What does mechanics study in physics. Classic mechanics

Mechanics

[from the Greek. mechanike (téchne) - the science of machines, the art of building machines], the science of the mechanical movement of material bodies and the interactions between bodies occurring during this. Mechanical motion is understood as a change over time in the relative position of bodies or their particles in space. Examples of such movements, studied by the methods of M., are: in nature - the movements of celestial bodies, vibrations of the earth's crust, air and sea currents, the thermal movement of molecules, etc., and in technology - the movements of various aircraft and vehicles, parts of all kinds engines, machines and mechanisms, deformation of elements of various structures and structures, the movement of liquids and gases, and many others.

The interactions considered in M. represent those actions of bodies on each other, the result of which are changes in the mechanical motion of these bodies. Their examples can be the attraction of bodies according to the law of universal gravitation, the mutual pressures of contacting bodies, the action of particles of a liquid or gas on each other and on bodies moving in them, etc. Usually, M. is understood as the so-called. classical M., which is based on Newton's laws of mechanics and the subject of which is the study of the motion of any material body (except for elementary particles), performed at speeds that are small in comparison with the speed of light. The motion of bodies with speeds of the order of the speed of light is considered in the theory of relativity (see Relativity theory), and intra-atomic phenomena and the motion of elementary particles are studied in quantum mechanics (see Quantum mechanics).

When studying the motion of material bodies, a number of abstract concepts are introduced into M., reflecting certain properties of real bodies; these are: 1) Material point - an object of negligible size, having a mass; this concept is applicable if in the studied motion it is possible to neglect the dimensions of the body in comparison with the distances traversed by its points. 2) An absolutely rigid body is a body, the distance between any two points of which always remains unchanged; this concept is applicable when the deformation of the body can be neglected. 3) Continuous variable environment; this concept is applicable when the molecular structure of the medium can be neglected when studying the motion of a variable medium (deformable body, liquid, gas).

When studying continuous media, one resorts to the following abstractions, reflecting, under given conditions, the most essential properties of the corresponding real bodies: an ideally elastic body, a plastic body, an ideal liquid, a viscous liquid, an ideal gas, etc. In accordance with this, M. is divided into: M. material points, the M. of a system of material points, the M. of an absolutely rigid body, and the M. of a continuous medium; the latter, in turn, is subdivided into the theory of elasticity, the theory of plasticity, hydromechanics, aeromechanics, gas dynamics, etc. geometric properties of the motion of bodies and dynamics - the doctrine of the motion of bodies under the action of forces. In dynamics, 2 main tasks are considered: finding the forces under the influence of which a given movement of the body can occur, and determining the movement of the body when the forces acting on it are known.

Mathematical methods are widely used to solve problems of mathematics, many of which owe mathematics to their very origin and development. The study of the basic laws and principles that govern the mechanical motion of bodies, and the general theorems and equations arising from these laws and principles, constitutes the content of the so-called. general, or theoretical, M. Sections of M., which have important independent significance, are also the theory of oscillations (see Oscillations), the theory of stability of equilibrium (see Stability of equilibrium) and stability of motion (see Stability of motion), the theory of gyroscope a, Mechanics bodies of variable mass, the theory of automatic control (see. Automatic control), the theory of Shock a. Experimental research carried out with the help of a variety of mechanical, optical, electrical, and other physical methods and devices occupies an important place in M., especially in M. of continuous media.

M. is closely connected with many other branches of physics. A number of concepts and methods of mechanics, with appropriate generalizations, find applications in optics, statistical physics, quantum mechanics, electrodynamics, the theory of relativity, and others (see, for example, Action, Lagrange function, Lagrange equations of mechanics, Mechanics canonical equations, Least action principle ). In addition, when solving a number of problems in gas dynamics (see Gas dynamics), the theory of Explosion a, heat transfer in moving liquids and gases, aerodynamics of rarefied gases (See aerodynamics of rarefied gases), magnetohydrodynamics (see Magnetic hydrodynamics), etc. simultaneously Methods and equations of both theoretical M. and, respectively, thermodynamics, molecular physics, theory of electricity, and others are used. M. is important for many branches of astronomy (see Astronomy), especially for celestial mechanics (see Celestial mechanics).

Part of M., directly related to technology, consists of numerous general technical and special disciplines, such as hydraulics, resistance of materials, kinematics of mechanisms, dynamics of machines and mechanisms, theory of gyroscopic devices (see gyroscopic devices), external ballistics, dynamics of missiles, theory of motion. various land, sea and air vehicles, the theory of regulation and control of the movement of various objects, construction M., a number of sections of technology, and much more. All these disciplines use the equations and methods of theoretical M.T.O., M. is one of the scientific foundations many areas of modern technology.

Basic concepts and methods of mechanics. The main kinematic measures of motion in M. are: for a point, its Velocity and Acceleration, and for a rigid body, the velocity and acceleration of translational motion and the angular velocity and angular acceleration of the rotational motion of the body. The kinematic state of a deformable solid is characterized by the relative elongations and displacements of its particles; the totality of these values ​​determines the so-called. strain tensor. For liquids and gases, the kinematic state is characterized by the strain rate tensor; in addition, when studying the velocity field of a moving fluid, the concept of a vortex, which characterizes the rotation of a particle, is used.

The main measure of the mechanical interaction of material bodies in M. is Force. At the same time, the concept of a moment of force (see Moment of force) relative to a point and relative to an axis is widely used in M. In a continuous medium, forces are specified by their surface or volumetric distribution, that is, by the ratio of the magnitude of the force to the surface area (for surface forces) or to the volume (for mass forces) on which the corresponding force acts. The internal stresses arising in a continuous medium are characterized at each point of the medium by tangential and normal stresses, the combination of which is a quantity called the stress tensor (see Stress). The arithmetic mean of three normal stresses, taken with the opposite sign, determines a quantity called the Pressure m at a given point in the medium.

In addition to the acting forces, the movement of a body depends on the degree of its inertia, that is, on how quickly it changes its movement under the action of the applied forces. For a material point, the measure of inertia is a quantity called the mass (see Mass) of the point. The inertia of a material body depends not only on its total mass, but also on the distribution of masses in the body, which is characterized by the position of the center of mass and quantities called axial and centrifugal moments of inertia (see Moment of inertia); the totality of these values ​​determines the so-called. tensor of inertia. The inertness of a liquid or gas is characterized by their density y.

M. is based on Newton's laws. The first two are true in relation to the so-called. inertial frame of reference (See Inertial frame of reference). The second law gives the basic equations for solving problems of the dynamics of a point, and together with the third - for solving problems of the dynamics of a system of material points. In the M. of a continuous medium, in addition to Newton's laws, laws are also used that reflect the properties of a given medium and establish a connection for it between the stress tensor and the strain or strain rate tensors. Such is Hooke's law for a linearly elastic body and Newton's law for a viscous fluid (see Viscosity). For the laws governing other media, see Plasticity Theory and Rheology.

The concepts of dynamic measures of motion, which are the Momentum, Moment of Moment (or kinetic moment), and kinetic energy, and the measures of force action, which are the Impulse of Force and Work, are of great importance for solving M.'s problems. The relationship between the measures of motion and the measures of action of the force is given by theorems about the change in the momentum, angular momentum and kinetic energy, which are called general theorems of dynamics. These theorems and the laws of conservation of momentum, angular momentum and mechanical energy resulting from them express the properties of motion of any system of material points and continuous medium.

Effective methods for studying the equilibrium and motion of a non-free system of material points, that is, a system whose motion is subject to predetermined constraints, called mechanical constraints (see Mechanical constraints), give the variational principles of mechanics, in particular, the principle of possible displacements, the principle of least action, and others, as well as the D "Alamber principle. In solving the problems of M., the differential equations of motion of a material point, a rigid body, and a system of material points, arising from its laws or principles, are widely used, in particular, the Lagrange equations, canonical equations, the Hamilton-Jacobi equation, and others. ., and in the M. of a continuous medium - the corresponding equations of equilibrium or motion of this medium, the equation of continuity (continuity) of the medium, and the equation of energy.

Historical sketch. M. is one of the most ancient sciences. Its emergence and development are inextricably linked with the development of the productive forces of society, the needs of practice. Earlier than other divisions of M., under the influence of requests, mainly of construction equipment, statics began to develop. It can be assumed that elementary information about statics (properties of the simplest machines) was known for several thousand years BC. e., which is indirectly evidenced by the remains of ancient Babylonian and Egyptian buildings; but no direct evidence of this has survived. The first surviving treatises on M., which appeared in Ancient Greece, include the natural philosophical works of Aristotle (see Aristotle) ​​(4th century BC), who introduced the term "M." into science. From these works it follows that at that time the laws of addition and balancing of forces applied at one point and acting along the same straight line, the properties of the simplest machines and the law of equilibrium of the lever were known. The scientific foundations of statics were developed by Archimedes (3rd century BC).

His works contain a rigorous theory of the lever, the concept of a static moment, the rule of addition of parallel forces, the theory of the balance of suspended bodies and the center of gravity, the beginning of hydrostatics. A further significant contribution to research on statics, which led to the establishment of the rule of the parallelogram of forces and the development of the concept of the moment of force, was made by I. Nemorarium (about the 13th century), Leonardo da Vinci (15th century), the Dutch scientist Stevin (16th century), and especially the French scientist P. Varignon (17th century), who completed these studies by constructing statics on the basis of the rules of addition and decomposition of forces and the theorem he proved about the moment of the resultant. The last stage in the development of geometric statics was the development by the French scientist L. Poinsot of the theory of pairs of forces and the construction of statics on its basis (1804). Dr. the direction in statics, based on the principle of possible displacements, developed in close connection with the theory of motion.

The problem of studying movement also arose in ancient times. Solutions to the simplest kinematic problems of the addition of movements are already contained in the writings of Aristotle and in the astronomical theories of the ancient Greeks, especially in the theory of epicycles, completed by Ptolemy (see Ptolemy) (2nd century AD). However, the dynamic doctrine of Aristotle, which prevailed almost until the 17th century, proceeded from the erroneous ideas that a moving body is always under the influence of some force (for an abandoned body, for example, this is the pushing force of air striving to take the place vacated by the body; the possibility of the existence of a vacuum it was denied) that the speed of the falling body is proportional to its weight, etc.

The 17th century was the period for the creation of the scientific foundations of dynamics, and with it the whole of M.. Already in the 15-16 centuries. in the countries of Western and Central Europe, bourgeois relations began to develop, which led to a significant development of crafts, merchant shipping and military affairs (improvement of firearms). This posed a number of important problems for science: the study of the flight of shells, the impact of bodies, the strength of large ships, oscillations of the pendulum (in connection with the creation of clocks), etc. ... The first important step in this direction was made by N. Copernicus (16th century), whose teachings had a tremendous influence on the development of all natural science and gave M. the concept of the relativity of motion and the necessity of choosing a frame of reference in his study. The next step was the discovery by I. Kepler empirically of the kinematic laws of planetary motion (early 17th century). G. Galileo, who laid the scientific foundations of modern M. Galileo established two basic principles of M. - the principle of relativity of classical M. and the law of inertia, which he, however, expressed only for the case of motion along a horizontal plane, but applied in his studies in complete generality. He was the first to find that in a vacuum the trajectory of a body thrown at an angle to the horizon is a parabola, applying the idea of ​​adding up movements: horizontal (by inertia) and vertical (accelerated). Having discovered the isochronism of small oscillations of a pendulum, he laid the foundation for the theory of oscillations. Investigating the equilibrium conditions for simple machines and solving some problems of hydrostatics, Galileo uses the so-called the golden rule of statics is the initial form of the principle of possible displacements. He was the first to investigate the strength of beams, which laid the foundation for the science of the strength of materials. An important merit of Galileo is the systematic introduction of a scientific experiment into medicine.

The merit of the final formulation of the fundamental laws of M. belongs to I. Newton in (1687). Having completed the research of his predecessors, Newton generalized the concept of force and introduced the concept of mass into M.. The basic (second) law of M., formulated by him, allowed Newton to successfully solve a large number of problems related mainly to celestial M., which was based on the law of universal gravitation discovered by him. He also formulates the third of the fundamental laws of M. - the law of equality of action and reaction, which is the basis of M. a system of material points. Newton's research completes the creation of the foundations of classical mismatch. The establishment of two initial positions of masses of a continuous medium belongs to the same period. Newton, who studied the resistance of a liquid by bodies moving in it, discovered the basic law of internal friction in liquids and gases, and the English scientist R. Hooke experimentally established a law expressing the relationship between stresses and deformations in an elastic body.

In the 18th century. General analytical methods for solving the problems of the theory of a material point, a system of points, and a rigid body, as well as celestial math, were intensively developed, based on the use of the infinitesimal calculus discovered by Newton and G.V. Leibniz. The main merit in the application of this calculus to the solution of M. problems belongs to L. Euler. He developed analytical methods for solving problems of the dynamics of a material point, developed the theory of moments of inertia, and laid the foundations of solid state physics. He also made the first studies on the theory of the ship, the theory of stability of elastic rods, the theory of turbines and the solution of a number of applied problems of kinematics. A contribution to the development of applied mathematics was the establishment by the French scientists G. Amonton and C. Coulomb of the experimental laws of friction.

An important stage in the development of mechanics was the creation of the dynamics of non-free mechanical systems. The starting point for solving this problem was the principle of possible displacements, which expresses the general condition of equilibrium of a mechanical system, the development and generalization of which in the 18th century. were devoted to the research of I. Bernoulli, L. Carnot, J. Fourier, J. L. Lagrange and others, and the principle expressed in the most general form by J. D'Alembert (see D "Alambert) and bearing his name. Using these two principles, Lagrange completed the development of analytical methods for solving problems of the dynamics of a free and non-free mechanical system and obtained the equations of motion of the system in generalized coordinates named after him. He also developed the foundations of the modern theory of oscillations. principle of least action in its form, which for one point was expressed by P. Maupertuis and developed by Euler, and generalized by Lagrange for the case of a mechanical system.

The application of analytical methods to the magnetic field of a continuous medium led to the development of the theoretical foundations of the hydrodynamics of an ideal fluid. The fundamental works here were the works of Euler, as well as D. Bernoulli, Lagrange, D'Alembert. The law of conservation of matter, discovered by MV Lomonosov, was of great importance for the magnetic field of a continuous medium.

In the 19th century. Intensive development of all branches of mechanics continued in the dynamics of a rigid body, the classical results of Euler and Lagrange, and then of S.V. Kovalevskaya, continued by other researchers, served as the basis for the theory of the gyroscope, which acquired especially great practical importance in the 20th century. The fundamental works of M.V. Ostrogradskii (See Ostrogradskii), W. Hamilton, K. Jacobi, G. Hertz, and others were devoted to the further development of the principles of M.

In solving the fundamental problem of M. and of all natural science - about the stability of equilibrium and motion, a number of important results were obtained by Lagrange, Eng. the scientist E. Rouse and N. Ye. Zhukovsky. A rigorous formulation of the problem of stability of motion and the development of the most general methods for its solution are due to A. M. Lyapunov. In connection with the demands of machine technology, research continued on the theory of oscillations and the problem of regulating the course of machines. The foundations of the modern theory of automatic control were developed by I.A.Vyshnegradskii (See Vyshnegradskii).

In parallel with the dynamics in the 19th century. kinematics also developed, acquiring more and more independent significance. Franz. the scientist G. Coriolis proved the theorem on the components of acceleration, which was the basis of the theory of relative motion. Instead of the terms “accelerating forces”, etc., a purely kinematic term “acceleration” appeared (J. Poncelet, A. Rezal). Poinsot gave a number of visual geometric interpretations of the motion of a rigid body. The importance of applied research in the kinematics of mechanisms increased, to which P. L. Chebyshev made an important contribution. In the second half of the 19th century. kinematics became an independent section of M.

Significant development in the 19th century. received also M. of a continuous medium. The works of L. Navier and O. Cauchy established the general equations of the theory of elasticity. Further fundamental results in this area were obtained by J. Green, S. Poisson, A. Saint-Venant, M. V. Ostrogradsky, G. Lame, W. Thomson, G. Kirchhoff, and others. Research by Navier and J. Stokes led to the establishment of differential equations of motion of a viscous fluid. A significant contribution to the further development of the dynamics of an ideal and viscous fluid was made by Helmholtz (the theory of vortices), Kirchhoff and Zhukovsky (separated flow around bodies), O. Reynolds (the beginning of the study of turbulent flows), L. Prandtl (the theory of the boundary layer), and others N. P. Petrov created the hydrodynamic theory of friction in lubrication, further developed by Reynolds, Zhukovsky, together with SA Chaplygin and others. Saint-Venant proposed the first mathematical theory of plastic flow of a metal.

In the 20th century. the development of a number of new sections of M. begins. Problems put forward by electrical and radio engineering, problems of automatic control, etc., caused the emergence of a new field of science - the theory of nonlinear oscillations, the foundations of which were laid by the works of Lyapunov and A. Poincaré. Another section of physics, on which the theory of jet propulsion is based, was the dynamics of bodies of variable mass; its foundations were laid back at the end of the 19th century. the works of I.V. Meshchersky (See. Meshchersky). The original research on the theory of the movement of missiles belongs to K.E. Tsiolkovsky (see Tsiolkovsky).

Two important new sections appear in continuous medium modeling: aerodynamics, the foundations of which, like all aviation science, were created by Zhukovsky, and gas dynamics, the foundations of which were laid by Chaplygin. The works of Zhukovsky and Chaplygin were of great importance for the development of all modern hydro-aerodynamics.

Modern problems of mechanics. Among the important problems of modern physics are the already noted problems of the theory of oscillations (especially nonlinear ones), the dynamics of a rigid body, the theory of stability of motion, and also the theory of bodies of variable mass and the dynamics of space flights. Problems in which, instead of "deterministic", that is, known in advance, quantities (for example, the acting forces or the laws of motion of individual objects), one has to consider "probabilistic" quantities, that is, quantities for which only the probability that they can have certain values ​​is known. In continuous medium modeling, the problem of studying the behavior of macroparticles upon changing their shape is highly topical, which is associated with the development of a more rigorous theory of turbulent fluid flows, the solution of problems of plasticity and creep, and the creation of a well-founded theory of the strength and fracture of solids.

A wide range of M.'s problems is also associated with the study of the motion of a plasma in a magnetic field (magnetohydrodynamics), that is, with the solution of one of the most pressing problems of modern physics — the implementation of a controlled thermonuclear reaction. In hydrodynamics, a number of the most important problems are associated with the problems of high speeds in aviation, ballistics, turbine building and engine building. Many new problems arise at the intersection of M. with other fields of science. These include the problems of hydrothermochemistry (i.e., the study of mechanical processes in liquids and gases that enter into chemical reactions), the study of forces that cause cell division, the mechanism of formation of muscle strength, etc.

Electronic computers and analogue machines are widely used in solving many of M.'s problems. At the same time, the development of methods for solving new problems of microscopy (especially micrometry of a continuous medium) using these machines is also a very urgent problem.

Research in various fields of M. is carried out at universities and in higher technical educational institutions of the country, at the Institute for Problems in Mechanics of the Academy of Sciences of the USSR, and also in many other research institutes both in the USSR and abroad.

International congresses on theoretical and applied M. The same committee, together with other scientific institutions, periodically organizes all-Union congresses and conferences devoted to research in various fields of M.

In any academic course, the study of physics begins with mechanics. Not from theoretical, not from applied and not computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, the scientist was walking in the garden, saw an apple falling, and it was this phenomenon that pushed him to the discovery of the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form that people understand, but his merit is priceless. In this article, we will not describe the laws of Newtonian mechanics in as much detail as possible, but we will outline the basics, basic knowledge, definitions and formulas that can always play into your hands.

Mechanics is a branch of physics, a science that studies the movement of material bodies and the interactions between them.

The word itself is of Greek origin and is translated as "the art of building machines." But before building machines, we are still like the Moon, so we will follow in the footsteps of our ancestors, and we will study the movement of stones thrown at an angle to the horizon, and apples falling on heads from a height of h.

Why does the study of physics begin with mechanics? Because it is completely natural, not to start it from thermodynamic equilibrium ?!

Mechanics is one of the oldest sciences, and historically the study of physics began precisely from the foundations of mechanics. Placed in the framework of time and space, people, in fact, could not start from something else, with all their desire. Moving bodies are the first thing we turn our attention to.

What is movement?

Mechanical movement is a change in the position of bodies in space relative to each other over time.

It is after this definition that we quite naturally come to the concept of a frame of reference. Changing the position of bodies in space relative to each other. Key words here: relative to each other ... After all, a passenger in a car moves relative to a person standing on the side of the road at a certain speed, and rests relative to his neighbor on the seat next to him, and moves at some other speed relative to a passenger in a car that overtakes them.

That is why, in order to normally measure the parameters of moving objects and not get confused, we need frame of reference - rigidly interconnected reference body, coordinate system and clock. For example, the earth moves around the sun in a heliocentric frame of reference. In everyday life, we carry out almost all of our measurements in a geocentric frame of reference associated with the Earth. Earth is a reference body, relative to which cars, airplanes, people, animals move.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of a body in space at any time. In other words, mechanics constructs a mathematical description of motion and finds connections between the physical quantities that characterize it.

In order to move further, we need the concept “ material point ”. They say physics is an exact science, but physicists know how many approximations and assumptions have to be made in order to agree on this very accuracy. Nobody has ever seen a material point or smelled ideal gas, but they are! It's just much easier to live with them.

Material point is a body, the size and shape of which can be neglected in the context of this problem.

