter the initial data into the standard program for solving differential equations by numerical methods.
At the beginning of the article, I mentioned that the measure mv is also used in quantum mechanics, but did not consider these cases in this article, and on this issue it turned out to be incomplete. And although a less-than-good theory in quantum mechanics is due to the wave properties of light and other radiations, but their properties as particles are used only for applied purposes to determine the mass and velocity of particles in their collisions (impacts), which we considered, but in those rare In cases when, in discussing theoretical questions, we are talking about an electron not as a wave or a cloud spread around the nucleus of an atom, but as a moving particle, then the measures of motion mv and mv 2 necessarily appear. But in quantum mechanics there is also a new measure of motion - the "action" with the joule dimension multiplied by a second, which corresponds to the dimension of the Planck constant, i.e. Quantum of action, which we have not talked about yet, and this simply forces us to continue the conversation that has begun, and I will certainly touch upon this question in the next article, "Mechanics for Quantum Mechanics" .
Bibliography
1. Butenin NV, Luntz YL, Merkin DR Course theoretical mechanics: T.2 .: Dynamics. - 3rd ed. - Moscow: Science. 1985, p. 496.
2. Butikov EI, Kondratiev AS Physics for in-depth study: Part 1: Mechanics. - , from. 352.
3. Irodov IE Basic laws of mechanics - 3rd ed. , from. 250.
4. Landau LD, Akhiezer AI, Lifshits EM General Physics Course: Mechanics and Molecular Physics. Moscow: Nauka, 1965. - 399 p.
5. LD Landau, EM Lifshits. Short course of theoretical physics: T.1. Mechanics. Electrodynamics. Moscow: Nauka, 1969. - 271 p.
6. LD Landau, EM Lifshitz. Theoretical physics: Proc. Allowance in 10 tt1. Mechanics. - 4 th ed. Moscow: Nauka, 1988. - 216 p.
7. Markeev A.P. Theoretical mechanics - 2 nd ed. - Moscow: CheRo. 1999, p. 572.
8. Savelyev M.V. Course of General Physics: T.1. Mechanics, oscillations and waves, molecular physics - 4 th ed. - Moscow: Nauka, 1969. - 271 p.
9. Sivukhin D.V. General course of physics: T.1. Mechanics. M .: 1979. - 520 p.
print version
Author: Candidate of Technical Sciences, Yudin Sergey Yurievich
PS The material is protected.
Date of publication 05.12.2004гг
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