Sections of classical mechanics

Mechanics consists of several sections

• Kinematics
• Dynamics
• Statics

Kinematics from a physical point of view, it studies exactly how the body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical kinematic problems

Dynamics solves the question of why it is moving that way. That is, it considers the forces acting on the body.

Statics studies the balance of bodies under the action of forces, that is, answers the question: why does it not fall at all?

The limits of applicability of classical mechanics

Classical mechanics no longer claims to be a science that explains everything (at the beginning of the last century, everything was completely different), and has a clear framework of applicability. In general, the laws of classical mechanics are valid for the world we are accustomed to in terms of size (macrocosm). They stop working in the case of the particle world, when quantum mechanics replaces the classical one. Also, classical mechanics is inapplicable to cases when bodies move at a speed close to the speed of light. In such cases, relativistic effects become pronounced. Roughly speaking, within the framework of quantum and relativistic mechanics - classical mechanics, this is a special case when the dimensions of the body are large and the speed is small.

Generally speaking, quantum and relativistic effects never go anywhere; they also take place during the ordinary motion of macroscopic bodies with a speed much less than the speed of light. Another thing is that the effect of these effects is so small that it does not go beyond the most accurate measurements. Thus, classical mechanics will never lose their fundamental importance.

We will continue to study the physical foundations of mechanics in future articles. For a better understanding of the mechanics, you can always refer to to our authors who individually shed light on the dark spot of the most difficult task.

Abstract on the topic:

HISTORY OF MECHANICS DEVELOPMENT

Completed: student of grade 10 "A"

Efremov A.V.

Checked by: O.P. Gavrilova

1. INTRODUCTION.

2. DEFINITION OF MECHANICS; ITS PLACE AMONG OTHER SCIENCES;

DEPARTMENTS OF MECHANICS.

4. HISTORY OF MECHANICS DEVELOPMENT:

The era preceding the establishment of the foundations of mechanics.

The period of creation of the foundations of mechanics.

Development of the methods of mechanics in the 18th century.

Mechanics of the 19th and early 20th centuries

Mechanics in Russia and the USSR.

6. CONCLUSION.

7. APPENDIX.

1. INTRODUCTION.

For each person there are two worlds: internal and external; the senses are the intermediaries between these two worlds. The external world has the ability to influence the senses, cause them a special kind of changes, or, as they say, excite irritation in them.

The inner world of a person is determined by the totality of those phenomena that absolutely cannot be accessible to the direct observation of another person.The irritation in the sense organ caused by the external world is transmitted to the inner world and, in turn, causes a subjective sensation in it, for the appearance of which the presence of consciousness is necessary. The subjective sensation perceived by the inner world is objectified, i.e. transported into outer space, as something belonging to a certain place and a certain time.

In other words, through such objectification, we transfer our sensations into the external world, and space and time serve as the background on which these objective sensations are located. In those places in the space where they are placed, we involuntarily assume the cause that generates them.

A person has the ability to compare perceived sensations with each other, to judge their similarity or dissimilarity, and, in the second case, to distinguish between qualitative and quantitative differences, and the quantitative dissimilarity can refer either to tension (intensity), or to length (extensiveness), or, finally, to the duration of the annoying objective reason.

Since the inferences accompanying any objectification are exclusively based on the perceived sensation, the complete sameness of these sensations will inevitably entail the identity of objective causes, and this identity, apart from, and even against our will, persists even in cases when other sense organs indisputably testify us about the diversity of reasons. Here lies one of the main sources of undoubtedly erroneous conclusions, leading to the so-called deceptions of sight, hearing, etc. Another source is the lack of skill with new sensations. reality that exists apart from our consciousness is called an external phenomenon. Changes in the color of bodies depending on lighting, the same level of water in the vessels, the swing of the pendulum are external phenomena.

One of the powerful levers moving humanity along the path of its development is curiosity, which has the last, unattainable goal - the knowledge of the essence of our being, the true relationship of our inner world to the outer world. The result of curiosity was acquaintance with a very large number of the most diverse phenomena that make up the subject of a number of sciences, among which physics occupies one of the first places, due to the vastness of the field it processes and the importance that it has for almost all other sciences.

2. DEFINITION OF MECHANICS; ITS PLACE AMONG OTHER SCIENCES; DEPARTMENTS OF MECHANICS.

Mechanics (from the Greek mhcanich - skill related to machines; the science of machines) is the science of the simplest form of movement of matter - mechanical movement, representing the change in the spatial arrangement of bodies over time, and the interactions between them associated with the movement of bodies. Mechanics explores the general laws connecting mechanical movements and interactions, accepting laws for the interactions themselves, obtained empirically and substantiated in physics. The methods of mechanics are widely used in various fields of natural science and technology.

Mechanics studies the movements of material bodies using the following abstractions:

1) A material point, like a body of negligible size, but of finite mass. The role of a material point can be played by the center of inertia of a system of material points, in which the mass of the entire system is considered to be concentrated;

2) An absolutely solid body, a set of material points located at constant distances from each other. This abstraction is applicable if the deformation of the body can be neglected;

3) Continuous medium. With this abstraction, a change in the relative position of elementary volumes is allowed. In contrast to a rigid body, to define the motion of a continuous medium, an infinite number of parameters are required. Continuous media include solid, liquid and gaseous bodies, reflected in the following abstract representations: ideally elastic body, plastic body, ideal fluid, viscous fluid, ideal gas and others. These abstract ideas about a material body reflect the actual properties of real bodies, essential in the given conditions.Accordingly, mechanics is divided into:

material point mechanics;

the mechanics of the system of material points;

mechanics of an absolutely rigid body;

continuum mechanics.

The latter, in turn, is subdivided into the theory of elasticity, hydromechanics, aeromechanics, gas mechanics and others (see Appendix). The term "theoretical mechanics" usually denotes a part of mechanics that deals with the study of the most general laws of motion, the formulation of its general provisions and theorems, as well as the application of methods mechanics to the study of the motion of a material point, a system of a finite number of material points and an absolutely rigid body.

In each of these sections, first of all, statics is highlighted, uniting issues related to the study of conditions for the balance of forces. Distinguish between statics of a rigid body and statics of a continuous medium: statics of an elastic body, hydrostatics and aerostatics (see Appendix). The motion of bodies in abstraction from the interaction between them is studied by kinematics (see Appendix). An essential feature of the kinematics of continuous media is the need to determine for each moment of time the distribution of displacements and velocities in space. The subject of dynamics is the mechanical motion of material bodies in connection with their interactions. The essential applications of mechanics are technical. The tasks posed by technology to mechanics are very diverse; these are questions of the movement of machines and mechanisms, the mechanics of vehicles on land, at sea and in the air, structural mechanics, various departments of technology, and many others. In connection with the need to satisfy the demands of technology, special technical sciences emerged from mechanics. Kinematics of mechanisms, dynamics of machines, theory of gyroscopes, external ballistics (see Appendix) are technical sciences using absolutely rigid body methods. Resistance of materials and hydraulics (see Appendix), which have common foundations with the theory of elasticity and hydrodynamics, develop calculation methods for practice, corrected by experimental data. All sections of mechanics have developed and continue to develop in close connection with the demands of practice; in the course of solving the problems of technology, mechanics as a branch of physics has developed in close connection with its other sections - with optics, thermodynamics and others. The foundations of the so-called classical mechanics were generalized at the beginning of the 20th century. in connection with the discovery of physical fields and the laws of motion of microparticles. The content of the mechanics of fast-moving particles and systems (with speeds of the order of the speed of light) are set forth in the theory of relativity, and the mechanics of micromotions - in quantum mechanics.

3. BASIC CONCEPTS AND METHODS OF MECHANICS.

The laws of classical mechanics are valid in relation to the so-called inertial, or Galilean, frames of reference (see Appendix). Within the limits within which Newtonian mechanics is valid, time can be considered independently of space. The time intervals are practically the same in all reporting systems, whatever their mutual motion, if their relative speed is small in comparison with the speed of light.

The main kinematic measures of motion are speed, which has a vector character, since it determines not only the rate of change of the path with time, but also the direction of motion, and acceleration is a vector, which is a measure of measuring the velocity vector in time. The vectors of angular velocity and angular acceleration serve as measures of the rotational motion of a rigid body. In the statics of an elastic body, the displacement vector and the corresponding deformation tensor, including the concepts of relative elongations and shears, are of primary importance. The main measure of the interaction of bodies, which characterizes the change in time of the mechanical movement of the body, is force. The aggregates of the magnitude (intensity) of the force, expressed in certain units, the direction of the force (line of action) and the point of application, determine quite unambiguously the force as a vector.

Mechanics is based on the following Newton's laws. The first law, or the law of inertia, characterizes the movement of bodies in conditions of isolation from other bodies, or when external influences are balanced. This law says: every body maintains a state of rest or uniform and rectilinear motion until the applied forces force it to change this state. The first law can serve to determine inertial reference frames.

The second law, which establishes a quantitative relationship between a force applied to a point and a change in the momentum caused by this force, says: the change in motion occurs in proportion to the applied force and occurs in the direction of the line of action of this force. According to this law, the acceleration of a material point is proportional to the force applied to it: a given force F causes the smaller the acceleration of the body, the greater its inertia. Mass is the measure of inertia. According to Newton's second law, the force is proportional to the product of the mass of a material point by its acceleration; with an appropriate choice of the unit of force, the latter can be expressed by the product of the mass of a point m by the acceleration a:

This vector equality represents the basic equation of the dynamics of a material point.

Newton's third law says: an action always corresponds to an equal and oppositely directed reaction, that is, the action of two bodies on each other is always equal and directed along one straight line in opposite directions. While the first two Newton's laws refer to one material point, the third law is fundamental for a system of points. Along with these three basic laws of dynamics, there is a law of independence of the action of forces, which is formulated as follows: if several forces act on a material point, then the acceleration of the point is made up of those accelerations that the point would have under the action of each force separately. The law of independence of the action of forces leads to the rule of the parallelogram of forces.

In addition to the previously named concepts, other measures of motion and action are used in mechanics.

The most important are the measures of motion: vector - momentum p = mv, equal to the product of mass by the velocity vector, and scalar - kinetic energy E k = 1/2 mv 2, equal to half of the product of mass and square of velocity. In the case of rotational motion of a rigid body, its inertial properties are set by the tensor of inertia, which determines the moments of inertia and centrifugal moments about three axes passing through this point at each point of the body. The measure of the rotational motion of a rigid body is the vector of the angular momentum, which is equal to the product of the moment of inertia and angular velocity. The measures of action of forces are: vector - elementary impulse of force F dt (product of force by element of time of its action), and scalar - elementary work F * dr (scalar product of vectors of force and elementary displacement of a point of position); in rotary motion, the measure of the impact is the moment of force.

The main measures of motion in the dynamics of a continuous medium are continuously distributed quantities and, accordingly, are specified by their distribution functions. Thus, density determines the distribution of mass; forces are given by their surface or volumetric distribution. The motion of a continuous medium, caused by external forces applied to it, leads to the emergence of a stress state in the medium, characterized at each point by a set of normal and tangential stresses, represented by a single physical quantity - the stress tensor. The arithmetic mean of the three normal stresses at a given point, taken with the opposite sign, determines the pressure (see Appendix).

The study of the equilibrium and motion of a continuous medium is based on the laws of the relationship between the stress tensor and the tensor of deformation or strain rates. Such are Hooke's law in the statics of a linear elastic body and Newton's law in the dynamics of a viscous fluid (see Appendix). These laws are the simplest; other relationships have been established that more accurately characterize the phenomena occurring in real bodies. There are theories that take into account the previous history of body movement and stress, theories of creep, relaxation, and others (see Appendix).

The relationships between the measures of motion of a material point or a system of material points and the measures of action of forces are contained in the general theorems of dynamics: the quantities of motion, angular momentum and kinetic energy. These theorems express the properties of motions of both a discrete system of material points and a continuous medium. When considering the equilibrium and motion of a non-free system of material points, that is, a system subject to predetermined constraints - mechanical connections (see Appendix), it is important to apply the general principles of mechanics - the principle of possible displacements and the D'Alembert principle. As applied to a system of material points, the principle of possible displacements is as follows: for the equilibrium of a system of material points with stationary and ideal connections, it is necessary and sufficient that the sum of elementary work of all active forces acting on the system with any possible displacement of the system is equal to zero (for non-liberating connections) or it was equal to zero or less than zero (for releasing links). D'Alembert's principle for a free material point says: at each moment of time, the forces applied to the point can be balanced by adding to them the force of inertia.

When formulating problems, mechanics proceeds from the basic equations expressing the found laws of nature. To solve these equations, mathematical methods are used, and many of them originated and received their development precisely in connection with the problems of mechanics. When setting a problem, it was always necessary to focus on those aspects of the phenomenon that seem to be the main ones. In cases where it is necessary to take into account side factors, as well as in those cases when the phenomenon, in its complexity, does not lend itself to mathematical analysis, experimental research is widely used.

Experimental methods of mechanics are based on the developed technique of physical experiment. To record movements, both optical methods and methods of electrical registration are used, based on the preliminary transformation of mechanical movement into an electrical signal.

To measure forces, various dynamometers and scales are used, supplied with automatic devices and tracking systems. For measuring mechanical vibrations, a variety of radio engineering schemes are widely used. The experiment in continuum mechanics has achieved particular success. To measure the voltage, an optical method is used (see Appendix), which consists in observing a loaded transparent model in polarized light.

In recent years, strain gauging with the help of mechanical and optical strain gauges (see Appendix), as well as resistance strain gauges, has been greatly developed for measuring deformation.

Thermoelectric, capacitive, induction and other methods are successfully used to measure velocities and pressures in moving liquids and gases.

4. HISTORY OF MECHANICS DEVELOPMENT.

The history of mechanics, like that of other natural sciences, is inextricably linked with the history of the development of society, with the general history of the development of its productive forces. The history of mechanics can be divided into several periods, differing both in the nature of the problems and in the methods of solving them.

The era preceding the establishment of the foundations of mechanics. The era of the creation of the first tools of production and artificial structures should be recognized as the beginning of the accumulation of that experience, which later served as the basis for the discovery of the basic laws of mechanics. While the geometry and astronomy of the ancient world were already quite developed scientific systems, in the field of mechanics, only a few provisions were known related to the simplest cases of equilibrium of bodies.

Statics was born earlier than all branches of mechanics. This section developed in close connection with the construction art of the ancient world.

The basic concept of statics - the concept of force - was initially closely associated with muscular effort caused by the pressure of an object on the arm. By about the beginning of the IV century. BC NS. the simplest laws of addition and balancing of forces applied to one point along the same straight line were already known. The problem of the lever attracted particular interest. The theory of leverage was created by the great scientist of antiquity Archimedes (III century BC) and is set forth in the work "On Levers". He established the rules for the addition and decomposition of parallel forces, gave a definition of the concept of the center of gravity of a system of two weights suspended from a rod, and clarified the equilibrium conditions for such a system. Archimedes also discovered the basic laws of hydrostatics.

He applied his theoretical knowledge in the field of mechanics to various practical issues of construction and military technology. The concept of the moment of force, which plays a major role in all modern mechanics, is already in a latent form in Archimedes' law. The great Italian scientist Leonardo da Vinci (1452 - 1519) introduced the concept of the shoulder of power under the guise of “potential leverage”.

The Italian mechanic Guido Ubaldi (1545 - 1607) applies the concept of moment in his block theory, where the concept of a chain hoist was introduced. Polyspast (Greek poluspaston, from polu - a lot and spaw - pull) - a system of movable and stationary blocks, bent around by a rope, are used to gain a gain in strength and, less often, to gain a gain in speed. Usually, it is customary to refer to statics as the doctrine of the center of gravity of a material body.

The development of this purely geometric doctrine (geometry of masses) is closely related to the name of Archimedes, who, using the famous method of exhaustion, indicated the position of the center of gravity of many regular geometric shapes, flat and spatial.

General theorems on the centers of gravity of bodies of revolution were given by the Greek mathematician Papp (3rd century AD) and the Swiss mathematician P. Gulden in the 17th century. Statics owes the development of its geometric methods to the French mathematician P. Varignon (1687); These methods were most fully developed by the French mechanic L. Poinsot, whose treatise "Elements of Statics" was published in 1804. Analytical statics, based on the principle of possible displacements, was created by the famous French scientist J. Lagrange. With the development of crafts, trade, navigation and military affairs and the associated accumulation of new knowledge, in the XIV and XV centuries. - in the Renaissance - the heyday of the arts and sciences begins. A major event that revolutionized the human worldview was the creation by the great Polish astronomer Nicolaus Copernicus (1473-1543) of the doctrine of the heliocentric system of the world, in which the spherical Earth occupies a central stationary position, and celestial bodies move around it in their circular orbits: the Moon, Mercury, Venus , Sun, Mars, Jupiter, Saturn.

The kinematic and dynamic studies of the Renaissance were focused mainly on clarifying the concepts of uneven and curvilinear motion of a point. Until that time, the generally accepted dynamic views of Aristotle, stated in his "Problems of Mechanics", were not consistent with reality.

So, he believed that in order to maintain a uniform and rectilinear movement of the body, a constantly acting force must be applied to it. This statement seemed to him to agree with everyday experience. Of course, Aristotle did not know anything about the fact that frictional force arises in this case. He also believed that the speed of free fall of bodies depends on their weight: "If half weight passes this much in some time, then the double weight passes the same amount in half time." Considering that everything consists of four elements - earth, water, air and fire, he writes: “Everything that is capable of rushing to the middle or center of the world is heavy; easily everything that rushes from the middle or center of the world ”. From this he concluded: since heavy bodies fall to the center of the Earth, this center is the focus of the world, and the Earth is motionless. Not yet possessing the concept of acceleration, which was later introduced by Galileo, the researchers of this era considered accelerated motion as consisting of separate uniform motions, each with their own speed in each interval. Galileo, at the age of 18, observing during the divine service the small damped oscillations of the chandelier and counting the time by the beats of the pulse, found that the period of the pendulum's oscillation does not depend on its span.

Having doubted the correctness of Aristotle's statements, Galileo began to carry out experiments, with the help of which, without analyzing the reasons, he established the laws of motion of bodies near the earth's surface. Dropping bodies from the tower, he found that the time of the body's fall does not depend on its weight and is determined by the height of the fall. He was the first to prove that in a free fall of a body, the distance traveled is proportional to the square of time.

Remarkable experimental studies of the free vertical fall of a heavy body were carried out by Leonardo da Vinci; these were probably the first specially organized experimental studies in the history of mechanics. The period of creation of the foundations of mechanics. Practice (mainly merchant shipping and military affairs)

puts before the mechanics of the XVI - XVII centuries. a number of important problems that occupied the minds of the best scientists of that time. “… Along with the emergence of cities, large buildings and the development of handicrafts, mechanics also developed. Soon it also becomes necessary for shipping and military affairs ”(F. Engels, Dialectics of Nature, 1952, p. 145). It was necessary to accurately investigate the flight of shells, the strength of large ships, the oscillations of the pendulum, the impact of the body. Finally, the victory of Copernicus' teachings raises the problem of the motion of celestial bodies. The heliocentric worldview by the beginning of the 16th century. created the preconditions for the establishment of the laws of planetary motion by the German astronomer I. Kepler (1571 - 1630).

He formulated the first two laws of planetary motion:

1. All planets move along ellipses, in one of the focuses of which is the Sun.

2. The radius vector drawn from the Sun to the planet describes equal areas in equal time intervals.

The founder of mechanics is the great Italian scientist G. Galilei (1564 - 1642). He experimentally established the quantitative law of falling bodies in a void, according to which the distances traversed by a falling body at equal intervals of time relate to each other as consecutive odd numbers.

Galileo established the laws of motion of heavy bodies on an inclined plane, showing that, whether heavy bodies fall vertically or along an inclined plane, they always acquire such speeds that need to be communicated to them in order to raise them to the height from which they fell. Passing to the limit, he showed that on the horizontal plane a heavy body will be at rest or will move uniformly and rectilinearly. Thus, he formulated the law of inertia. By adding the horizontal and vertical movements of the body (this is the first addition of finite independent movements in the history of mechanics), he proved that a body thrown at an angle to the horizon describes a parabola, and showed how to calculate the length of flight and the maximum height of the trajectory. For all his conclusions, he always emphasized that we are talking about movement in the absence of resistance. In dialogues about the two systems of the world, very figuratively, in the form of an artistic description, he showed that all the movements that can occur in the ship's cabin do not depend on whether the ship is at rest or moves in a straight line and evenly.

By this he established the principle of relativity of classical mechanics (the so-called Galileo-Newton principle of relativity). In the particular case of the force of weight, Galileo closely linked the constancy of weight with the constancy of the acceleration of falling, but only Newton, introducing the concept of mass, gave an exact formulation of the relationship between force and acceleration (second law). Exploring the conditions of equilibrium of simple machines and floating bodies, Galileo, in essence, applies the principle of possible displacements (albeit in a rudimentary form). To him, science owes the first study of the strength of beams and the resistance of a fluid to bodies moving in it.

The French geometer and philosopher R. Descartes (1596 - 1650) expressed the fruitful idea of ​​conservation of momentum. He applies mathematics to the analysis of motion and, by introducing variable quantities into it, establishes a correspondence between geometric images and algebraic equations.

But he did not notice the essential fact that the momentum is a directional quantity, and he added the momentum arithmetically. This led him to erroneous conclusions and reduced the significance of his applications of the law of conservation of momentum, in particular, to the theory of impact of bodies.

Galileo's follower in the field of mechanics was the Dutch scientist H. Huygens (1629 - 1695). He is responsible for the further development of the concepts of acceleration in the curvilinear motion of a point (centripetal acceleration). Huygens also solved a number of the most important problems of dynamics - the motion of a body in a circle, oscillations of a physical pendulum, the laws of elastic impact. He was the first to formulate the concepts of centripetal and centrifugal force, moment of inertia, center of oscillation of a physical pendulum. But his main merit is that he was the first to apply a principle that is essentially equivalent to the principle of living forces (the center of gravity of a physical pendulum can only rise to a height equal to the depth of its fall). Using this principle, Huygens solved the problem of the center of oscillation of a pendulum - the first problem of the dynamics of a system of material points. Based on the idea of ​​conservation of momentum, he created a complete theory of the impact of elastic balls.

The merit of formulating the basic laws of dynamics belongs to the great English scientist I. Newton (1643 - 1727). In his treatise "Mathematical Principles of Natural Philosophy", published in the first edition in 1687, Newton summed up the achievements of his predecessors and indicated the ways for the further development of mechanics for centuries to come. Completing the views of Galileo and Huygens, Newton enriches the concept of force, indicates new types of forces (for example, gravitational forces, resistance forces of the medium, viscous forces and many others), studies the laws of the dependence of these forces on the position and motion of bodies. The basic equation of dynamics, which is the expression of the second law, allowed Newton to successfully solve a large number of problems related mainly to celestial mechanics. In it, he was most interested in the reasons for moving in elliptical orbits. Back in his student years, Newton pondered over the issues of gravitation. In his papers they found the following entry: “From Kepler's rule that the periods of the planets are in one and a half proportion to the distance from the centers of their orbits, I deduced that the forces holding the planets in their orbits should be in the inverse ratio of the squares of their distances from the centers around which they revolve. From here I compared the force required to keep the Moon in its orbit with the force of gravity on the surface of the Earth and found that they almost correspond to each other. "

In the above passage, Newton does not provide a proof, but I can assume that his reasoning was as follows. If we roughly assume that the planets move uniformly in circular orbits, then according to Kepler's third law, which Newton refers to, I get:

T 2 2 / T 2 1 = R 3 2 / R 3 1, (1.1) where T j and R j are the periods of revolution and radii of the orbits of two planets (j = 1, 2) With uniform motion of planets in circular orbits with velocities V j their periods of circulation are determined by the equalities T j = 2 p R j / V j

Therefore, T 2 / T 1 = 2 p R 2 V 1 / V 2 2 p R 1 = V 1 R 2 / V 2 R 1

Now relation (1.1) is reduced to the form V 2 1 / V 2 2 = R 2 / R 1. (1.2)

By the years under consideration, Huygens had already established that the centrifugal force is proportional to the square of the velocity and inversely proportional to the radius of the circle, that is, F j = kV 2 j / R j, where k is the proportionality coefficient.

If we now introduce into equality (1.2) the ratio V 2 j = F j R j / k, then I will get F 1 / F 2 = R 2 2 / R 2 1, (1.3) which establishes the inverse proportionality of the centrifugal forces of the planets to the squares of their distances before the Sun, Newton also carried out studies of the resistance of fluids to moving bodies; he established the law of resistance, according to which the resistance of a fluid to the movement of a body in it is proportional to the square of the body's velocity. Newton discovered the basic law of internal friction in liquids and gases.

By the end of the 17th century. the fundamentals of mechanics have been elaborated. If the ancient centuries are considered the prehistory of mechanics, then the 17th century. can be considered as the period of creation of its foundations. Development of methods of mechanics in the XVIII century. In the XVIII century. production needs - the need to study the most important mechanisms, on the one hand, and the problem of the motion of the Earth and the Moon, put forward by the development of celestial mechanics, on the other, - led to the creation of general methods for solving problems of the mechanics of a material point, a system of points of a rigid body, developed in "Analytical Mechanics" (1788) J. Lagrange (1736 - 1813).

In the development of the dynamics of the post-Newtonian period, the main merit belongs to the St. Petersburg academician L. Euler (1707 - 1783). He developed the dynamics of a material point in the direction of applying the methods of analysis of the infinitesimal to the solution of the equations of motion of a point. Euler's treatise “Mechanics, that is, the science of motion, expounded by the analytical method”, published in St. Petersburg in 1736, contains general uniform methods for the analytical solution of problems of the dynamics of a point.

L. Euler - the founder of rigid body mechanics.

He owns the generally accepted method for the kinematic description of the motion of a rigid body using three Euler angles. A fundamental role in the further development of dynamics and many of its technical applications was played by the basic differential equations of the rotational motion of a rigid body around a fixed center established by Euler. Euler established two integrals: the integral of the angular momentum

A 2 w 2 x + B 2 w 2 y + C 2 w 2 z = m

and integral of living forces (integral of energy)

A w 2 x + B w 2 y + C w 2 z = h,

where m and h are arbitrary constants, A, B and C are the main moments of inertia of the body for a fixed point, and w x, w y, w z are the projections of the angular velocity of the body onto the main axes of inertia of the body.

These equations were an analytical expression of the theorem of the angular momentum discovered by him, which is a necessary addition to the law of momentum, formulated in general form in Newton's "Principles". Euler's “Mechanics” gives a close to modern formulation of the law of “living forces” for the case of rectilinear motion and notes the presence of such motions of a material point in which the change in living force when a point moves from one position to another does not depend on the shape of the trajectory. This laid the foundation for the concept of potential energy. Euler is the founder of fluid mechanics. They were given the basic equations of the dynamics of an ideal fluid; he is credited with creating the foundations of the theory of a ship and the theory of stability of elastic rods; Euler laid the foundation for the theory of calculating turbines by deriving the turbine equation; in applied mechanics, Euler's name is associated with the kinematics of figured wheels, the calculation of friction between a rope and a pulley, and many others.

Celestial mechanics was largely developed by the French scientist P. Laplace (1749 - 1827), who in his extensive work "Treatise on Celestial Mechanics" combined the results of the study of his predecessors - from Newton to Lagrange - by his own research on the stability of the solar system, by solving the three-body problem , the motion of the moon and many other questions of celestial mechanics (see Appendix).

One of the most important applications of Newton's theory of gravitation was the question of the figures of equilibrium of rotating liquid masses, the particles of which gravitate towards each other, in particular, the figure of the Earth. The foundations of the theory of equilibrium of rotating masses were set forth by Newton in the third book of the "Elements".

The problem of the figures of equilibrium and stability of a rotating liquid mass played a significant role in the development of mechanics.

The great Russian scientist MV Lomonosov (1711 - 1765) highly appreciated the importance of mechanics for natural science, physics and philosophy. He owns the materialistic interpretation of the processes of interaction between two bodies: "when one body accelerates the movement of the other and imparts to it a part of its motion, then only in such a way that it itself loses the same part of the motion." He is one of the founders of the kinetic theory of heat and gases, the author of the law of conservation of energy and motion. Let's quote Lomonosov's words from a letter to Euler (1748): “All changes that occur in nature take place in such a way that if something is added to something, the same amount will be subtracted from something else. So, how much matter joins to some body, the same amount will be taken away from another; how many hours I spend in sleep, as much I take away from vigil, etc. Since this law of nature is universal, it even extends to the rules of motion, and a body that impels another to move with its impetus loses its motion as much as it communicates to another, moved by him. "

Lomonosov was the first to predict the existence of absolute zero temperature, and expressed the idea of ​​a connection between electrical and light phenomena. As a result of the activities of Lomonosov and Euler, the first works of Russian scientists appeared, who creatively mastered the methods of mechanics and contributed to its further development.

The history of the creation of the dynamics of a non-free system is associated with the development of the principle of possible displacements, which expresses the general conditions for the equilibrium of the system. This principle was first applied by the Dutch scientist S. Stevin (1548 - 1620) when considering the equilibrium of a block. Galileo formulated the principle in the form of the “golden rule” of mechanics, according to which “what is gained in strength is lost in speed”. The modern formulation of the principle was given at the end of the 18th century. on the basis of the abstraction of “ideal connections”, reflecting the idea of ​​an “ideal” machine, devoid of internal losses for harmful resistances in the transmission mechanism. It looks as follows: if in the position of isolated equilibrium of a conservative system with stationary bonds the potential energy has a minimum, then this equilibrium position is stable.

The creation of the principles of the dynamics of a non-free system was facilitated by the problem of the motion of a non-free material point. A material point is called non-free if it cannot occupy an arbitrary position in space.

In this case, D'Alembert's principle sounds as follows: the active forces and reactions of bonds acting on a moving material point can be balanced at any time by adding to them the force of inertia.

An outstanding contribution to the development of the analytical dynamics of a non-free system was made by Lagrange, who in his fundamental two-volume work “Analytical Mechanics” indicated the analytical expression of D'Alembert's principle - “the general formula of dynamics”. How did Lagrange get it?

After Lagrange outlined the various principles of statics, he proceeds to establish "a general statics formula for the equilibrium of any system of forces." Starting with two forces, Lagrange establishes by induction the following general formula for the equilibrium of any system of forces:

P dp + Q dq + R dr +… = 0. (2.1)

This equation represents a mathematical notation of the principle of possible displacements. In modern notation, this principle has the form

е n j = 1 F j d r j = 0 (2.2)

Equations (2.1) and (2.2) are practically the same. The main difference is, of course, not in the form of notation, but in the definition of variation: today it is an arbitrarily conceivable movement of the point of application of the force, compatible with constraints, and in Lagrange it is a small movement along the line of action of the force and in the direction of its action Lagrange introduces into consideration the function P (now it is called the potential energy), defining it by equality.

d П = P dp + Q dq + R dr + ..., (2.3) in Cartesian coordinates, the function П (after integration) has the form

P = A + Bx + Cy + Dz + ... + Fx 2 + Gxy + Hy 2 + Kxz + Lyz +

Mz 2 +… (2.4)

To further prove his point, Lagrange invents the famous indefinite multiplier method. Its essence is as follows. Consider the equilibrium of n material points, each of which is acted upon by the force F j. There are m links j r = 0 between the coordinates of the points, depending only on their coordinates. Taking into account that d j r = 0, equation (2.2) can immediately be reduced to the following modern form:

å n j = 1 F j d r j + å m r = 1 l r d j r = 0, (2.5) where l r are undefined factors. Hence, the following equilibrium equations are obtained, called the Lagrange equations of the first kind:

X j + å m r = 1 l r j r / x j = 0, Y j + å m r = 1 l r j r / y j = 0,

Z j + å m r = 1 l r j r / z j = 0 (2.6) To these equations it is necessary to add m equations of constraints j r = 0 (X j, Y j, Z j are the projections of the force F j)

Let us show how Lagrange uses this method to derive the equilibrium equations for an absolutely flexible and inextensible thread. First of all, referred to the unit of length of the thread (its dimension is equal to F / L).

The constraint equation for an inextensible thread has the form ds = const, and, therefore, d ds = 0. In equation (2.5), the sums transform into integrals over the length of the thread l ò l 0 F d rds + ò l 0 ld ds = 0. (2.7 ) Taking into account the equality (ds) 2 = (dx) 2 + (dy) 2 + (dz) 2, we find

d ds = dx / ds d dx + dy / ds d dy + dz / ds d dz.

ò l 0 l d ds = ò l 0 (l dx / ds d dx + l dy / ds d dy + l dz / ds d dz)

or, rearranging the operations d and d and integrating by parts,

ò l 0 l d ds = (l dx / ds d x + l dy / ds d y + l dz / ds d z) -

- ò l 0 d (l dx / ds) d x + d (l dy / ds) d y + d (l dz / ds) d z.

Assuming that the thread is fixed at the ends, we get d x = d y = d z = 0 for s = 0 and s = l, and, therefore, the first term vanishes. We introduce the rest into equation (2.7), open the scalar product F * dr and group the terms:

ò l 0 [Xds - d (l dx / ds)] d x + [Yds - d (l dy / ds)] d y + [Zds

- d (d dz / ds)] d z = 0

Since the variations d x, d y and d z are arbitrary and independent, all square brackets must equal zero, which gives three equations of equilibrium of an absolutely flexible inextensible thread:

d / ds (l dx / ds) - X = 0, d / ds (l dy / ds) - Y = 0,

d / ds (l dz / ds) - Z = 0. (2.8)

Lagrange explains the physical meaning of the factor l as follows: “Since the value ld ds can represent the moment of some force l (in modern terminology -“ virtual (possible) work ”) tending to reduce the length of the element ds, then the term ò ld ds of the general equilibrium equation of the thread will express the sum of the moments of all forces l, which we can imagine acting on all the elements of the thread. Indeed, due to its inextensibility, each element resists the action of external forces, and this resistance is usually considered as an active force, which is called tension. Thus, l represents the tension of the thread "

Turning to dynamics, Lagrange, taking bodies as points of mass m, writes that “the quantities md 2 x / dt 2, md 2 y / dt 2, md 2 z / dt 2 (2.9) express the forces applied directly to move the body m parallel to the x, y, z axes ”.

The given accelerating forces P, Q, R,…, according to Lagrange, act along the lines p, q, r,…, proportional to the masses, directed to the corresponding centers and tend to decrease the distance to these centers. Therefore, the variations of the lines of action will be - d p, - d q, - d r, ..., and the virtual work of the applied forces and forces (2.9) will be respectively equal

е m (d 2 x / dt 2 d x + d 2 y / dt 2 d y + d 2 z / dt 2 d z), - е (P d p

Q d q + R d r + ...). (2.10)

Equating these expressions and transferring all terms to one side, Lagrange obtains the equation

е m (d 2 x / dt 2 d x + d 2 y / dt 2 d y + d 2 z / dt 2 d z) + е (P d p

Q d q + R d r +…) = 0, (2.11) which he called “the general formula of dynamics for the motion of any system of bodies”. It was this formula that Lagrange made the basis for all further conclusions - both general theorems of dynamics and theorems of celestial mechanics and dynamics of liquids and gases.

After deriving equation (2.11), Lagrange decomposes the forces P, Q, R, ... along the axes of rectangular coordinates and reduces this equation to the following form:

е (m d 2 x / dt 2 + X) d x + (m d 2 y / dt 2 + Y) d y + (m d 2 z / dt 2

Z) d z = 0. (2.12)

Equation (2.12) completely coincides with the modern form of the general equation of dynamics up to signs:

е j (F j - m j d 2 r j / dt 2) d r j = 0; (2.13) if we expand the scalar product, then we get equation (2.12) (except for the signs in brackets)

Thus, continuing the work of Euler, Lagrange completed the analytical formulation of the dynamics of a free and non-free system of points and gave numerous examples illustrating the practical power of these methods. Proceeding from the “general formula of dynamics”, Lagrange indicated two main forms of differential equations of motion of a non-free system, which now bear his name: “Lagrange equations of the first kind” and equations in generalized coordinates, or “Lagrange's equation of the second kind”. What led Lagrange to equations in generalized coordinates? Lagrange in his works on mechanics, including celestial mechanics, determined the position of a system, in particular, a rigid body, with various parameters (linear, angular, or their combination). For such a brilliant mathematician as Lagrange was, the problem of generalization naturally arose - to go to arbitrary, not concretized parameters.

This led him to differential equations in generalized coordinates. Lagrange called them “differential equations for solving all problems in mechanics,” now we call them Lagrange equations of the second kind:

d / dt L / q j - L / q j = 0 (L = T - P)

The overwhelming majority of the problems solved in "Analytical Mechanics" reflect the technical problems of that time. From this point of view, it is necessary to especially highlight the group of the most important problems of dynamics, united by Lagrange under the general name “On small vibrations of any system of bodies”. This section provides the basis for modern vibration theory. Considering small movements, Lagrange showed that any such movement can be represented as the result of the superposition of simple harmonic vibrations.

Mechanics of the 19th and early 20th centuries Lagrange's “Analytical Mechanics” summed up the achievements of theoretical mechanics in the 18th century. and identified the following main directions of its development:

1) expansion of the concept of connections and generalization of the basic equations of the dynamics of a non-free system for new types of connections;

2) the formulation of the variational principles of dynamics and the principle of conservation of mechanical energy;

3) development of methods for integrating the equations of dynamics.

In parallel with this, new fundamental problems of mechanics were put forward and resolved. For the further development of the principles of mechanics, the works of the outstanding Russian scientist M.V. Ostrogradsky (1801 - 1861) were fundamental. He was the first to consider connections that depend on time, introduced a new concept of unstoppable connections, that is, connections expressed analytically using inequalities, and generalized the principle of possible displacements and the general equation of dynamics to the case of such connections. Ostrogradskiy also has priority in the consideration of differential relationships that impose restrictions on the speed of points in the system; analytically, such connections are expressed using non-integrable differential equalities or inequalities.

A natural addition, expanding the area of ​​application of the D'Alembert principle, was the application of the principle proposed by Ostrogradsky to systems subject to the action of instantaneous and impulsive forces arising from impacts on the system. Ostrogradsky considered such impact phenomena as the result of the instant destruction of connections or the instant introduction of new connections into the system.

In the middle of the XIX century. the principle of conservation of energy was formulated: for any physical system, it is possible to determine a quantity called energy and equal to the sum of kinetic, potential, electrical and other energies and heat, the value of which remains constant regardless of what changes occur in the system. Significantly accelerated by the beginning of the 19th century. the process of creating new machines and the desire for their further improvement caused the appearance of applied, or technical, mechanics in the first quarter of the century. In the first treatises on applied mechanics, the concepts of the work of forces were finally formed.

D'Alembert's principle, containing the most general formulation of the laws of motion of a non-free system, does not exhaust all the possibilities of posing dynamics problems. In the middle of the 18th century. arose, and in the XIX century. new general principles of dynamics - variational principles - were developed.

The first variational principle was the principle of least action, put forward in 1744 without any proof, as a general law of nature, by the French scientist P. Maupertuis (1698 - 1756). The principle of least action states, “that the path that it (the light) follows is the path for which the number of actions will be the least”.

The development of general methods for integrating differential equations of dynamics refers mainly to the middle of the 19th century. The first step in reducing the differential equations of dynamics to a system of equations of the first order was made in 1809 by the French mathematician S. Poisson (1781 - 1840). The problem of reducing the equations of mechanics to the "canonical" system of equations of the first order for the case of time-independent constraints was solved in 1834 by the English mathematician and physicist W. Hamilton (1805 - 1865). Its final completion belongs to Ostrogradsky, who extended these equations to cases of nonstationary constraints.The largest problems of dynamics, the formulation and solution of which relate mainly to the 19th century, are: the motion of a heavy rigid body, the theory of elasticity (see Appendix) of equilibrium and motion, and also closely related to this theory, the problem of fluctuations of a material system. The first solution to the problem of the rotation of a heavy rigid body of arbitrary shape around a fixed center in the special case when the fixed center coincides with the center of gravity belongs to Euler.

Kinematic representations of this movement were given in 1834 by L. Poinsot. The case of rotation, when the stationary center, which does not coincide with the center of gravity of the body, is placed on the axis of symmetry, was considered by Lagrange. The solution of these two classical problems formed the basis for the creation of a rigorous theory of gyroscopic phenomena (a gyroscope is a device for observing rotation). Outstanding research in this area belongs to the French physicist L. Foucault (1819-1968), who created a number of gyroscopic instruments.

Examples of such devices are gyroscopic compass, artificial horizon, gyroscope and others. These studies indicated the fundamental possibility, without resorting to astronomical observations, to establish the daily rotation of the Earth and to determine the latitude and longitude of the observation site. After the works of Euler and Lagrange, despite the efforts of a number of outstanding mathematicians, the problem of rotation of a heavy rigid body around a fixed point did not receive further development for a long time.

The foundations of the theory of motion of a rigid body in an ideal fluid were given by the German physicist G. Kirchhoff in 1869. rifled guns, which was intended to give the projectile the rotation necessary for stability in flight, the task of external ballistics turned out to be closely related to the dynamics of a heavy rigid body. This formulation of the problem and its solution belongs to the outstanding Russian scientist - artilleryman N.V. Maevsky (1823 - 1892).

One of the most important problems in mechanics is the problem of the stability of the equilibrium and motion of material systems. The first general theorem on the stability of the equilibrium of a system under the action of generalized forces belongs to Lagrange and is stated in “Analytical Mechanics”. According to this theorem, a sufficient condition for equilibrium is the presence of a minimum of potential energy in the equilibrium position. The method of small oscillations, applied by Lagrange to prove the theorem on the stability of equilibrium, turned out to be fruitful for studying the stability of steady motions. In “Treatise on the stability of a given state of motion”.

The English scientist E. Routh, published in 1877, the study of stability by the method of small oscillations was reduced to considering the distribution of the roots of a certain "characteristic" equation and indicated the necessary and sufficient conditions under which these roots have negative real parts.

From a point of view different from that of Routh, the problem of stability of motion was considered in the work of NE Zhukovsky (1847 - 1921) "On the strength of motion" (1882), in which orbital stability was studied. The criteria for this stability, established by Zhukovsky, are formulated in a visual geometric form, so characteristic of the entire scientific work of the great mechanic.

A rigorous formulation of the problem of stability of motion and an indication of the most general methods of its solution, as well as a specific consideration of some of the most important problems of the theory of stability, belong to A. M. Lyapunov, and were presented by him in his fundamental essay "The General Problem of Stability of Motion" (1892). They gave the definition of a stable equilibrium position, which looks as follows: if for a given r (radius of the sphere) one can choose such an arbitrarily small but not equal to zero value of h (initial energy) that in all subsequent time the particle does not go beyond the limits sphere of radius r, then the equilibrium position at this point is called stable. Lyapunov connected the solution of the stability problem with the consideration of some functions, from the comparison of the signs of which with the signs of their time derivatives, one can conclude about the stability or instability of the considered state of motion (“the second Lyapunov method”). With the help of this method, Lyapunov, in his stability theorems in the first approximation, indicated the limits of applicability of the method of small oscillations of a material system about the position of its stable equilibrium (first described in Lagrange's “Analytical Mechanics”).

The subsequent development of the theory of small fluctuations in the XIX century. was associated mainly with the influence of resistances leading to damping of oscillations, and external disturbing forces that create forced oscillations. The theory of forced vibrations and the theory of resonance appeared in response to the demands of machine technology and, first of all, in connection with the construction of railway bridges and the creation of high-speed steam locomotives. Another important branch of technology, the development of which required the application of the methods of the theory of oscillations, was regulator construction. The founder of the modern dynamics of the regulation process is the Russian scientist and engineer I.A.Vyshnegradskiy (1831 - 1895). In 1877, in his work “On Direct Controllers”, Vyshnegradskiy was the first to formulate the well-known inequality that must be satisfied by a stably operating machine equipped with a controller.

The further development of the theory of small oscillations was closely related to the emergence of individual major technical problems. The most important works on the theory of the pitching of a ship in waves belong to the outstanding Soviet scientist

A.N. Krylov, whose entire activity was devoted to the application of modern achievements of mathematics and mechanics to the solution of the most important technical problems. In the XX century. problems of electrical engineering, radio engineering, the theory of automatic control of machines and production processes, technical acoustics and others gave rise to a new field of science - the theory of nonlinear oscillations. The foundations of this science were laid in the works of A.M. Lyapunov and the French mathematician A. Poincaré, and further development, as a result of which a new, rapidly growing discipline was formed, is due to the achievements of Soviet scientists. By the end of the XIX century. a special group of mechanical problems was distinguished - the motion of bodies of variable mass. The fundamental role in the creation of a new area of ​​theoretical mechanics - the dynamics of variable mass - belongs to the Russian scientist I.V. Meshchersky (1859 - 1935). In 1897 he published the fundamental work "Dynamics of a point of variable mass".

In the XIX and early XIX centuries. the foundations were laid for two important branches of hydrodynamics: viscous fluid dynamics and gas dynamics. The hydrodynamic theory of friction was created by the Russian scientist N.P. Petrov (1836 - 1920). The first rigorous solution of problems in this area was indicated by N. Ye. Zhukovsky.

By the end of the XIX century. mechanics has reached a high level of development. XX century brought a deep critical revision of a number of basic provisions of classical mechanics and was marked by the emergence of the mechanics of fast motions proceeding with speeds close to the speed of light. The mechanics of fast movements, as well as the mechanics of microparticles, were further generalizations of classical mechanics.

Newtonian mechanics retained a vast field of activity in the basic questions of mechanics engineering in Russia and the USSR. Mechanics in pre-revolutionary Russia, thanks to the fruitful scientific activity of M.V. Ostrogradsky, N.E. Zhukovsky, S.A.Chaplygin, A.M. Lyapunov, A.N. to cope with the tasks put before it by domestic technology, but also to contribute to the development of technology throughout the world. The works of the “father of Russian aviation” N. Ye. Zhukovsky laid the foundations of aerodynamics and aviation science in general. The works of N. Ye. Zhukovsky and S. A. Chaplygin were of fundamental importance in the development of modern hydro-aeromechanics. SA Chaplygin is the author of fundamental research in the field of gas dynamics, which indicated the development of high-speed aerodynamics for many decades ahead. A. N. Krylov's work on the theory of stability of a ship's roll on waves, research on the buoyancy of their hull, and the theory of compass deviation put him among the founders of modern science of shipbuilding.

One of the important factors that contributed to the development of mechanics in Russia was the high level of teaching it in higher education. Much has been done in this respect by M. V. Ostrogradskii and his followers. Problems of motion stability are of the greatest technical importance in problems of the theory of automatic control. I. N. Voznesenskiy (1887 - 1946) played an outstanding role in the development of the theory and technology of regulation of machines and production processes. Problems of rigid body dynamics developed mainly in connection with the theory of gyroscopic phenomena.

Soviet scientists have achieved significant results in the field of elasticity theory. They carried out research on the theory of plate bending and general solutions of problems in the theory of elasticity, on the plane problem of the theory of elasticity, on the variational methods of the theory of elasticity, on structural mechanics, on the theory of plasticity, on the theory of an ideal fluid, on the dynamics of a compressible fluid and gas dynamics, on the theory filtration of movements, which contributed to the rapid development of Soviet hydro-aerodynamics, dynamic problems in the theory of elasticity were developed. The results of paramount importance, obtained by scientists of the Soviet Union on the theory of nonlinear oscillations, confirmed the USSR's leading role in this field. The formulation, theoretical consideration and organization of the experimental study of nonlinear oscillations are an important achievement of L.I. Mandel'shtam (1879 - 1944) and N.D. Papaleksi (1880 - 1947) and their school (A.A.

The foundations of the mathematical apparatus of the theory of nonlinear oscillations are contained in the works of A. M. Lyapunov and A. Poincaré. Poincaré's “limit cycles” were formulated by A. A. Andronov (1901 - 1952) in connection with the problem of continuous oscillations, which he called self-oscillations. Along with the methods based on the qualitative theory of differential equations, the analytical direction of the theory of differential equations developed.

5. PROBLEMS OF MODERN MECHANICS.

The main problems of modern mechanics of systems with a finite number of degrees of freedom include, first of all, the problems of the theory of oscillations, the dynamics of a rigid body and the theory of stability of motion. In the linear theory of oscillations, it is important to create effective methods for studying systems with periodically changing parameters, in particular, the phenomenon of parametric resonance.

To study the motion of nonlinear oscillatory systems, both analytical methods and methods based on the qualitative theory of differential equations are being developed. The problems of vibrations are closely intertwined with the issues of radio engineering, automatic regulation and control of movements, as well as with the tasks of measuring, preventing and eliminating vibrations in transport devices, machines and building structures. In the field of rigid body dynamics, the greatest attention is paid to the problems of the theory of oscillations and the theory of stability of motion. These tasks are posed by the dynamics of flight, the dynamics of the ship, the theory of gyroscopic systems and instruments, which are mainly used in air navigation and ship navigation. In the theory of motion stability, the first place is given to the study of Lyapunov's “special cases”, the stability of periodic and unsteady motions, and the main research tool is the so-called “second Lyapunov method”.

In the theory of elasticity, along with problems for a body obeying Hooke's law, the greatest attention is paid to the issues of plasticity and creep in the details of machines and structures, the calculation of the stability and strength of thin-walled structures. The direction that aims to establish the basic laws of the relationship between stresses and deformations and strain rates for models of real bodies (rheological models) is also acquiring great importance. In close connection with the theory of plasticity, the mechanics of a free-flowing medium is being developed. The dynamic problems of the theory of elasticity are associated with seismology, the propagation of elastic and plastic waves along the rods and dynamic phenomena arising from impact. The most important problems of hydroaerodynamics are associated with the problems of high speeds in aviation, ballistics, turbine and engine building.

This includes, first of all, the theoretical definition of the aerodynamic characteristics of bodies at sub-, near- and supersonic speeds, both in steady and unsteady motions.

The problems of high-speed aerodynamics are closely intertwined with the issues of heat transfer, combustion and explosions. The study of the motion of a compressible gas at high speeds presupposes the main problem of gas dynamics, and at low speeds it is associated with problems of dynamic meteorology. The problem of turbulence, which has not yet received a theoretical solution, is of fundamental importance for hydroaerodynamics. In practice, they continue to use numerous empirical and semi-empirical formulas.

The hydrodynamics of a heavy fluid is faced with the problems of the spatial theory of waves and wave drag of bodies, wave formation in rivers and canals, and a number of problems associated with hydraulic engineering.

Problems of filtration movement of liquids and gases in porous media are of great importance for the latter, as well as for issues of oil production.

6. CONCLUSION.

Galileo - Newton mechanics has come a long way of development and did not immediately win the right to be called classical. Her successes, especially in the 17th and 18th centuries, established experiment as the main method for testing theoretical constructions. Almost until the end of the 18th century, mechanics occupied a leading position in science, and its methods had a great influence on the development of all natural science.

In the future, the mechanics of Galileo - Newton continued to develop intensively, but its leading position gradually began to be lost. Electrodynamics, the theory of relativity, quantum physics, nuclear energy, genetics, electronics, and computer technology began to emerge at the forefront of science. Mechanics gave way to a leader in science, but did not lose its significance. As before, all dynamic calculations of any mechanisms operating on the ground, under water, in the air and in space are based to one degree or another on the laws of classical mechanics. On the basis of far from obvious consequences from its basic laws, devices are built, autonomously, without human intervention, determining the location of submarines, surface ships, aircraft; systems have been built that autonomously orient spacecraft and direct them to the planets of the solar system, Halley's comet. Analytical mechanics - an integral part of classical mechanics - retains the "inconceivable efficiency" in modern physics. Therefore, no matter how physics and technology develop, classical mechanics will always take its rightful place in science.

7. APPENDIX.

Hydromechanics is a branch of physics dealing with the study of the laws of motion and equilibrium of a fluid and its interaction with washed solids.

Aeromechanics is the science of equilibrium and movement of gaseous media and solids in a gaseous medium, primarily in air.

Gas mechanics is a science that studies the movement of gases and liquids under conditions when the property of compressibility is essential.

Aerostatics is a part of mechanics that studies the equilibrium conditions for gases (especially air).

Kinematics is a branch of mechanics in which the movements of bodies are studied without taking into account the interactions that determine these movements. Basic concepts: instantaneous speed, instantaneous acceleration.

Ballistics is the science of projectile movement. External ballistics studies the movement of a projectile in the air. Internal ballistics studies the movement of a projectile under the action of propellant gases, the mechanical freedom of which is limited by any effort.

Hydraulics is the science of the conditions and laws of equilibrium and motion of fluids and the ways of applying these laws to solving practical problems. Can be defined as applied fluid mechanics.

An inertial coordinate system is a coordinate system in which the law of inertia is fulfilled, i.e. in which the body, when compensating for external influences exerted on it, moves uniformly and rectilinearly.

Pressure is a physical quantity equal to the ratio of the normal component of the force with which the body acts on the surface of the support in contact with it, to the contact area, or otherwise - the normal surface force acting per unit area.

Viscosity (or internal friction) is the property of liquids and gases to resist when one part of the liquid moves relative to another.

Creep is a process of small continuous plastic deformation that occurs in metals under conditions of prolonged static loading.

Relaxation is the process of establishing static equilibrium in a physical or physicochemical system. In the process of relaxation, the macroscopic quantities characterizing the state of the system asymptotically approach their equilibrium values.

Mechanical connections are restrictions imposed on the movement or position of a system of material points in space and carried out using surfaces, threads, rods, and others.

Mathematical relations between the coordinates or their derivatives, which characterize the mechanical constraints of motion, are called constraint equations. For the system to move, the number of constraint equations must be less than the number of coordinates that determine the position of the system.

The optical method for studying stresses is a method for studying stresses in polarized light, based on the fact that particles of an amorphous material become optically anisotropic upon deformation. In this case, the main axes of the refractive index ellipsoid coincide with the main directions of deformation, and the main light oscillations, passing through the deformed plate of polarized light, receive a path difference.

Strain gauge - a device for measuring tensile or compressive forces applied to any system due to deformations caused by these forces

Celestial mechanics is a section of astronomy dedicated to the study of the motion of cosmic bodies. Now the term is used differently and the subject of celestial mechanics is usually considered only the general methods of studying the motion and force field of the bodies of the solar system.

The theory of elasticity is a branch of mechanics that studies displacements, elastic deformations and stresses that arise in a solid under the action of external forces, from heating and from other influences. The task is to determine quantitative relationships characterizing the deformation or internal relative displacements of particles of a solid body that is under the influence of external influences in a state of equilibrium or small internal relative motion.

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Mechanics is the science of moving bodies and the interactions between them during motion. At the same time, attention is paid to those interactions as a result of which the movement has changed or the deformation of the bodies has occurred. In this article we will tell you about what mechanics is.

Mechanics can be quantum, applied (technical) and theoretical.

1. What is Quantum Mechanics? This is a branch of physics that describes physical phenomena and processes, the actions of which are comparable to the magnitude of Planck's constant.
2. What is technical mechanics? This is a science that reveals the principle of operation and the structure of mechanisms.
3. What is theoretical mechanics? This is the science and motion of bodies and the general laws of motion.

Mechanics studies the movement of all kinds of machines and mechanisms, aircraft and celestial bodies, oceanic and atmospheric currents, plasma behavior, deformation of bodies, the movement of gases and liquids in natural conditions and technical systems, a polarizing or magnetizing medium in electric and magnetic fields, the stability and strength of technical and building structures, movement along the respiratory tract of air and blood through the vessels.

Newton's law lies at the foundations, with the help of it they describe the motion of bodies with small speeds in comparison with the speed of light.

There are the following sections in mechanics:

• kinematics (about the geometric properties of moving bodies without taking into account their mass and acting forces);
• statics (about finding bodies in equilibrium using external influences);
• dynamics (about moving bodies under the influence of force).

In mechanics, there are concepts that reflect the properties of bodies:

• material point (body, the size of which can be ignored);
• an absolutely rigid body (a body in which the distance between any points is unchanged);
• continuous medium (a body whose molecular structure is neglected).

If the rotation of the body with respect to the center of mass under the conditions of the problem under consideration can be neglected or it moves translationally, the body is equated to a material point. If you do not take into account the deformation of the body, then it must be considered absolutely non-deformable. Gases, liquids and deformable bodies can be considered as solid media in which particles continuously fill the entire volume of the medium. In this case, when studying the movement of the medium, the apparatus of higher mathematics is used, which is used for continuous functions. Equations describing the behavior of a continuous medium follow from the fundamental laws of nature - the laws of conservation of momentum, energy and mass. Continuum mechanics contains a number of independent sections - aerodynamics and hydrodynamics, theory of elasticity and plasticity, gas dynamics and magnetohydrodynamics, dynamics of the atmosphere and water surface, physicochemical mechanics of materials, mechanics of composites, biomechanics, space hydro-aeromechanics.

Now you know what mechanics are!

# 1 Mechanics. Mechanical movement.

Mechanics- the science of the movement of material objects and the interaction between them. The most important sections of mechanics are classical mechanics and quantum mechanics. The objects studied by mechanics are called mechanical systems. A mechanical system has a certain number k of degrees of freedom and is described using generalized coordinates q1,… qk. The task of mechanics is to study the properties of mechanical systems, and, in particular, to clarify their evolution in time.

The most important mechanical systems are: 1) material point 2) harmonic oscillator 3) mathematical pendulum 4) torsion pendulum 5) absolutely rigid body 6) deformable body 7) absolutely elastic body 8) continuous medium

Mechanical movement body is called the change in its position in space relative to other bodies over time. In this case, the bodies interact according to the laws of mechanics.

Types of mechanical movement

Mechanical movement can be considered for different mechanical objects:

Material point movement is completely determined by the change in its coordinates in time (for example, two on a plane). The study of this is the kinematics of the point.

1) Rectilinear movement of a point (when it is always on a straight line, the speed is parallel to this straight line)

2) Curvilinear movement is the movement of a point along a trajectory that is not a straight line, with arbitrary acceleration and arbitrary speed at any time (for example, movement in a circle).

Solid body movement consists of the movement of any of its points (for example, the center of mass) and rotational movement around this point. It is studied by the kinematics of a rigid body.

1) If there is no rotation, then the movement is called translational and is completely determined by the movement of the selected point. Note that it is not necessarily straightforward.

2) To describe the rotational motion - the movement of a body relative to a selected point, for example, fixed at a point, Euler angles are used. Their number in the case of three-dimensional space is three.

3) Also, for a rigid body, a plane motion is distinguished - a movement in which the trajectories of all points lie in parallel planes, while it is completely determined by one of the sections of the body, and the section of the body by the position of any two points.

Continuous medium motion... It is assumed here that the motion of individual particles of the medium is quite independent of each other (usually limited only by the conditions of continuity of the velocity fields), therefore, the number of defining coordinates is infinite (functions become unsettled).

№4 Basic laws of the dynamics of a material point

Newton's second law can be written in a different form. By definition:

Then or

The vector is called the impulse or momentum of the body and coincides in direction with the velocity vector, and expresses the change in the impulse vector. Let's transform the last expression to the following form: The vector is called the impulse of force. This equation is an expression of the basic law of the dynamics of a material point: the change in the momentum of the body is equal to the momentum of the force acting on it.

Dynamics- a section of mechanics, which studies the laws of motion of material bodies under the action of forces. Basic laws of mechanics (Galileo-Newton's laws): the law of inertia (1st law): a material point maintains a state of rest or uniform rectilinear motion until the action of other bodies changes this state; the basic law of dynamics (2nd law (Newton's)): the acceleration of a material point is proportional to the force applied to it and has the same direction as it; the law of equality of action and reaction (3rd law (Newton's)): every action corresponds to an equal and oppositely directed reaction; the law of the independence of forces: several forces simultaneously acting on a material point impart to the point such acceleration as would be imparted to it by one force equal to their geometric sum. In classical mechanics, the mass of a moving body is taken to be equal to the mass of a body at rest, a measure of the body's inertia and its gravitational properties. Mass = body weight divided by the acceleration due to gravity. m = G / g, g9.81m / s2. g depends on the geographical latitude of the location and the altitude above sea level - not constant. Force - 1N (Newton) = 1kgm / s2. The frame of reference in which the 1st and 2nd laws are manifested, name. inertial frame of reference. Differential equations of motion of a material point:, in projection on the Cartesian axes coord.:, On the axis of the natural trihedron: ma = Fi; man = Fin; mab = Fib (ab = 0 is the projection of the acceleration onto the binormal), i.e. ( is the radius of curvature of the trajectory at the current point). In the case of a plane movement of a point in polar coordinates :. Two main tasks of dynamics: the first task of dynamics - knowing the law of motion of a point, determine the force acting on it; the second task of dynamics (the main one) - knowing the forces acting on the point, determine the law of motion of the point. - differential ur-ye of rectilinear motion of a point. Integrating it twice, we find the general solution x = f (t, C1, C2).

Integration constants C1, C2 are sought from the initial conditions: t = 0, x = x0, = Vx = V0, x = f (t, x0, V0) - a particular solution - the law of motion of a point.

No. 6 The law of change in the impulse of a mechanical system

The physical content of the concept of impulse or momentum is determined by the purpose of this concept. Impulse is one of the parameters that describe qualitatively and quantitatively the movement of a mechanical system.

The theorem on the change in the momentum of an open-loop system: If the system is open, then its momentum is not conserved, and the change in the momentum of such a system over time is expressed by the formula:

Vector K is called the main vector of external acting forces.

(Proof) Differentiate (4):

Let's use the equation of motion of an open system:

Momentum The momentum of a body (material point) is a vector quantity equal to the product of the mass of a body (material point) by its speed. The momentum of a system of bodies (material points) is the vector sum of the momenta of all points. The impulse of force is the product of the force and the time of its action (or the integral over time, if the force changes with time). The law of conservation of momentum: in the inertial frame of reference, the momentum of a closed-loop system is conserved.

Change in the momentum of a system of material points - in the inertial frame of reference, the rate of change in the momentum of a mechanical system is equal to the vector sum of external forces acting on the material points of the system. The forces acting on a particle in a mechanical system can be divided into internal and external forces (Fig. 5.2). Internal forces are called forces that are caused by the interaction of the particles of the system with each other. External forces characterize the action of bodies not included in the system (i.e. external) bodies on the particles of the system. A system that is not acted upon by external forces is called closed.

№10 Mechanical work Mechanical work or simply the work of a constant force on displacement is a scalar physical quantity equal to the product of the modulus of force, modulus of displacement and the cosine of the angle between these vectors. If the work is denoted by the letter A, then by definition A = Fscos (a) α is the angle between force and displacement. Work Fcosa represents the projection of the force onto the direction of travel. It is on the magnitude of this projection that what will be the work of force on a given displacement depends. If, in particular, the force F is perpendicular to the displacement, then this projection is zero and no work, while the force F does not. For other values ​​of the angle, the work of the force can be both positive (when 0 ° ≤α<90°), так и отрицательной (когда 90°<α≤180°). Единицей работы в СИ является 1 Дж (joule). 1 J is the work performed by a constant force of 1 N on a displacement of 1 m in the direction coinciding with the line of action of this force.

The work of any constant force has the following two remarkable properties: 1. The work of a constant force on any closed trajectory is always zero. 2. The work of constant force, performed when a particle moves from one point to another, does not depend on the shape of the trajectory connecting these points. According to the formula A = Fscos (a), you can find a job only permanent strength. If the force acting on the body varies from point to point, then the work throughout the entire territory is determined by the formula: A = A1 + A2 + ... + An work for which this device (mechanism) is used. The efficiency is equal to:

Power To characterize the process of performing work, it is also important to know the time it takes to complete it. The speed of doing work is characterized by a special quantity called power . Power is a scalar physical quantity equal to the ratio of work to the time during which it was performed. Denoted by a letter R: P = A / t = Fv The SI unit of power is 1 W (watt). 1 W is the power at which 1 J of work is done in 1 s.

№11 Kinetic energy Another fundamental physical concept is closely related to the concept of work - the concept energy. Since mechanics studies, firstly, the motion of bodies, and secondly, the interaction of bodies with each other, it is customary to distinguish between two types of mechanical energy: kinetic energy, due to body movement, and potential energy, due to the interaction of the body with other bodies. Kinetic energy, obviously, should depend on the speed of movement of the body v , and potential - from the mutual arrangement of interacting bodies. Kinetic energy particle is called a scalar physical quantity equal to half the product of the mass of this particle by the square of its speed.

Kinetic energy theorem: The change in the kinetic energy of a body is equal to the work of all forces acting on this body,

If is the final kinetic energy, and is the initial kinetic energy, then.

If the body moving at the beginning gradually stops, for example, hitting any obstacle, and its kinetic energy Ek vanishes, then the work done by him will be completely determined by his initial kinetic energy.

The physical meaning of kinetic energy: the kinetic energy of a body is equal to the work that it is able to perform in the process of reducing its speed to zero. The more "reserve" of kinetic energy a body has, the more work it is able to perform.

No. 12 Potential energy

The second type of energy is potential energy-energy, due to the interaction of bodies.

The value equal to the product of the body's mass m by the acceleration of gravity g and by the height h of the body above the Earth's surface is called the potential energy of interaction between the body and the Earth. Let us agree to denote potential energy by the letter Er.

Ep = mgh. A value equal to half the product of the coefficient of elasticity k bodies per strain square NS are called potential energy of an elastically deformed body :

In both cases, the potential energy is determined by the arrangement of the bodies of the system or parts of one body relative to each other.

By introducing the concept of potential energy, we are able to express the work of any conservative forces through a change in potential energy. The change in value is understood as the difference between its final and initial values

This formula allows you to give a general definition of potential energy. The potential energy of the system is called the quantity depending on the position of the bodies, the change of which during the transition of the system from the initial state to the final state is equal to the work of the internal conservative forces of the system, taken with the opposite sign. The minus sign in the formula does not mean that the work of conservative forces is always negative. It only means that the change in potential energy and the work of forces in the system always have opposite signs. Zero level is the level of potential energy counting. Since work determines only the change in potential energy, then only the change in energy in mechanics has a physical meaning. Therefore, one can arbitrarily choose the state of the system in which its potential energy is considered to be zero. This state corresponds to the zero level of potential energy. Not a single phenomenon in nature or technology is determined by the value of the potential energy itself. Only the difference in the values ​​of the potential energy in the final and initial states of the system of bodies is important. Usually, the state of the system with the minimum energy is chosen as the state with zero potential energy. Then the potential energy is always positive.

№25 Fundamentals of Molecular Kinetic Theory Molecular kinetic theory (MKT) explains the properties of macroscopic bodies and thermal processes in them, based on the idea that all bodies are composed of separate, randomly moving particles. Basic concepts of molecular kinetic theory: Atom (from the Greek atomos - indivisible) - the smallest part of a chemical element, which is the carrier of its properties. The dimensions of an atom are of the order of 10-10 m. A molecule is the smallest stable particle of a given substance, which has its basic chemical properties and consists of atoms connected by chemical bonds. The size of the molecules is 10-10 -10-7 m. A macroscopic body is a body consisting of a very large number of particles. Molecular kinetic theory (abbreviated as MKT) is a theory that considers the structure of matter from the point of view of three main approximately correct positions:

1) all bodies consist of particles, the size of which can be neglected: atoms, molecules and ions; 2) particles are in continuous chaotic motion (thermal); 3) particles interact with each other through absolutely elastic collisions.

Basic equation of MKT

where k is the ratio of the gas constant R to Avogadro's number, and i - the number of degrees of freedom of molecules. The basic equation of the MKT connects macroscopic parameters (pressure, volume, temperature) of a gas system with microscopic ones (mass of molecules, average speed of their movement).

Derivation of the basic equation of the MKT

Let there be a cubic vessel with an edge of length l and one particle of mass m in him. Let's designate the speed of movement vx, then before collision with the vessel wall the momentum of the particle is mvx, and after - - mvx, therefore, the impulse is transferred to the wall p = 2mvx... The time after which the particle collides with the same wall is equal.

This implies:

therefore pressure.

Accordingly, and.

Thus, for a large number of particles, the following is true:, similarly for the y and z axes.

Since, then.

Let be the average kinetic energy of molecules, and Ek is the total kinetic energy of all molecules, then:

The equation of the rms velocity of a molecule The equation of the rms velocity of a molecule is easily derived from the basic equation of the MKT for one mole of gas.

For 1 mole N = Na, where Na- Avogadro's constant Na m = Mr, where Mr is the molar mass of the gas.

Isoprocesses are processes that take place at the value of one of the macroscopic parameters. There are three isoprocesses: isothermal, isochoric, isobaric.

26 Thermodynamic system. Thermodynamic process A thermodynamic system is any area of ​​space bounded by real or imaginary boundaries chosen for the analysis of its internal thermodynamic parameters. The space adjacent to the system boundary is called the external environment. All thermodynamic systems have a medium with which energy and matter can be exchanged. The boundaries of a thermodynamic system can be fixed or movable. Systems can be large or small, depending on the boundaries. For example, the system can cover the entire refrigeration system or the gas in one of the compressor cylinders. The system can exist in a vacuum or it can contain several phases of one or more substances. Thermodynamic systems can contain dry air and water vapor (two substances) or water and water vapor (two stages of the same substance). A homogeneous system consists of one substance, one phase or a homogeneous mixture of several components. Systems are isolated (closed) or open. In an isolated system, there are no exchange processes with the external environment. In an open system, both energy and matter can pass from the system to the environment and vice versa. When analyzing pumps and heat exchangers, an open system is required as liquids must cross boundaries during analysis. If the mass flow rate of an open system is stable and uniform, the system is called an open system with a constant flow rate. The state of a thermodynamic system is determined by the physical properties of a substance. Temperature, pressure, volume, internal energy, enthalpy and entropy are thermodynamic quantities that determine certain integral parameters of the system. These parameters are strictly determined only for systems in a state of thermodynamic equilibrium.

A thermodynamic process is any change that occurs in a thermodynamic system and is associated with a change in at least one of its state parameters.

36 Reversible and irreversible processes

If the external influence on the system is carried out in the forward and reverse directions, for example, alternating expansion and contraction, moving the piston in the cylinder, then the parameters of the state of the system will also change in the forward and reverse directions. Externally set state parameters are called external parameters. In the simplest case under consideration, the role of the external parameter is played by the volume of the system. Reversible such processes are called for which, with direct and reverse changes in external parameters, the system will go through the same intermediate states. Let us explain with an example that this is not always true. If we move the piston up and down very quickly, so that the uniformity of the gas concentration in the cylinder will not have time to be established, then during compression under the piston there will be gas compaction, and during expansion - rarefaction, that is, intermediate states of the system (gas) with one and the same position of the piston will be different depending on the direction of its movement. That's an example irreversible process. If the piston moves slowly enough so that the gas concentration has time to equalize, then during forward and reverse movements the system will pass through states with the same parameters at the same position of the piston. This is a reversible process. It can be seen from the given example that for reversibility it is necessary that the change in external parameters is carried out sufficiently slowly, so that the system has time to return to the state of equilibrium (establishment of a uniform distribution of the gas density), or, in other words, that all intermediate states are equilibrium (more precisely, quasi-equilibrium ). Note that in the given example, the concepts of "slow" and "fast" in relation to the movement of the piston must be taken in comparison with the speed of sound in gas, since it is this speed that is the characteristic speed of concentration equalization (recall that sound is a wave-like propagation of alternating seals and dilutions of the medium). So most of the engines used in technology satisfy the criterion of "slowness" of the piston movement in terms of the reversibility of the ongoing processes. It is in this sense that we talked about the "slow" movement of the piston when introducing the concept of work. Let's consider other examples of irreversible processes.
Let the vessel be divided into two parts by a septum. There is gas on one side and vacuum on the other. At some point, the tap opens and the irreversible flow of gas into the void begins. Here we are also dealing with non-equilibrium intermediate states. After reaching equilibrium, the gas flow will stop. Let us bring into thermal contact two bodies with different temperatures. The resulting system will be nonequilibrium until the temperatures of the bodies equalize, which will be accompanied by an irreversible transfer of heat from a more heated body to a less heated one.

39. II - the law of thermodynamics.

The first law of thermodynamics means the impossibility of existence perpetual motion machine of the first kind- a machine that would create energy. However, this law does not impose restrictions on the conversion of energy from one type to another. Mechanical work can always be converted to heat (for example, by friction), but there are limitations to converting it back. Otherwise, it would be possible to turn into work the heat taken from other bodies, i.e. create perpetual motion machine of the second kind. The second law of thermodynamics excludes the possibility of creating a perpetual motion machine of the second kind. There are several different but equivalent formulations of this law. Here are two of them. 1. Clausius' postulate. The process, in which no other changes occur, except for the transfer of heat from a hot body to a cold one, is irreversible, i.e. heat cannot pass from a cold body to a hot one without any other changes in the system. 2. Kelvin's postulate. The process in which work turns into heat without any other changes in the system is irreversible, i.e. it is impossible to turn into work all the heat taken from a source with a uniform temperature, without making other changes in the system. In these postulates, it is essential that no other changes take place in the system except those indicated. In the presence of changes, the transformation of heat into work is, in principle, possible. So, with isothermal expansion of an ideal gas enclosed in a cylinder with a piston, its internal energy does not change, since it depends only on temperature. Therefore, it follows from the first law of thermodynamics that all the heat received by the gas from the environment is converted into work. This does not contradict Kelvin's postulate, since the conversion of heat into work is accompanied by an increase in the volume of gas. The impossibility of the existence of a perpetual motion machine of the second kind follows directly from Kelvin's postulate. Therefore, the failure of all attempts to build such an engine is an experimental proof of the second law of thermodynamics. Let us prove the equivalence of Clausius' and Kelvin's postulates. To do this, it is necessary to show that if Kelvin's postulate is incorrect, then Clausius's postulate is also incorrect, and vice versa. If Kelvin's postulate is incorrect, then the heat taken from a source with temperature T 2, you can turn a work, and then, for example, with the help of friction, turn this work into heat and heat a body with a temperature T 1 >T 2. The only result of such a process will be the transfer of heat from a cold body to a hot one, which contradicts Clausius's postulate.

The second part of the proof of the equivalence of the two postulates is based on considering the possibility of converting heat into work. The next section is devoted to a discussion of this issue.

No. 32 Barometric formula. Boltzmann distribution Barometric formula - the dependence of the pressure or density of a gas on the height in the gravity field. For ideal gas at constant temperature T and located in a uniform gravity field (at all points of its volume, the gravitational acceleration g the same), the barometric formula is as follows:

where p- gas pressure in a layer located at a height h, p 0- pressure at zero level ( h = h 0), M- molar mass of gas, R- gas constant, T- absolute temperature. It follows from the barometric formula that the concentration of molecules n(or gas density) decreases with height according to the same law:

where M- molar mass of gas, R- gas constant. The barometric formula can be obtained from the law of distribution of ideal gas molecules by velocities and coordinates in a potential force field. In this case, two conditions must be met: the constancy of the gas temperature and the uniformity of the force field. Similar conditions can be met for the smallest solid particles suspended in a liquid or gas. Based on this, the French physicist J. Perrin in 1908 applied the barometric formula to the distribution of emulsion particles over the height, which allowed him to directly determine the value of the Boltzmann constant. The barometric formula shows that the density of a gas decreases exponentially with height. The quantity that determines the rate of decrease in density is the ratio of the potential energy of particles to their average kinetic energy, which is proportional to kT... The higher the temperature T, the more slowly the density decreases with height. On the other hand, an increase in gravity mg(at a constant temperature) leads to a significantly greater compaction of the lower layers and an increase in the density gradient (gradient). Gravity acting on particles mg can be changed due to two values: acceleration g and particle masses m... Consequently, in a mixture of gases in a gravity field, molecules of different masses are distributed in different ways along the height. The actual distribution of air pressure and density in the earth's atmosphere does not follow the barometric formula, since within the atmosphere, the temperature and the acceleration of gravity change with altitude and latitude. In addition, atmospheric pressure increases with the concentration of water vapor in the atmosphere. The barometric formula is the basis of barometric leveling - a method for determining the height difference Δ h between two points according to the pressure measured at these points ( p 1 and p 2). Since atmospheric pressure depends on the weather, the time interval between measurements should be as short as possible and the measurement points should not be too far apart. The barometric formula is written in this case in the form: Δ h = 18400(1 + at) lg ( p 1 / p 2) (in m), where t- the average temperature of the air layer between the measurement points, a- temperature coefficient of volumetric expansion of air. The error in calculations using this formula does not exceed 0.1-0.5% of the measured height. More accurate is the Laplace formula, which takes into account the influence of air humidity and the change in the acceleration of gravity. Boltzmann distribution- the distribution of probabilities of various energy states of an ideal thermodynamic system (ideal gas of atoms or molecules) under conditions of thermodynamic equilibrium; discovered by L. Boltzmann in 1868-1871. According to Boltzmann distribution the average number of particles with total energy is

where is the multiplicity of the state of a particle with energy - the number of possible states of a particle with energy. The constant Z is found from the condition that the sum over all possible values ​​is equal to the given total number of particles in the system (normalization condition):

In the case when the motion of particles obeys classical mechanics, the energy can be considered consisting of 1) the kinetic energy (kin) of a particle (molecule or atom), 2) internal energy (hn) (for example, the excitation energy of electrons) and 3) potential energy (sweat ) in an external field, depending on the position of the particle in space:

45.46. Phase transitions of the first and second kind

Phase transition(phase transformation) in thermodynamics - the transition of a substance from one thermodynamic phase to another when external conditions change. From the point of view of the movement of the system along the phase diagram when its intensive parameters (temperature, pressure, etc.) change, the phase transition occurs when the system crosses the line separating the two phases. Since different thermodynamic phases are described by different equations of state, it is always possible to find a quantity that changes abruptly during the phase transition. Since the division into thermodynamic phases is a finer classification of states than the division according to the aggregate states of matter, not every phase transition is accompanied by a change in the aggregate state. However, any change in the state of aggregation is a phase transition. Phase transitions are most often considered when the temperature changes, but at a constant pressure (usually equal to 1 atmosphere). That is why the terms "point" (and not a line) of a phase transition, melting point, etc. are often used. crystals of salt in solution, which has reached saturation). Phase transition classification During a phase transition of the first kind, the most important, primary extensive parameters change abruptly: specific volume (i.e. density), the amount of stored internal energy, concentration of components, etc. and so on, and not an abrupt change in time (about the latter, see the section Dynamics of phase transitions below). Most common examples phase transitions of the first order: 1) melting and solidification 2) boiling and condensation 3) sublimation and desublimation During a phase transition of the second kind, the density and internal energy do not change, so that such a phase transition can be invisible to the naked eye. A jump is experienced by their second derivatives with respect to temperature and pressure: heat capacity, coefficient of thermal expansion, various susceptibilities, etc. Phase transitions of the second kind occur when the symmetry of the structure of a substance changes (the symmetry can completely disappear or decrease). The description of a second-order phase transition as a consequence of a change in symmetry is given by the Landau theory. At present, it is customary to speak not about a change in symmetry, but about the appearance at the transition point of an order parameter equal to zero in a less ordered phase and changing from zero (at the transition point) to nonzero values ​​in a more ordered phase. The most common examples of second-order phase transitions: 1) the passage of the system through the critical point 2) paramagnet-ferromagnet or paramagnet-antiferromagnet transition (order parameter - magnetization) 3) transition of metals and alloys to the state of superconductivity (order parameter - density of superconducting condensate) 4) the transition of liquid helium to the superfluid state (a.p. is the density of the superfluid component) 5) the transition of amorphous materials to the glassy state Modern physics also studies systems with phase transitions of the third or higher order. Recently, the concept of a quantum phase transition has become widespread, i.e. a phase transition controlled not by classical thermal fluctuations, but by quantum ones, which exist even at absolute zero temperatures, where the classical phase transition cannot be realized due to the Nernst theorem.

47 ... Liquid structure

A liquid occupies an intermediate position between a solid and a gas. What is its similarity to gas? Liquid, like gases, is isotopic. In addition, the liquid is fluid. In it, as in gases, there are no tangential stresses (shear stresses). Perhaps, it is only these properties that limit the similarity of a liquid to a gas. The similarity of liquid with solids is much more significant. Liquids are heavy, i.e. their specific gravity is comparable to the specific gravity of solids. Liquids, like solids, are poorly compressible. Near the crystallization temperature, their heat capacity and other thermal characteristics are close to the corresponding characteristics of solids. All this suggests that in their structure, liquids should in some way resemble solid bodies. The theory should explain this similarity, although it should also find an explanation for the differences between liquids and solids. In particular, it should explain the reason for the anisotropy of crystal bodies and the isotropy of liquids. A satisfactory explanation of the structure of liquids was proposed by the Soviet physicist J. Fraenkel. According to Frenkel's theory, liquids have the so-called quasi-crystal structure. The crystal structure is characterized by the correct arrangement of atoms in space. It turns out that in liquids, to a certain extent, the correct arrangement of atoms is also observed, but only in small regions. In a small area, a periodic arrangement of atoms is observed, but as the area under consideration in a liquid increases, the correct, periodic arrangement of atoms is lost and completely disappears in large areas. It is customary to say that in solids there is a "long-range order" in the arrangement of atoms (the regular crystal structure in large areas of space, covering a very large number of atoms), in liquids, "short-range order". The liquid, as it were, is broken down into small cells, within which a crystalline, regular structure is observed. There are no clear boundaries between cells, the boundaries are blurred. This structure of liquids is called quasi-crystalline.
The nature of the thermal motion of atoms in liquids also resembles the motion of atoms in solids. In a solid body, atoms perform oscillatory motion around the nodes of the crystal lattice. To a certain extent, a similar picture takes place in liquids. Here, the atoms also perform oscillatory motion near the nodes of the quasi-crystal cell, but unlike the atoms of a solid body, they jump from one node to another from time to time. As a result, the movement of atoms will be very complex: it is oscillatory, but at the same time the center of oscillations moves from time to time in space. This movement of atoms can be likened to the movement of a “nomad”. Atoms are not tied to one place, they "wander", but at each place they are held up for a certain, very short time, while making random fluctuations. It is possible to introduce the notion of the “sedentary life” of the atom. By the way, atoms in solids also wander from time to time, but unlike atoms in liquids, their "average sedentary life" is very long. Due to the small values ​​of the "average sedentary life" of atoms in liquids, there are no tangential stresses (shear stresses). If in a solid body the tangential force acts for a long time, then some "fluidity" is also observed in it. On the other hand, if the tangential load acts in a liquid for a very short time, then the liquid in relation to such loads is "elastic", i.e. discovers the shear resistance of deformation.
Thus, the notions of the "short order" in the arrangement of atoms and the "nomadic" movement of atoms bring the theory of the liquid state of a body to the theory of a solid, crystalline state.

Rotational dynamics material point -

has no special features. As usual, the central relation is Newton's second law for a body moving (in a circle). It should, of course, be remembered that with rotational motion, the vector equality that grows this law

F i = m a ,

you should almost always project in radial (normal) and tangential (tangential) directions:

Fn = man (*)

F t = ma t (**)

In this case, аn = v2 / R - here v is the speed of the body at a given time, and R is the radius of rotation. Normal acceleration is responsible for changing the speed in the direction only.

Sometimes an = v2 / R is called centripetal acceleration. The origin of this name is clear: this acceleration is always directed towards the center of rotation.

№3 Movement of a point in a circle

The movement of a point in a circle can be very difficult (fig. 17).

Let us consider in detail the movement of a point along a circle, at which v = const. This movement is called uniform circular motion. Naturally, the velocity vector cannot be constant (v is not equal to const), since the direction of the velocity is constantly changing.

The time it takes for the trajectory of a point to describe a circle is called the period of revolution of the point (T). The number of revolutions of a point in one second is called the frequency of revolution (v). The period of circulation can be found by the formula: T = 1 / v

Naturally, the movement of a point in one revolution will be equal to zero. However, the distance traveled will be equal to 2PiR, and at the number of revolutions n, the path will be equal to 2PiRn or 2PiRt / T, where t is the time of movement.

Acceleration with uniform motion of a point along a circle is directed to its center and is numerically equal to a = v2 / R.

This acceleration is called centripetal (or normal). The conclusion of this equality can be as follows. Let us bring the velocity vectors to one point at least behind - T (it is possible for T / 2 or T) (Fig. 18).

Then the sum of the changes in the velocity vectors for small time intervals will be equal to the length of the arc AB, which is equal to the modulus | v2 - v1 | for time t = 1/4 * T.

Determine the length of the arc. Since the radius for the arc will be the modulus of the vector v1 = v2 = v, the length of the arc l can be calculated as the length of a quarter circle with radius v:

After reduction we get: If the motion is uniformly variable, then v Ф const, then another component of the acceleration is considered, providing a change in the velocity modulus. This acceleration is called tangential: Tangential acceleration is directed tangentially to the trajectory, it can coincide in direction with the speed (uniformly accelerated motion) or be oppositely directed (equally slowed motion).

Consider the movement of a material point along a circle with a constant value with speed. In this case, called uniform motion along a circle, the tangential component of the acceleration is absent (ak = 0) and the acceleration coincides with its centripetal component. In a small time interval ^ t, the point passed the path ^ S, and the radius vector of the moving point turned by a small angle

The speed is constant in magnitude and the angle ^ AOB and ^ BCD are similar, therefore (48) and (49). Then, (50) or taking into account that v and R are constant and a = an (51), we obtain (52). When striving, therefore (53). Therefore, (54).
The uniform movement of a material point along a circle is characterized by angular velocities. It is determined with the ratio of the angle of rotation to the time interval during which this rotation occurred: (55).

Measurement unit in SI [rad / s]. Linear and angular velocity is related to the relation: (56). Uniform motion along a circle is described by a periodic function: f = (f + T) (57). Here the shortest repetition time T is called the period of this process. In our case, T is the time of one complete revolution. If N complete revolutions are made during time t, then the time of one revolution is N times less than t: T = t / N (58). To characterize such a movement, the number of complete revolutions per unit of time v (rotation frequency) is entered. Obviously, T and v are mutually inverse quantities: T = t / N (59). The unit of measurement of frequency in SI [Hz]. With an uneven movement of a material point along a circle, the angular one changes along with the linear speed. Therefore, the concept of angular acceleration is introduced. The average angular acceleration is the ratio of the change in the angular velocity to the time interval during which this change occurred: (60). With an equally variable motion of a material point along a circle and. Therefore, the angular velocity and the angle of rotation of the radius is determined by the equation: (61) where is the initial angular velocity of the material point.

Uniform motion of a material point in a circle is the movement of a material point in a circle, at which the modulus of its velocity does not change. With such a movement, the material point has centripetal acceleration.

No. 2 Characteristics of the movement of a material point Mechanical movement of a material point.

The simplest form of motion of matter is mechanical motion, which consists in the movement of bodies or their parts relative to each other. The main characteristics of motion.

The position of the material point M in the Cartesian coordinate system is determined by three coordinates (x, y, z) (Fig. 1) Otherwise, the position of the point can be specified by the radius - vector r drawn from the origin of coordinates 0 to point M. When its motion, point M describes a curve called the trajectory of motion. Depending on the section of the trajectory, traversed by a point in time t, is called the length of the path S. The forms of the trajectory of movement are rectilinear and curvilinear.
The traversed path S is associated with the time of movement by the functional dependence S = f (t) (1), which is the equation of motion.

The simplest types of mechanical movement of the body are translational and rotational movements. In this case, any straight line connecting two arbitrary points of the body moves, remaining parallel to itself. For example, a piston in a cylinder of an internal combustion engine moves progressively.

During the rotational movement of the body, its points describe circles located in parallel planes. The centers of all circles lie on one straight line perpendicular to the planes of the circles and called the axis of rotation.

The simplest case of mechanical movement is the movement of a point along a straight line, in which it travels equal path segments at equal time intervals. With uniform movement, the speed of the point, i.e. a value equal to the ratio of the distance traveled S to the corresponding time interval t: V = S / t (2) does not change with time (V = const). With an uneven movement, the speed changes from one point of the trajectory to another. To assess the uneven movement, the concept of average speed is introduced. For this, the ratio of the entire path s to the time t during which it has been traversed is taken: Vav = S / t (3).
Consequently, the average speed of non-uniform motion is equal to the speed of uniform motion at which the body travels the same path S and for the same time t as for a given motion.

Consider the movement of point M along an arbitrary trajectory (Fig. 2). Let its position at time t be characterized by the radius vector r0. After a time interval ^ t, the point will occupy a new position M1 on the trajectory, characterized by the radius vector r. At the same time, she traveled a path of length (4), and the radius vector received the transformation: ^ r = r-ro (5).

A directed segment of a straight line connecting some initial position of a point with its subsequent position is called a displacement. The displacement vector of the point ^ r is the vector difference of the radius vectors of the initial r0 and final positions r of the point. In a straight-line motion of a point, the displacement is equal to the distance traveled; in a curvilinear motion, it is in absolute value less than the path. Average speed in the section MM1, equal to the ratio (6)

The movement in the section MM1 is characterized by the direction of the vector MM1 and the value of the velocity Vcp. Therefore, it is possible to introduce a vector numerically equal to the average speed and having the direction of the displacement vector: (7)

Taking an infinitely small time interval (^ t-> 0) during which motion occurs, we obtain that the ratio ^ r / ^ t tends to the limit, and then lim (^ r / ^ t) = V (8)

Will express the instantaneous velocity vector, i.e. speed at a given time. With an infinite decrease in ^ t, the difference between ^ S and ^ r will decrease in the limit. They will coincide, then on the basis of (4) we can write that the modulus of speed: V = lim (^ S / ^ t) = dS / dt (9) i.e. the instantaneous speed with non-uniform motion is numerically equal to the first derivative of the path with respect to time.

In case of uneven movement, it is necessary to find out the pattern of speed changes over time. For this, a value is introduced that characterizes the rate of change in speed with time, i.e. acceleration. Acceleration, like speed, is a vector quantity. The ratio of the velocity increment ^ V to the time interval ^ t, expresses the average acceleration: acp = ^ V / ^ t (10). The instantaneous speed is numerically equal to the limit of the average acceleration when the time interval ^ t tends to zero: d = lim (^ V / ^ t) = dV / dt = d ^ 2S / dt ^ 2 (11)
Uniform rectilinear movement. With uniform rectilinear motion of a material point, the instantaneous speed does not depend on time and is directed along the trajectory at each point of the trajectory. The average speed for any period of time is equal to the instantaneous speed of the point: (12). Thus, (13). Graph (15) with uniform movement is represented by a straight line parallel to the time axis Ot Fig. The form of graphs (16), (17) and (18) depends on the direction of the vector V and on the choice of the positive direction of one or another coordinate axis. With uniform and rectilinear motion with a speed V, the vector of displacement ^ t of a material point over a period of time: ^ t = t-t0 (19) is equal to: (20)

The path S traversed by a material point with uniform rectilinear motion in a time interval ^ t = t-t0 (21) is equal to the modulus ^ t of the point displacement vector over the same time interval. Therefore, (22) or, if t0 = 0, (23)

Equally variable rectilinear motion. Equally variable rectilinear motion is a special case of non-uniform motion in which the acceleration remains constant both in magnitude and in direction (a = const). In this case, the average acceleration acp is equal to the instantaneous acceleration (24). If the direction of acceleration a coincides with the direction of the velocity V point, the motion is called uniformly accelerated. The module of the speed of uniformly accelerated motion of a point increases with time. If the directions of vectors a and V are opposite, the movement is called equally slow. The speed module at uniform slow motion decreases over time. The change in speed (25) over a period of time with an equally variable rectilinear motion is equal to (26) or (27). If at the moment of the beginning of the countdown the speed of the point is equal to V0 (initial speed) and the acceleration a is known, then the speed V at an arbitrary moment of time t: (28). The projection of the velocity vector onto the OX axis of the rectangular Cartesian coordinate system is associated with the corresponding projections of the vectors of the initial velocity and acceleration by the equation: (29).
The displacement vector Dr of a point over a period of time with an equally variable rectilinear motion with an initial speed and acceleration a is equal to: (30), and its projection onto the OX axis of a rectangular Cartesian coordinate system at is equal to: (31). The path S, traversed by a point in a period of time in uniformly accelerated rectilinear motion with an initial speed and acceleration a, is equal to: (32). When the path is equal to: (33).
With equidistant rectilinear motion, the path formula is: (34).

No. 9 Moment of inertia of a rigid body

Consider a rigid body that can rotate about a certain axis (Fig.). Moment of impulse i th point of the body relative to this axis is determined by the formula:

... (1.84) Expressing the linear velocity of a point through the angular velocity of the body and using the properties of the vector product, we obtain

(1.85) Let's project the moment of impulse on the axis of rotation: - this projection defines the moment about this axis. We get

(1.86) where zi, - coordinate i-points along the axis Z, a Ri, is the distance of the point from the axis of rotation. Summing over all particles of the body, we obtain the angular momentum of the whole body relative to the axis of rotation:

(1.87) The quantity

(1.88) is the moment of inertia of the body about the axis of rotation. The moment of momentum of the body relative to the given axis of rotation thus takes the form: Mz =J· Ω. (1.89) The resulting formula is similar to the formula Pz = mVz for translational motion. The role of mass is played by the moment of inertia, the role of linear speed is played by the angular velocity. Substituting expression (1.89) into the equation for the angular momentum (2.74), we obtain

J ·β z = Nz... (1.90) where βz. - projection on the axis of rotation of the angular acceleration. This equation is equivalent in form to Newton's second law. In the general case of an asymmetric body, the vector M does not coincide in direction with the axis of rotation of the body and rotates around this axis together with the body, describing a cone. From considerations of symmetry, it is clear that for a homogeneous body symmetric about the axis of rotation, the angular momentum relative to a point lying on the axis of rotation coincides with the direction of the axis of rotation. In this case, the following relationship takes place:

... (1.91) From expression (1.90) it follows that when the moment of external forces is equal to zero, the product Jω remains constant Jω = const and a change in the moment of inertia entails a corresponding change in the angular velocity of rotation of the body. This explains the well-known phenomenon that a person standing on a rotating bench, spreading his arms to the sides or pressing them to the body, changes the frequency of rotation. From the expressions obtained above, it is clear that the moment of inertia is the same characteristic of the inertia property of a macroscopic body with respect to rotational motion, as the inertial mass of a material point with respect to translational motion. From expression (1.88) it follows that the moment of inertia is calculated by summing over all particles of the body. In the case of a continuous distribution of body mass over its volume, it is natural to go from summation to integration, introducing the body density. If the body is homogeneous, then the density is determined by the ratio of the mass to the volume of the body: p = m / V (1.92) For a body with an unevenly distributed mass, the density of the body at some point is determined by the derivative p = dm / dV (1.93) The moment of inertia is represented as:

where  V is the microscopic volume occupied by a point mass. Since a solid consists of a large number of particles that almost continuously fill the entire volume occupied by the body, in expression (1.94) the microscopic volume can be considered infinitely small, while at the same time assuming that the point mass is "smeared" over this volume. In fact, we are now making a transition from a model of a point mass distribution to a model of a continuous medium, which in reality is a solid body due to its high density. The performed transition allows in formula (2.94) to replace the summation over individual particles by integration over the entire volume of the body: (1.95)

Rice. Calculation of the moment of inertia of a homogeneous disc Here the quantities ρ and r are functions of a point, for example, its Cartesian coordinates. Formula (1.95) allows you to calculate the moments of inertia of bodies of any shape. Let us calculate, as an example, the moment of inertia of a homogeneous disk about an axis perpendicular to the plane of the disk and passing through its center (Fig.). Since the disk is homogeneous, the density can be removed from the integral sign. Disk volume element dV= 2πr b · dr, where b is the thickness of the disc. Thus,

, (1.96) where R is the radius of the disc. Introducing the mass of the disk equal to the product of the density and the volume of the disk π R2 b, we get:

... (1.97) Finding the moment of inertia of the disk in the considered example was facilitated by the fact that the body was homogeneous and symmetric, and the moment of inertia was calculated relative to the axis of symmetry of the body. In the general case of rotation of a body of arbitrary shape around an arbitrary axis, the calculation of the moment of inertia can be performed using Steiner's theorem: the moment of inertia about an arbitrary axis is equal to the sum of the moment of inertia J0 relative to the axis parallel to the given one and passing through the center of inertia of the body, and the product of the body mass by the square of the distance between the axes: J =J +ma 2 . (1.98)

№24 The basic law of relativistic dynamics.

Relativistic energy According to the concepts of classical mechanics, the mass of a body is a constant value. However, at the end of the 19th century. it was found in experiments with electrons that the mass of a body depends on the speed of its movement, namely, it increases with increasing v according to law

where - rest mass, i.e. the mass of a material point, measured in that inertial frame of reference, relative to which the point is at rest; m- the mass of a point in the frame of reference, relative to which it moves with speed v.
From Einstein's principle of relativity, which asserts the invariance of all laws of nature in the transition from one inertial frame of reference to another, it follows that the fundamental law of Newton's dynamics

turns out to be invariant with respect to the Lorentz transformations if the derivative of relativistic momentum:

From the above formulas it follows that at speeds much lower than the speed of light in vacuum, they transform into the formulas of classical mechanics. Consequently, the condition for the applicability of the laws of classical mechanics is the condition. Newton's laws are obtained as a consequence of the STR for the limiting case. Thus, classical mechanics is the mechanics of macro-bodies moving at low (compared to the speed of light in vacuum) speeds.
Due to the homogeneity of space in relativistic mechanics, relativistic momentum conservation law: the relativistic momentum of a closed system of bodies is conserved, i.e. does not change over time.
A change in the speed of a body in relativistic mechanics entails a change in mass, and, consequently, in total energy, i.e. there is a relationship between mass and energy. This universal addiction law of relationship of mass and energy- established A. Einstein:

From (5.13) it follows that any mass (moving m or at rest) corresponds to a certain value of energy. If the body is at rest, then its rest energy

Rest energy is the internal energy of the body, which consists of the kinetic energies of all particles, the potential energy of their interaction and the sum of the rest energies of all particles.
In relativistic mechanics, the rest mass conservation law is not valid. It is on this concept that the explanation of the nuclear mass defect and nuclear reactions is based.
The service station carries out conservation law for relativistic mass and energy: a change in the total energy of a body (or system) is accompanied by an equivalent change in its mass:

Thus, the mass of a body, which in classical mechanics is a measure of inertia or gravity, in relativistic mechanics is also a measure of the energy content of a body.
The physical meaning of expression (5.14) is that there is a fundamental possibility of the transition of material objects that have a rest mass into electromagnetic radiation, which does not have a rest mass; in this case, the law of conservation of energy is fulfilled.
A classic example of this is the annihilation of an electron-positron pair and, conversely, the formation of an electron-positron pair from quanta of electromagnetic radiation:

In relativistic dynamics, the value of kinetic energy Ek is defined as the difference between the energies of the moving E and resting E 0 bodies:

For, equation (5.15) becomes the classical expression

From formulas (5.13) and (5.11) we find the relativistic relation between the total energy and momentum of the body:

The law of the relationship between mass and energy is fully confirmed by experiments on the release of energy during nuclear reactions. It is widely used to calculate the energy effect in nuclear reactions and transformations of elementary particles.

№30 Velocity distribution of molecules. Maxwell distribution

The velocity distribution of molecules is the functional dependence of the relative number of gas molecules on their velocity during thermal motion.

Maxwell distribution. Let us fix the values ​​of the velocities that the gas molecules currently possess, and then depict them in the velocity space. This is an ordinary three-dimensional space, but the axes of which are not spatial coordinates, but the projections of velocities in the corresponding directions (see Fig. 14.5). Due to the equality of all directions of motion, the location of points in this space will be spherically symmetric and should depend only on the modulus of velocity or the value of v2. The probability that molecules have a velocity in the range from v to v + dv will be equal to the ratio of the number of molecules with given velocities dNv to the total number of molecules N:

dPv = dNv / N. (14.23)

Based on the definition of the probability density, we have:

dNv / N = f (v) dV = f (v) 4  v2 dv, (14.24)
where dV is a volume element in the velocity space equal to the volume of the spherical layer (see Fig. 14.5).

Therefore, the probability that the molecules have a velocity in the range from v to v + dv can be calculated using the expression:

dPv = F (v) dv, (14.25)
where F (v) = f (v) · 4 · · v2 is the velocity distribution function of molecules.

Maxwell, proceeding from the assumption of the independence of the distribution of the projections of the velocity from its direction, received the form of the function F (v), called the Maxwell distribution function (see Fig. 14.6). (14.26) The form of the Maxwell function depends on the temperature and on the mass of the molecules. Note that the exponent is equal to the ratio of the kinetic energy of the molecule to the thermal energy (m · v2 / 2) / (k · T).

That. the higher the temperature, the more probable the growth of the number of molecules at high speeds becomes, the greater the mass of the molecule, the higher the temperature with the corresponding probability the molecule reaches a given speed.

The area under the curve in Fig. 14.6 is equal to the probability that the velocity of a molecule at a given temperature has an arbitrary value from zero to infinity is equal to 1. Knowing the expression for the Maxwell function, one can find the most probable, mean and root-mean-square velocities.

We suggest you get these expressions yourself. The average value of the velocity of gas molecules under normal conditions is about 103 m / s. Rice. 14.8. Experimental verification of the velocity distribution of molecules... One of the classical experiments confirming the presence of a velocity distribution of molecules is Stern's experience... A schematic of the experiment is shown in Fig. 14.7.

The installation consists of two coaxial (having one axis of symmetry) cylinders between which a vacuum was created. A platinum thread covered with silver is stretched along the axis of the cylinders. When an electric current was passed through it, the silver atoms evaporated. A slit was cut in the inner cylinder through which silver atoms penetrated onto the surface of the outer cylinder, leaving a trace on it in the form of a narrow vertical strip.

When the cylinders were brought into rotation with a constant angular velocity w, the trace left by the silver molecules was displaced and washed out (see Fig. 14.8). Indeed, the Coriolis force Fk acts on silver atoms in a non-inertial frame of reference associated with rotating cylinders

Fк = 2 m

This force deflects the silver atoms from straight propagation. The average displacement of atoms s is equal to:

s = w R t = w2 R / . (14.28)

By measuring the value of s from the experiment, proceeding from formula (14.28), one can find the average velocity of the molecules. Its value coincides with the theoretical value obtained using Maxwell's formula.

More precisely, the velocity distribution law of molecules was verified in Lammert's experiment .

48. Wetting. Capillary phenomena

It is known from practice that a drop of water spreads on glass and takes the form shown in Fig. 98, while mercury on the same surface turns into a somewhat flattened drop (Fig. 99). In the first case, they say that the liquid wets hard surface, in the second - does not wet her. Wetting depends on the nature of the forces acting between the molecules of the surface layers of the contacting media. For a wetting liquid, the forces of attraction between the molecules of the liquid and the solid are greater than between the molecules of the liquid itself, and the liquid tends to increase the surface of contact with the solid. For a non-wetting liquid, the forces of attraction between the molecules of the liquid and the solid are less than between the molecules of the liquid, and the liquid tends to reduce the surface of its contact with the solid.

To the line of contact of three media (point O there is its intersection with the plane of the drawing) three surface tension forces are applied, which are directed tangentially inside the contact surface of the corresponding two media (Fig. 98 and 99). These forces attributed to unit of length contact lines are equal to the corresponding surface

tension s12 , s 13, s23. The angle q between the tangents to the surface of the liquid and the solid is called edge angle. The condition for the equilibrium of the drop (Fig. 98) is the equality to zero of the sum of the projections of the surface tension forces on the direction of the tangent to the surface of the solid, that is,

S13 + s12 + s23 cosq = 0,

cosq = (s13 -s12) / s23. (67.1)

It follows from condition (67.1) that the contact angle can be acute or obtuse, depending on the values ​​of s13 and s12. If s13> s12, then cosq> 0 and the angle q is acute (Fig. 98), that is, liquid wets a solid surface. If s13

The contact angle satisfies condition (67.1) if

| s13 -s12 | / s23<1. (67.2)

If condition (67.2) is not satisfied, then the liquid drop 2 at no value can 6 be in equilibrium. If s13> s12 + s23, then the liquid spreads over the surface of the solid, covering it with a thin film (for example, kerosene on the glass surface), - there is complete wetting(in this case q = 0). If s12> s13 + s23, then the liquid contracts into a spherical drop, in the limit having only one point of contact with it (for example, a drop of water on the surface of paraffin), - takes place complete non-wetting(in this case q = p).

Wetting and non-wetting are relative concepts, that is, a liquid that wets one solid surface does not wet another. For example, water wets glass but does not wet paraffin; mercury does not wet glass, but it does wet clean metal surfaces.

Capillary phenomena

If you put a narrow tube (capillary) with one end into a liquid poured into a wide vessel, then due to wetting or non-wetting of the capillary walls by the liquid, the curvature of the liquid surface in the capillary becomes significant. If the liquid wets the material of the tube, then inside its surface of the liquid - meniscus- has a concave shape, if not wetting - convex (fig. 101).

A negative excess pressure will appear under the concave surface of the liquid, determined by the formula (68.2). The presence of this pressure leads to the fact that the liquid in the capillary rises, since there is no excess pressure under the flat surface of the liquid in a wide vessel. If the liquid does not wet the walls of the capillary, then the positive overpressure will lead to the lowering of the liquid in the capillary. The phenomenon of a change in the height of the liquid level in the capillaries is called capillarity. The liquid in the capillary rises or falls to such a height h , at which the pressure of the liquid column (hydrostatic pressure) r gh counterbalanced by excess pressure Dp, i.e.

where r is the density of the liquid, g- acceleration of free fall.

If m - capillary radius, q is the contact angle, then from Fig. 101 it follows that (2scosq) / r = r gh , where

h = (2scosq) / (rgr). (69.1)

In accordance with the fact that the wetting liquid rises through the capillary, and the non-wetting liquid goes down, from the form

mules (69.1) for q

0) we obtain positive values ​​of A, and for 0> p / 2 (cosq<0) -отрицательные. Из выражения (69.1) видно также, что высо­та поднятия (опускания) жидкости в ка­пилляре обратно пропорциональна его ра­диусу. В тонких капиллярах жидкость под­нимается достаточно высоко. Так, при полном смачивании (6 = 0) вода (r=1000 кг/м3, s=0,073 Н/м) в капилляре диаметром 10 мкм поднимается на высоту h»3 м.

38. Cyclic processes. Carnot's theorem

1. Working body (working agent) is called a thermodynamic system that performs a process and is designed to transform one form of energy transfer - heat or work - into another. For example, in a heat engine, the working fluid, receiving energy in the form of heat, transfers part of it in the form of work.
2. Heater (heat sink) is called a system that imparts energy to the considered thermodynamic system in the form of heat.
Refrigerator (heat sink) is called a system that receives energy from the considered thermodynamic system in the form of heat.
3. Circular processes are depicted in thermodynamic diagrams in the form of closed curves. The work against external pressure performed by the system in a reversible circular process is measured by the area bounded by the curve of this process in the V - p diagram.
Direct cycle is called a circular process in which the system does positive work: A> 0 . In the V - p diagram, the direct cycle is depicted as a closed curve traversed by the working fluid clockwise.
Reverse, cycle is called a circular process in which the work done by the system is negative A < 0. В диаграмме V - p обратный цикл изображается в виде замкнутой кривой, проходимой рабочим телом против часовой стрелки.
In a heat engine, the working fluid performs a direct cycle, and in a refrigerating machine, a reverse cycle.
4. Thermal (thermodynamic) efficiency(efficiency)  is the ratio of the thermal equivalent A of the work performed by the working fluid in the considered direct circular process to the sum Q1 of all the amounts of heat imparted to the working fluid by the heaters:

 = A / Q1 = (Q1 - Q2) / Q1

Where Q2 - the absolute value of the sum of the amounts of heat given by the working fluid to the refrigerators. Thermal efficiency characterizes the degree of perfection of the transformation of internal energy into mechanical energy that occurs in a heat engine that operates according to the cycle under consideration.
5. Carnot cycle is called a direct circular process (Fig. 1), consisting of two isothermal processes 1 - 1 "and 2 - 2" and two adiabatic processes 1 "- 2 and 2" - 1. In the process 1 - 1 ", the working fluid receives from the heater a quantity heat Q1 and in the process 2 - 2 "the working fluid gives the refrigerator the amount of heat Q2.

Fig. 1. Carnot cycle

Carnot's theorem: thermal K. and. the reversible Carnot cycle does not depend on the nature of the working fluid and is a function of only the absolute temperatures of the heater (T1) and cooler (T2):

 = (T1 - T2) / T1

40. The third law of thermodynamics

The value of the additive constant arising in the definition of entropy is established by the Nernst theorem, which is often called the third law of thermodynamics: the entropy of any system at absolute zero temperature can always be taken to be zero.

The physical meaning of the theorem is that for T= 0 all possible states of the system have the same entropy. Therefore, the state of the system at T= 0 it is convenient to take O as the initial state and set the entropy of this state equal to zero. Then the entropy of an arbitrary state A can be defined by the integral (63) where the integration is performed along a reversible process starting from the state at T= 0 and ending in state A.

In thermodynamics, Nernst's theorem is accepted as a postulate. It is proved by the methods of quantum statistics.

An important conclusion follows from the Nernst theorem about the behavior of the heat capacity of bodies at T→ 0. Consider the heating of a solid. When its temperature changes T on dT the body absorbs the amount of heat δ Q = C (T) dT, (64) where C (T) is its heat capacity. Therefore, according to definition (63), the entropy of a body at a temperature T can be presented in the form

It can be seen from this formula that if the heat capacity of the body at absolute zero, C(0) differed from zero, then integral (65) would diverge at the lower limit. Therefore, at T= 0 heat capacity should be zero: C(0) = 0 (66) This conclusion is in agreement with experimental data on the heat capacity of bodies at T→ 0. It should be noted that (66) refers not only to solids, but also to gases. The earlier statement that the heat capacity of an ideal gas does not depend on temperature is valid only for not too low temperatures. In this case, two circumstances must be borne in mind. 1. At low temperatures, the properties of any gas are very different from those of an ideal gas; near absolute zero, no substance is an ideal gas. 2. Even if an ideal gas could exist near zero temperature, then a rigorous calculation of its heat capacity by the methods of quantum statistics shows that it would tend to zero at T → 0.

15. Non-inertial frames of reference. Forces of inertia

Newton's laws are fulfilled only in inertial frames of reference. Reference frames moving with acceleration relative to the inertial frame are called non-inertial. In non-inertial systems, Newton's laws, generally speaking, are already unfair. However, the laws of dynamics can be applied to them, if, in addition to the forces caused by the action of bodies on each other, we introduce into consideration forces of a special kind - the so-called forces of inertia.

If we take into account the forces of inertia, then Newton's second law will be valid for any frame of reference: the product of the mass of a body and the acceleration in the considered frame of reference is equal to the sum of all forces acting on the given body (including the forces of inertia). Forces of inertia F in this case must be such that, together with the forces F caused by the action of bodies on each other, they impart acceleration to the body a"as it possesses in non-inertial frames of reference, ie.

m a " = F +F in. (27.1)

Because F= m a (a is the acceleration of the body in the inertial reference frame), then

m a"= m a +F in.

The forces of inertia are due to the accelerated motion of the frame of reference relative to the measured system, therefore, in the general case, the following cases of manifestation of these forces should be taken into account: 1) forces of inertia during accelerated translational motion of the frame of reference; 2) forces of inertia acting on a body at rest in a rotating frame of reference; 3) forces of inertia acting on a body moving in a rotating frame of reference.

Let's consider these cases.

1. Forces of inertia during accelerated translational motion of the frame of reference. Let a ball of mass be suspended on a trolley to a tripod on a thread T(fig. 40). As long as the cart is at rest or is moving evenly and in a straight line, the thread holding the ball is vertical and gravity R is balanced by the reaction of the thread T. If the carriage is set in translational motion with acceleration a 0, then the thread will begin to deviate from the vertical back to such an angle a, until the resulting force F =P +T will not provide the acceleration of the ball equal to a0. Thus, the resulting force F directed towards the acceleration of the cart a 0 and for the steady motion of the ball (the ball now moves with the trolley with acceleration a 0) is equal to

F = mg tga = ma0,

whence the angle of deflection of the thread from the vertical tga = a0 / g,

that is, the more, the greater the acceleration of the cart. The ball is at rest with respect to the frame of reference associated with an accelerated moving cart, which is possible if the force F balanced by an equal and opposite force F and, which is nothing more than the force of inertia, since no other forces act on the ball. Thus,

F and = -m a 0. (27.2)

The manifestation of inertial forces during translational motion is observed in everyday phenomena. For example, when the train picks up speed, the passenger sitting in the direction of the train is pressed against the back of the seat by the force of inertia. Conversely, when the train is braking, the inertia force is directed in the opposite direction and the passenger is separated from the seat back. These forces are especially noticeable during sudden train braking. The forces of inertia are manifested in the overloads that arise during the launch and deceleration of spaceships.

2. Forces of inertia acting on a body at rest in a rotating frame of reference. Let the disk rotate uniformly with an angular velocity w (w = const) around a vertical axis passing through its center. Pendulums are installed on the disk, at different distances from the axis of rotation (balls with a mass of m ). When the pendulums rotate together with the disk, the balls deviate from the vertical by a certain angle (Fig. 41).

In an inertial reference system, associated, for example, with the room where the disc is installed, the ball rotates uniformly around a circle with a radius R(distance from the point of attachment of the pendulum to the disk to the axis of rotation). Therefore, it is acted upon by a force equal to F = mw2 R and directed perpendicular to the axis of rotation of the disk. It is the resultant of gravity R and thread tension T: F = P + T , When the movement of the ball sets -

Xia, then F = mgtgalfa = mw2 R, whence tgalfa = w 2 R / g ,

i.e. the angles of deflection of the threads of the pendulums will be the greater, the greater the distance TO from the ball to the axis of rotation of the disk and the greater the angular velocity of rotation w.

The ball is at rest with respect to the frame of reference associated with the rotating disk, which is possible if the force F balanced by an equal and opposite force directed to it F and, which is nothing more than the force of inertia, since no other forces act on the ball. Force F c, called centrifugal force of inertia, is directed horizontally from the axis of rotation of the disk and is equal to

Fц = -mw2 R. (27.3)

Centrifugal forces of inertia are applied, for example, to passengers in moving vehicles at turns, pilots when performing aerobatics; centrifugal forces of inertia are used in all centrifugal mechanisms: pumps, separators, etc., where they reach enormous values. When designing rapidly rotating machine parts (rotors, aircraft propellers, etc.), special measures are taken to balance the centrifugal forces of inertia.

It follows from formula (27.3) that the centrifugal force of inertia acting on bodies in rotating frames of reference in the direction of the radius from the axis of rotation depends on the angular velocity of rotation and the frame of reference and radius R , but does not depend on the speed of bodies relative to rotating frames of reference. Consequently, the centrifugal force of inertia acts in rotating frames of reference on all bodies located at a finite distance from the axis of rotation, regardless of whether they are at rest in this frame (as we have assumed so far) or moving relative to it with some speed.

3. Forces of inertia acting on the body, moving in a rotating frame of reference. Let the ball weigh T moving at a constant speed v " along the radius of a uniformly rotating disk (v '= const, w = const, v "┴w). If the disk does not rotate, then the ball directed along the radius moves along a radial straight line and hits the point A, if the disk is brought into rotation in the direction indicated by the arrow, then the ball rolls along a curve 0V(Fig. 42, a), and its speed v " changes its direction relative to the disk. This is possible only if the ball is acted upon by a force perpendicular to the velocity v ".

In order to make the ball roll on a rotating disk along the radius, we use a rod rigidly fixed along the radius of the disk, on which the ball moves without friction evenly and rectilinearly with a velocity v "(Fig. 42, b). When the ball is deflected, the rod acts on it with a certain force F. Relative to the disk (rotating frame of reference), the ball moves uniformly and rectilinearly, which can be explained by the fact that the force F balanced by the inertial force applied to the ball F K perpendicular to the velocity v ". This force is called Coriolis force of inertia. It can be shown that the Coriolis force

Vector f k is perpendicular to the velocity vectors v "of the body and the angular velocity of rotation w of the reference frame in accordance with the right-hand screw rule.

The Coriolis force acts only on bodies moving relative to a rotating frame of reference, for example, relative to the Earth. Therefore, the action of these forces explains a number of phenomena observed on Earth. So, if the body moves in the northern hemisphere to the north (Fig. 43), then the Coriolis force acting on it, as follows from expression (27.4), will be directed to the right with respect to the direction of motion, that is, the body will deviate somewhat to the east ... If the body moves south. then the Coriolis force also acts to the right when viewed in the direction of motion, i.e., the body deviates to the west. Therefore, in the northern hemisphere, there is a stronger erosion of the right banks of the rivers; right rails of railway tracks on the movement of wear

move faster than the left, etc. Similarly, it can be shown that in the southern hemisphere the Coriolis force acting on moving bodies will be directed to the left with respect to the direction of motion.

Due to the Coriolis force, bodies falling on the Earth's surface are deflected to the east (at 60 ° latitude, this deflection should be 1 cm when falling from a height of 100 m). The Coriolis force is associated with the behavior of the Foucault pendulum, which at one time was one of the proofs of the rotation of the Earth. If this force did not exist, then the plane of oscillations of the pendulum swinging near the surface of the Earth would remain unchanged (relative to the Earth). The action of the Coriolis forces leads to the rotation of the vibration plane around the vertical direction.

(27.1), we get basic law of dynamics for non-inertial reference systems:

m a "=F +F and + F c + F K, where the forces of inertia are given by the formulas

(27.2) - (27.4).

35 Basic processes in ideal gas Isothermal process Boyle's law - Mariotte is valid for any gases, as well as their mixtures, for example, for air. Only at pressures several hundred times greater than atmospheric does the deviation from this law become significant. The dependence of gas pressure on volume at constant temperature is graphically depicted by a curve called isotherm. Isothermagas shows an inversely proportional relationship between pressure and volume. A curve of this kind in mathematics is called hyperbola. Isobaric process This law was established experimentally in 1802 by the French scientist J. Gay-Lussac (1778 - 1850) and is called the Gay-Lussac law. According to the equation, the volume of gas linearly depends on temperature at constant pressure: V = const T. This dependence is graphically depicted by a straight line, which is called an isobar. Different isobars correspond to different pressures. With increasing pressure, the volume of gas at a constant temperature decreases according to the Boyle-Mariotte law. Therefore, the isobar corresponding to the higher pressure p2 lies below the isobar corresponding to the lower pressure p1. At low temperatures, all ideal gas isobars converge at the point T = 0. But this does not mean that the volume of real gas really vanishes. All gases turn into a liquid upon strong cooling, and the equation of state is inapplicable to liquids. Isobaric expansion of a gas can be considered when it is heated in a cylinder with a movable piston. The constant pressure in the cylinder is ensured by atmospheric pressure on the outer surface of the piston. Isochoric process This gas law was established in 1787 by the French physicist J. Charles (1746 - 1823) and is called Charles's law. According to the equation = const at V = const, the gas pressure linearly depends on the temperature at a constant volume: p = const T. This dependence is depicted by a straight line, called an isochore. Different isochores correspond to different volumes. With an increase in the volume of gas at a constant temperature, its pressure decreases according to the Boyle-Mariotte law. Therefore, the isochore corresponding to a larger volume V2 lies below the isochore corresponding to a smaller volume V1. According to the equation, all isochores start at the point T = 0, which means that the pressure of an ideal gas at absolute zero is zero. The increase in gas pressure in any container or in a light bulb during heating is an isochoric process. The isochoric process is used in constant volume gas thermostats.

Isoprocess is called the process that occurs with a given mass of gas at one constant parameter - temperature, pressure or volume. Laws for isoprocesses are obtained from the equation of state as special cases.
Isothermal is called a process that takes place at a constant temperature. T = const. It is described by the Boyle-Mariotte law: pV = const.
Isochorny is called a process that occurs at a constant volume. Charles's law is valid for him: V = const, p / T = const.
Isobaric is called a process that takes place at constant pressure. The equation of this process has the form V / T = const at p = const and is called the Gay-Lussac law. All processes can be represented graphically (Fig. 15).
Real gases satisfy the equation of state of an ideal gas at not too high pressures (as long as the intrinsic volume of molecules is negligible in comparison with the volume of the vessel,

In which the gas is located) and at not too low temperatures (while the potential energy of intermolecular interaction can be neglected in comparison with the kinetic energy of the thermal motion of molecules), i.e., for a real gas, this equation and its consequences are a good approximation.

41. THERMODYNAMIC POTENTIALS, functions status parameters macroscopic systems (t-ry T, pressure R, volume V, entropy S, the number of moles of the components ni, chem. potentials of components m, etc.), which are mainly used to describe thermodynamic equilibrium. To each thermodynamic potentials the set of state parameters corresponds. called natural variables. The most important thermodynamic potentials: internal energy U(natural variables S, V, ni); enthalpy H = U - (- pV) (natural variables S, p, ni); Helmholtz energy (Helmholtz free energy, Helmholtz function) F = = U - TS(natural variables V, T, ni); Gibbs energy (Gibbs free energy, Gibbs f-tion) G = U - - TS - (- pV) (natural variables p, T, ni); great thermodynamic. potential (natural variables V, T, mi). thermodynamic potentials can be represented by a common f-loy

where Lk- intensive parameters. mass-independent systems (these are T, p, m i), Xk - extensive parameters proportional to the mass of the system ( V, S, ni). Index l= 0 for internal energy U, 1-for H and F, 2-for G and W. thermodynamic potentials are f-tions of the state of a thermodynamic system, i.e. their change in any process of transition between two states is determined only by the initial and final states and does not depend on the path of the transition. Full differentials thermodynamic potentials look like:

Ur-nie (2) called. fundamental ur-ni Gibbs in energetic. expression. Everything thermodynamic potentials have the dimension of energy. Equilibrium conditions thermodynamic. systems are formulated as the equality to zero of the total differentials thermodynamic potentials with the constancy of the corresponding natural variables:

Thermodynamic. the stability of the system is expressed by the inequalities:

Decrease thermodynamic potentials in an equilibrium process with constancy of natural variables is equal to the maximum useful work of the process A :

At the same time, work A produced against any generalized force Lk acting on the system, except for ext. pressure (see. Maximum reaction work). thermodynamic potentials, taken as functions of their natural variables, are characteristic functions of the system. This means that any thermodynamic. property (compressibility, heat capacity, etc.) m b. expressed by a ratio including only this thermodynamic potentials, its natural variables and derivatives thermodynamic potentials of different orders in natural variables. In particular, using thermodynamic potentials the equations of state of the system can be obtained. Derivatives have important properties thermodynamic potentials The first partial derivatives with respect to natural extensive variables are equal to intensive variables, for example:

[in general: ( 9 Y l /9Xi)= Li]. Conversely, derivatives with respect to natural intensive variables are equal to extensive variables, for example:

[in general: ( 9 Y l /9Li)= Xi]. The second partial derivatives with respect to natural variables define fur. and thermal. system properties, for example:

Because differentials thermodynamic potentials are complete, cross second partial derivatives thermodynamic potentials are equal, for example for G (T, p, ni):

Relations of this type are called Maxwell's relations. thermodynamic potentials can also be represented as functions of variables other than natural ones, for example G (T, V, ni), but in this case the properties thermodynamic potentials as characteristic. functions will be lost. In addition to thermodynamic potentials characteristic f-tions are entropy S(natural variables U, V, ni), f-tion Massier F1 = (natural variables 1 / T, V ,ni), the Planck function (natural variables 1 / T, p / T, ni). thermodynamic potentials are interconnected by the Gibbs-Helmholtz equations. For example, for H and G

In general:

thermodynamic potentials are homogeneous functions of the first degree of their natural extensive variables. For example, with increasing entropy S or the number of moles ni the enthalpy increases proportionally N. According to Euler's theorem, the homogeneity thermodynamic potentials leads to relations of the type:

№5 Types of forces in mechanics The law of universal gravitation. Gravity. Body weight. Weightlessness.

Isaac Newton put forward the assumption that there are forces of mutual attraction between any bodies in nature. These forces are called the forces of gravity, or the forces of gravity. The force of universal gravity manifests itself in Space, the Solar System and on Earth. Newton generalized the laws of motion of celestial bodies and found out

That the force F is equal to:

The masses of the interacting bodies, R is the distance between them, G is the proportionality coefficient, which is called the gravitational constant. The numerical value of the gravitational constant was experimentally determined by Cavendish by measuring the force of interaction between lead balls. As a result, the law of universal gravitation sounds like this: between any material points there is a force of mutual attraction, directly proportional to the product of their masses and inversely proportional to the square of the distance between them, acting along the line connecting these points.
A particular type of the force of universal gravity is the force of attraction of bodies to the Earth (or to another planet). This force is called gravity. Under the influence of this force, all bodies acquire the acceleration of free fall. In accordance with Newton's second law, g = Ft * m, therefore, Ft = mg. The force of gravity is always directed towards the center of the earth. Depending on the height h above the Earth's surface and the geographic latitude of the position of the body, the acceleration of gravity takes on different values. On the surface of the Earth and in mid-latitudes, the acceleration due to gravity is 9.831 m / s2.
In technology and everyday life, the concept of body weight is widely used. The weight of the body is the force with which the body presses on the support or suspension as a result of gravitational attraction to the planet (Fig. 6). The weight of a body is denoted by R. The unit of weight is N. Since weight is equal to the force with which the body acts on the support, in accordance with Newton's third law, the weight of the body is equal to the reaction force of the support. Therefore, in order to find the body weight, it is necessary to determine what the reaction force of the support is equal to.

Elastic forces During deformations of a solid, its particles (atoms, molecules, ions) located in the nodes of the crystal lattice are displaced from their equilibrium positions. This displacement is counteracted by the forces of interaction between the particles of the solid, which keep these particles at a certain distance from each other. Therefore, for any type of elastic deformation, internal forces arise in the body that prevent its deformation. The forces arising in the body during its elastic deformation and directed against the direction of displacement of the body particles caused by deformation are called elastic forces. Elastic forces act in any section of a deformed body, as well as in the place of its contact with the body, causing deformations. In the case of unilateral stretching or compression, the elastic force is directed along a straight line along which an external force acts, causing deformation of the body, opposite to the direction of this force and perpendicular to the surface of the body. The nature of elastic forces is electrical Friction forces. Considering the forces so far, we have not been interested in their origin. However, various forces act in mechanical processes: friction, elasticity, gravity. Consider the forces of friction. It is known from experience that any body moving along the horizontal surface of another body, in the absence of other forces acting on it, slows down its motion over time and eventually stops. From a mechanical point of view, this can be explained by the existence of some force that impedes movement. This is the friction force - a resistance force directed opposite to the relative displacement of a given body and applied tangentially to the contacting surfaces. Static friction force. It is determined by the projection of the resultant force on the direction of the contacting surfaces. Increases in proportion to this force until movement begins. The graph of the dependence of the friction force on the projection of the resultant force is as follows. Internal friction is the friction between parts of the same body, for example, between different layers of liquid or gas, the velocities of which vary from layer to layer.

Unlike external friction, there is no static friction here. If the bodies slide relative to each other and are separated by a layer of viscous liquid (lubricant), then friction occurs in the lubricant layer. In this case, one speaks of hydrodynamic friction (the lubricant layer is quite thick) and boundary friction (the thickness of the lubricant layer is ~ 0.1 μm or less). Let's consider some regularities of external friction. This friction is due to the roughness of the contacting surfaces, while in the case of very smooth surfaces, the friction is due to the forces of intermolecular attraction.

Consider a body lying on a plane (figure), to which a horizontal force is applied. The body will begin to move only when the applied force is greater than the friction force. French physicists G. Amonton and S. Coulomb experimentally established the following law: the force Ffr of sliding friction is proportional to the force N of normal pressure:

Ftr = f N, where f is the sliding friction coefficient depending on the properties of the contacting surfaces.

A rather radical way to reduce the friction force is to replace sliding friction with rolling friction (ball and roller bearings, etc.). The rolling friction coefficient is tens of times less than the sliding friction coefficient. The rolling friction force is determined by Coulomb's law:

The radius of the rolling body, fк is the rolling friction coefficient having dimension = L. From this formula it follows that the rolling friction force is inversely proportional to the radius of the rolling body.

Postulates of the special theory of relativity.
Lorentz transformations Special theory of relativity is a modern physical theory of space and time. In SRT, as in classical mechanics, it is assumed that time is homogeneous (invariance of physical laws with respect to the choice of the origin of time), and space is homogeneous and isotropic (symmetric). The special theory of relativity is also called the relativistic theory, and the phenomena described by this theory are called relativistic effects.
SRT is based on the position that no energy, no signal can propagate at a speed exceeding the speed of light in a vacuum, and the speed of light in a vacuum is constant and does not depend on the direction of propagation.
This position is formulated in the form of two postulates of A. Einstein: the principle of relativity and the principle of the constancy of the speed of light.
The first postulate is a generalization of Galileo's mechanical principle of relativity to any physical processes and asserts that the laws of physics have the same form (invariant) in all inertial reference frames: any process proceeds the same way in an isolated material system in a state of rest, and in the same system, in a state of uniform rectilinear motion. The state of rest or motion is defined here with respect to an arbitrarily chosen inertial frame of reference; physically these states are equal.
The second postulate states: the speed of light in a vacuum does not depend on the speed of movement of the light source or the observer and is the same in all inertial reference frames.

The analysis of phenomena in inertial reference frames, carried out by A. Einstein on the basis of the postulates formulated by him, showed that Galileo's transformations are incompatible with them and, therefore, must be replaced by transformations that satisfy the SRT postulates.
Consider two inertial reference systems: K (with coordinates x, y, z) and K΄ (with coordinates x΄, y΄, z΄), moving relative to K along the x-axis with velocity = const. Let at the initial moment of time (t = t΄ = 0), when the origin of the coordinate systems coincide (0 = 0΄), a light pulse is emitted. According to Einstein's second postulate, the speed of light in both systems is the same and is equal to c. Therefore, if in time t in frame K the signal reaches some point A, having passed the distance

then in the K΄ system the coordinate of the light pulse at the moment of reaching point A will be equal to

where t΄ is the transit time of a light pulse from the origin to point A in the K΄ system. Subtracting (5.6) from (5.7), we get:

Since (the K΄ system moves relative to K), it turns out that, i.e. the timing in the K΄ and K systems is different or has a relative character(in classical mechanics, it is believed that time in all inertial frames of reference flows in the same way, i.e. t = t΄).
A. Einstein showed that in SRT, the classical Galilean transformations in the transition from one inertial frame of reference to another are replaced by the Lorentz transformations (1904), which satisfy the first and second postulates

It follows from the Lorentz transformations that at low speeds (compared to the speed of light) they transform into Galileo transformations. For v> c, expressions for x, t, x΄, and t΄ lose their physical meaning, i.e. movement with a speed greater than the speed of light in a vacuum is impossible. In addition, from table. 5.1 it follows that both spatial and temporal Lorentz transformations are not independent: time enters the coordinate transformation law, and spatial coordinates enter the time transformation law, i.e. the relationship between space and time is established. Thus, Einstein's relativistic theory does not operate with a three-dimensional space, to which the concept of time is added, but considers inextricably linked spatial and temporal coordinates that form a four-dimensional space-time.

34 Specific heat body (denoted by C) is a physical quantity that determines the ratio of the infinitesimal amount of heat ΔQ received by the body to the corresponding increment of its temperature ΔT:

The unit for measuring the heat capacity in the SI system is J / K. Specific heat of the substance is the heat capacity of a unit mass of a given substance. Units of measurement - J / (kg K). Molar heat capacity of a substance- heat capacity of 1 mole of a given substance. Units of measurement - J / (mol K). If we talk about the heat capacity of an arbitrary system, then it is appropriate to formulate it in terms of thermodynamic potentials - the heat capacity is the ratio of a small increase in the amount of heat Q to a small change in temperature T:

The concept of heat capacity is defined both for substances in various states of aggregation (solids, liquids, gases) and for ensembles of particles and quasiparticles (in the physics of metals, for example, they speak of the heat capacity of an electron gas). If we are not talking about any body, but about some substance as such, then we distinguish between specific heat capacity - the heat capacity of a unit mass of this substance and molar - the heat capacity of one mole of it. For example, in the molecular kinetic theory of gases, it is shown that the molar heat capacity of an ideal gas with i degrees of freedom at constant volume is equal to:

R = 8.31 J / (mol K) - universal gas constant. And at constant pressure The specific heat capacities of many substances are given in reference books, usually for a process at constant pressure. For example, the specific heat of liquid water under normal conditions is 4200 J / (kg K). Ice - 2100 J / (kg K) There are several theories of the heat capacity of a solid: 1) Dulong-Petit law and Joule-Kopp's law. Both laws are derived from classical concepts and with a certain accuracy are valid only for normal temperatures (approximately from 15 ° C to 100 ° C). 2) Quantum theory of heat capacities of Einstein. The first very successful attempt to apply quantum laws to the description of heat capacity. 3) Debye's quantum theory of heat capacities. Contains the most complete description and agrees well with the experiment. The heat capacity of a system of non-interacting particles (for example, a gas) is determined by the number of degrees of freedom of the particles.

# 21 Galileo's principle of relativity The laws of nature that determine the change in the state of motion of mechanical systems do not depend on which of the two inertial frames of reference they refer to. That's what it is Galileo's principle of relativity... From Galileo's transformations and the principle of relativity, it follows that interactions in classical physics should be transmitted with an infinitely high speed c = ∞, since otherwise one inertial reference system could be distinguished from another by the nature of the physical processes in them.
The fact is that principle relativity Galileo allows you to distinguish between absolute and relative motion. This is possible only within the framework of a certain interaction in a system consisting of two bodies. If extraneous interactions do not interfere in an isolated (quasi-isolated) system of two bodies interacting with each other, or there are interactions that can be neglected, then their movements can be considered absolute with respect to their center of gravity. Such systems can be considered the Sun - the planets (each separately), the Earth - the Moon, etc. And, moreover, if the center of gravity of the interacting bodies practically coincides with the center of gravity of one of the bodies, then the motion of the second body can be considered absolute in relation to the first. So, the center of gravity can be taken as the beginning of the absolute reference frame of the solar system Suns and the motions of the planets are considered absolute. And then: the Earth revolves around the Sun, but not the Sun around Of the earth(remember J. Bruno), a stone falls on the Earth, but not the Earth on a stone, etc. Galileo's principle of relativity and Newton's laws were confirmed hourly when considering any motion, and dominated physics for over 200 years.
But in 1865 J. Maxwell's theory appeared, and Maxwell's equations did not obey Galileo's transformations. Few people accepted her immediately, she did not receive recognition during Maxwell's life. But soon everything changed a lot, when in 1887, after the discovery of electromagnetic waves by Hertz, all the consequences arising from Maxwell's theory were confirmed - it was recognized. Numerous works have appeared that develop Maxwell's theory.
The fact is that in Maxwell's theory the speed of light (the speed of propagation of electromagnetic waves) is finite and equal to c = 299792458 m / s. (Based on Galileo's principle of relativity, the speed of signal transmission is infinite and depends on the frame of reference z = z ’). The first guesses about the finiteness of the propagation of the speed of light were expressed by Galileo. Astronomer Roemer in 1676 tried to find the speed of light. According to his approximate calculations, it was equal to c = 214300000 m / s.
An experimental test of Maxwell's theory was needed. He himself proposed the idea of ​​experience - to use the Earth as a moving system. (It is known that the speed of the Earth's movement is relatively high :).

In the 80s of the XIX century, experiments were carried out that proved the independence of the speed of light from the speed of the source or observer.
The device necessary for the experiment was invented by the brilliant US naval officer A. Michelson (Fig. 8.3).

The device consisted of an interferometer with two "arms" located perpendicular to each other. Due to the relatively high speed of the Earth's movement, light had to have different speeds in the vertical and horizontal directions. Therefore, the time spent on the passage of the vertical path of the source S - the semitransparent mirror (sr) - mirror (s1) - (ns) and the horizontal path of the source - (ns) - mirror (s2) - (ns) should be different. As a result, the light waves, having passed the indicated paths, should have changed the interference pattern on the screen.

Rice. 8.3

Michelson conducted experiments for seven years from 1881 in Berlin and from 1887 in the United States together with the chemist Professor Morley. The accuracy of the first experiments was low: ± 5 km / s. However, the experiment gave a negative result: it was not possible to detect a shift in the interference pattern. Thus, the results of the Michelson-Morley experiments showed that the magnitude of the speed of light is constant and does not depend on the movement of the source and the observer. These experiments were repeated and rechecked many times. At the end of the 60s, C. Townes brought the measurement accuracy to ± 1 m / s. The speed of light remained unchanged c = 3 · 108 m / s. The independence of the speed of light from the movement of the source and from the direction was recently demonstrated with record accuracy in experiments carried out by researchers from the Universities of Konstanz and Dusseldorf (modern version of the Michelson – Morley experiment), in which the best to date accuracy of 1.7 × 1015 was established. This accuracy is 3 times higher than previously achieved. A standing electromagnetic wave was investigated in the cavity of a sapphire crystal cooled by liquid helium. Two such resonators were oriented at right angles to each other. The entire installation could rotate, which made it possible to establish the independence of the speed of light from direction. There have been many attempts to explain the negative result of the Michelson-Morley experiment. The most famous hypothesis of Lorentz about the reduction of the size of bodies in the direction of motion. He even calculated these cancellations using a coordinate transformation called "Lorentz-Fitzgerald cancellations". J. Larmor in 1889 proved that Maxwell's equations are invariant under the Lorentz transformations. Henri Poincaré was very close to the creation of the theory of relativity. But Albert Einstein was the first to articulate the basic ideas of the theory of relativity clearly and clearly.

27,28,29 Ideal gas, average energy of molecules, gas pressure on the wall An ideal gas is a mathematical model of a gas, in which it is assumed that the potential energy of molecules can be neglected in comparison with their kinetic energy. There are no forces of attraction or repulsion between molecules, collisions of particles with each other and with the walls of the vessel are absolutely elastic, and the interaction time between molecules is negligible compared to the average time between collisions. Distinguish between a classical ideal gas (its properties are derived from the laws of classical mechanics and are described by the Boltzmann statistics) and a quantum ideal gas (the properties are determined by the laws of quantum mechanics, described by the Fermi - Dirac or Bose - Einstein statistics). Classical ideal gas The properties of an ideal gas based on molecular kinetic representations are determined based on a physical model of an ideal gas, in which the following assumptions are made: 1) the volume of a gas particle is zero (that is, the diameter of a molecule d is negligible compared to the average distance between them,) ; 2) momentum is transmitted only during collisions (that is, the forces of attraction between molecules are not taken into account, and repulsive forces arise only during collisions); 3) the total energy of the gas particles is constant (that is, there is no energy transfer due to the transfer of heat or radiation) In this case, the gas particles move independently of each other, the gas pressure on the wall is equal to the sum of impulses per unit time transferred when the particles collide with the wall, the energy - the sum of the energies of the gas particles. The properties of an ideal gas are described by the Mendeleev - Clapeyron equation

where p is the pressure, n is the concentration of particles, k is the Boltzmann constant, and T is the absolute temperature. The equilibrium distribution of particles of a classical ideal gas over states is described by the Boltzmann distribution:

where is the average number of particles in the jth state with energy, and the constant a is determined by the normalization condition:

Where N is the total number of particles. The Boltzmann distribution is the limiting case (quantum effects are negligible) of the Fermi - Dirac and Bose - Einstein distributions, and, accordingly, the classical ideal gas is the limiting case of the Fermi gas and Bose gas. For any ideal gas, the Mayer relation is valid:

where R is the universal gas constant, Cp is the molar heat capacity at constant pressure, Cv is the molar heat capacity at constant volume. Ideal gas equation of state(sometimes clapeyron equation or Clapeyron - Mendeleev equation) is a formula that establishes the relationship between pressure, molar volume and the absolute temperature of an ideal gas. The equation is:

where p is pressure, Vm is molar volume, T is absolute temperature, R is universal gas constant. Since, where is the amount of substance, and, where m is the mass, is the molar mass, the equation of state can be written:

This form of notation is named after the Mendeleev-Clapeyron equation (law). In the case of constant gas mass, the equation can be written as:

p * V / T = vR, p * V / T = const

The last equation is called unified gas law... From it the laws of Boyle - Mariotte, Charles and Gay-Lussac are obtained: T = const => P * V = const- Boyle's law - Mariotte .

P = const => V / T = const- law Gay - Lussac .

V = const => P / T = const- law Charles(Gay-Lussac's second law, 1808)

From the point of view of a chemist, this law may sound a little differently: The volumes of gases that enter into the reaction under the same conditions (temperature, pressure) relate to each other and to the volumes of the formed gaseous compounds as simple integers.

In some cases (in gas dynamics), the equation of state for an ideal gas can be conveniently written in the form

where is the adiabatic exponent, is the internal energy of a unit mass of a substance. On the one hand, in highly compressed gases, the sizes of the molecules themselves are comparable to the distances between the molecules. Thus, the free space in which the molecules move is less than the total volume of the gas. This circumstance increases the number of impacts of molecules on the wall, since it reduces the distance that a molecule must fly to reach the wall.

On the other hand, in a highly compressed and therefore denser gas, molecules are noticeably attracted to other molecules much more of the time than molecules in a rarefied gas. This, on the contrary, reduces the number of impacts of molecules on the wall, since in the presence of attraction to other molecules, gas molecules move towards the wall at a lower speed than in the absence of attraction. At not too high pressures. the second circumstance is more significant and the work is slightly reduced. At very high pressures, the first circumstance plays an important role and the product P * V increases.

Is the average kinetic energy of gas molecules (per molecule). at thermal equilibrium, the average kinetic energy of the translational motion of the molecules of all gases is the same. The pressure is directly proportional to the average kinetic energy of the translational motion of the molecules:
In thermal equilibrium, if the pressure of a gas of a given mass and its volume are fixed, the average kinetic energy of gas molecules must have a strictly defined value, like the temperature. The quantity
grows with increasing temperature and does not depend on anything other than temperature. Therefore, it can be considered a natural measure of temperature. The average kinetic energy of the translational motion of molecules is:

T is the temperature on the Kelvin scale, k is the Boltzmann constant, k = 1.4 * 10-23 J / K. A quantity proportional to the average kinetic energy of the translational motion of particles is called body temperature :

Where k= 1.38 * 10-23 J / K - Boltzmann's constant. Temperature is a measure of the average kinetic energy of molecules. From this it can be seen that. The temperature determined in this way is called thermodynamic or absolute, it is measured in Kelvin (K).

33 The first law of thermodynamics In fig. 3.9.1 conventionally depicts energy flows between the selected thermodynamic system and surrounding bodies. The value of Q> 0, if the heat flux is directed towards the thermodynamic system. The value A> 0 if the system does positive work on the surrounding bodies.

Figure 3.9.1.

Energy exchange between a thermodynamic system and surrounding bodies as a result of heat exchange and work performed.

If the system exchanges heat with surrounding bodies and performs work (positive or negative), then the state of the system changes, that is, its macroscopic parameters (temperature, pressure, volume) change. Because internal energy U is unambiguously determined by the macroscopic parameters characterizing the state of the system, then it follows that the processes of heat exchange and performance of work are accompanied by a change in ΔU of the internal energy of the system.

The first law of thermodynamics is a generalization of the law of conservation and transformation of energy for a thermodynamic system. It is formulated as follows:

The change ΔU of the internal energy of a non-isolated thermodynamic system is equal to the difference between the amount of heat Q transferred to the system and the work A performed by the system over external bodies. ΔU = Q - A.

The relationship expressing the first law of thermodynamics is often written in a different form: Q = ΔU + A.

The amount of heat received by the system is used to change its internal energy and work on external bodies.

The first law of thermodynamics is a generalization of experimental facts. According to this law, energy cannot be created or destroyed; it is transferred from one system to another and changes from one form to another. An important consequence of the first law of thermodynamics is the statement about the impossibility of creating a machine capable of performing useful work without the consumption of energy from the outside and without any changes inside the machine itself. Such a hypothetical machine was called a perpetual motion machine ( perpetuum mobile) first kind... Numerous attempts to create such a machine invariably ended in failure. Any machine can do positive work A on external bodies only by receiving a certain amount of heat Q from the surrounding bodies or by decreasing ΔU of its internal energy.

Let us apply the first law of thermodynamics to isoprocesses in gases. V isochoric process(V = const) the gas does not work, A = 0. Therefore, Q = ΔU = U (T2) - U (T1). Here U (T1) and U (T2) are the internal energies of the gas in the initial and final states. The internal energy of an ideal gas depends only on temperature (Joule's law). With isochoric heating, heat is absorbed by the gas (Q> 0), and its internal energy increases. When cooled, heat is transferred to external bodies (Q< 0). В isobaric process(p = const) the work done by the gas is expressed by the relation A = p (V2 - V1) = pΔV. The first law of thermodynamics for an isobaric process gives: Q = U (T2) - U (T1) + p (V2 - V1) = ΔU + pΔV. With isobaric expansion Q> 0, heat is absorbed by the gas, and the gas does positive work. At isobaric compression Q< 0 – тепло отдается внешним телам. В этом случае A < 0. Температура газа при изобарном сжатии уменьшается, T2 < T1; внутренняя энергия убывает, ΔU < 0. В isothermal process the gas temperature does not change, therefore, the internal energy of the gas does not change, ΔU = 0. The first law of thermodynamics for the isothermal process is expressed by the relation Q = A. The amount of heat Q received by the gas in the process of isothermal expansion turns into work on external bodies. With isothermal compression, the work of external forces produced on the gas turns into heat, which is transferred to the surrounding bodies. Along with isochoric, isobaric and isothermal processes in thermodynamics, processes are often considered that occur in the absence of heat exchange with surrounding bodies. Vessels with heat-resistant walls are called adiabatic shells, and the processes of expansion or contraction of gas in such vessels are called adiabatic... V adiabatic process Q = 0; therefore, the first law of thermodynamics takes the form A = –ΔU, that is, the gas does work due to the loss of its internal energy. In thermodynamics, the equation of the adiabatic process for an ideal gas is derived. In coordinates (p, V), this equation has the form pVγ = const. This ratio is called Poisson's equation. 37 entropy entropy(from the Greek εντροπία - turn, turn) - a concept that first appeared in thermodynamics as a measure of irreversible energy dissipation; it is widely used in other areas: in statistical mechanics - as a measure of the probability of the system state realization; in information theory - as a measure of the uncertainty of messages; in probability theory - as a measure of the uncertainty of experience, tests with different outcomes; its alternative interpretations have a deep internal connection: for example, all the most important provisions of statistical mechanics can be derived from probabilistic concepts of information. In thermodynamics, the concept of entropy was introduced by the German physicist R. Clausis (1865), when he showed that the process of converting heat into work obeys regularities - the second law of thermodynamics, which is formulated strictly mathematically, if we introduce the function of the state of the system - entropy... Clausis also showed the importance of the concept entropy for the analysis of irreversible (nonequilibrium) processes, if the deviations from the thermodynamics of equilibrium are small and it is possible to introduce the concept of local thermodynamic equilibrium in small, but still macroscopic volumes. Generally entropy non-equilibrium system is equal to the sum entropy its parts that are in local equilibrium. In statistical mechanics Statistical mechanics connects entropy with the probability of the macroscopic state of the system being realized by the famous Boltzmann relation "entropy - probability" S = kB ln W, where W is the thermodynamic probability of the realization of a given state (the number of ways in the realization of the state), and kB is the Boltzmann constant. Unlike thermodynamics, statistical mechanics considers a special class of processes - fluctuations, in which the system passes from more probable states to less probable and, as a result, its entropy decreases. The presence of fluctuations shows that the law of increase entropy performed only statistically: on average for a long period of time. The adiabatic process can also be referred to as isoprocesses. In thermodynamics, an important role is played by a physical quantity called entropy (see §3.12). The change in entropy in any quasi-static process is equal to the reduced heat ΔQ / T obtained by the system. Since at any part of the adiabatic process ΔQ = 0, the entropy in this process remains unchanged. An adiabatic process (like other isoprocesses) is a quasi-static process. All intermediate states of the gas in this process are close to the states of thermodynamic equilibrium (see §3.3). Any point on the adiabat describes the equilibrium state. Not every process carried out in an adiabatic shell, that is, without heat exchange with surrounding bodies, satisfies this condition. An example of a non-quasi-static process in which intermediate states are nonequilibrium is the expansion of a gas into a void. In fig. 3.9.3 shows a rigid adiabatic shell, consisting of two communicating vessels separated by a valve K. In the initial state, the gas fills one of the vessels, and in the other vessel - a vacuum. After opening the valve, the gas expands, fills both vessels, and a new equilibrium state is established. In this process, Q = 0, since there is no heat exchange with the surrounding bodies, and A = 0, because the shell is not deformable. From the first law of thermodynamics follows: ΔU = 0, that is, the internal energy of the gas remains unchanged. Since the internal energy of an ideal gas depends only on temperature, the temperatures of the gas in the initial and final states are the same - the points on the plane (p, V) representing these states lie on one isotherm... All intermediate gas states are nonequilibrium and cannot be plotted on a diagram. Expansion of a gas into a void - an example irreversible process. It cannot be swiped in the opposite direction.