From the previous chapter we learned that the mass of a body increases as its speed increases. But we did not provide any evidence of this, similar to the reasoning with the clock that we used to justify time dilation. Now, however, we can prove that (as a consequence of the principle of relativity and other reasonable considerations) mass should change in exactly this way. (We have to talk about “other considerations” for the reason that nothing can be proven, nothing can be hoped for in a meaningful way, without relying on some laws that are assumed to be true.) In order not to study

laws of force transformation, let us turn to collisions particles. Here we do not need the law of action of force, but only the assumption of conservation of energy and momentum will suffice. In addition, we will assume that the momentum of a moving particle is a vector always directed along its motion. But we won't count the momentum proportional speed, as Newton did. For us it will just be some function speed. We will write the momentum vector in the form of a velocity vector multiplied by a certain coefficient

p=m 0 v . (16.8)

Index v the coefficient will remind us that it is a function of speed v. We will call this coefficient “mass”. It is clear that at low speeds this is exactly the same mass that we are used to measuring. Now, based on the principle that the laws of physics are the same in all coordinate systems, let’s try to show that the formula for m v should have the form m 0 /(1- v 2 /c 2 ).

Let us have two particles (for example, two protons), which are completely identical to each other and move towards each other at the same speeds. Their total momentum is zero. What will happen to them? After the collision, their directions of motion should still remain opposite, because if this is not the case, then their total momentum vector will be non-zero, i.e., will not be conserved. Since the particles are the same, then their speeds must be the same; Moreover, they simply must remain the same, otherwise the energy during the collision will change. This means that the diagram of such an elastic reversible collision will look like in Fig. 16.2a: all arrows are the same, all speeds are equal. Let us assume that such collisions can always be prepared, that any angles of 0 are permissible in them, and that the initial velocities of the particles can be any.

Fig. 16.2. Elastic collision identical bodies moving at equal speeds in opposite directions, with different choices of coordinate systems.

Next, recall that the same collision looks different depending on how the axes are rotated. For convenience, we will rotate the axes so that the horizontal bisects the angle between the directions of the particles before and after the collision (Fig. 16.2b). This is the same collision as in Fig. 16.2,a, but with rotated axes.

T Now comes the most important thing: let's look at this collision from the position of an observer moving in a car with a speed that coincides with the horizontal component of the speed of one of the particles. What will it look like? It will seem to the observer that the particle 1 rises straight up (its horizontal component has disappeared), and after a collision falls straight down for the same reason (Fig. 16.3, A).

Fig. 16.3. Two more pictures of the same collision (visible from moving cars).

But the particle 2 moves completely differently, it rushes past with colossal speed and at a small angle (but this angle both before and after the collision is the same). Let us denote the horizontal component of the particle velocity 2 through And, and the vertical velocity of the particle 1 - through w.

What is the vertical speed utg of particle 2? Knowing this, one can obtain the correct expression for momentum using the conservation of momentum in the vertical direction. (Conservation of the horizontal component of momentum is already ensured: for both particles before and after the collision this component is the same, and for the particle 1 it is generally equal to zero. So you should only require maintaining vertical speed utga.) But the vertical speed Can gain by simply looking at this encounter from a different perspective! Look at the collision depicted in Fig. 16.3, A from a car that is now moving to the left at speed And. You will see the same collision, but turned upside down (Fig. 16.3, b). Now it's a particle 2 will fall and jump up at speed w, and horizontal speed And the particle will acquire 1. You, of course, already guess what the horizontal speed is equal to utg; it is equal w (1- u 2 /c 2) [see equation (16.7)]. In addition, we know that the change in vertical momentum of a vertically moving particle is equal to

p=2m w w

(two here because the upward movement turned into a downward movement). A particle moving obliquely has a speed equal to v, its components are equal to u And w (1-u 2 /c 2 ), and its mass m v . Change vertical momentum of this particle  p"=2t v w ( 1-u 2 /с 2), since, in accordance with our assumption (16.8), any momentum component is equal to the product of the velocity component of the same name and the mass corresponding to this velocity. But the total impulse is zero. This means that the vertical impulses must cancel each other out, and the ratio of the mass moving at speed w, to a mass moving with speed v, should be equal

m w /m v =(1-u 2 /c 2). (16.9).

Let's move on to the limiting case when w tends to zero. At very small w quantities v And u will almost coincide, m w  m 0 , a m v  m u . The final result is this:

Now do this interesting exercise: check whether condition (16.9) is satisfied for arbitrary w , when the mass obeys formula (16.10). At the same time, the speed v, in equation (16.9) can be found from the right triangle

IN You will see that (16.9) is satisfied identically, although above we only needed the limit of this equality at w->0. Now let's move on to further consequences, already assuming that, according to (16.10), mass depends on speed. Let's consider the so-called inelastic collision. For simplicity, we assume that of two identical bodies colliding with equal velocities w, a new body is formed, which no longer disintegrates (Fig. 16.4, a).

F ig. 16.4. Two pictures of an inelastic collision of bodies of equal mass.

The masses of the bodies before the collision are equal, as we know, m 0 /  (1- w 2 /c 2 ). By assuming conservation of momentum and accepting the principle of relativity, we can demonstrate an interesting property of the mass of a newly formed body. Let's imagine an infinitesimal speed And, transverse to speeds w(it would be possible to work with a finite speed And, but with an infinitesimal value And it’s easier to understand everything), and let’s look at this collision, moving in the elevator at a speed - u. We will see the picture shown in Fig. 16.4, a. Composite body has unknown mass M. At the body 1, like the body 2, there is a velocity component And, upward, and a horizontal component, almost equal w. After the collision, the mass remains M, moving upward at speed u, much less than the speed of light and speed w. The momentum must remain the same; Let us therefore see what he was like before the collision and what he became after. Before the collision it was equal p~=2m w u,A then he became p"=M u u. But M u due to the smallness of u , essentially coincides with M 0 . Thanks to conservation of momentum

M 0 =2m w. (16.11)

So, The mass of a body formed in the collision of two identical bodies is equal to their double mass. You might actually say, “Well, that’s just conservation of mass.” But don’t be so quick to exclaim, “So what!” because The masses of the bodies themselves were greater than when the bodies were motionless. They contribute to the total mass M not the rest mass, but more. Isn't it amazing? It turns out that conservation of momentum in a collision of two bodies requires that the mass they form be greater than their rest masses, although after the collision these bodies themselves will come to a state of rest!

Figure 1. Relativistic mechanics of a material point. Author24 - online exchange of student works

At such ultra-high speeds, completely unexpected and magical processes begin to happen to physical things, such as time dilation and relativistic length contraction.

Within the framework of the study of relativistic mechanics, the formulations of some well-established physical quantities in physics change.

This formula, which almost every person knows, shows that mass is an absolute measure of the energy of a body, and also demonstrates the fundamental probability of the transition of the energy potential of a substance into radiation energy.

The basic law of relativistic mechanics in the form of a material point is written in the same way as Newton’s second law: $F=\frac(dp)(dT)$.

The principle of relativity in relativistic mechanics

Figure 2. Postulates of Einstein's theory of relativity. Author24 - online exchange of student works

Einstein's principle of relativity implies the invariance of all existing laws of nature with respect to the gradual transition from one inertial concept of reference to another. This means that all formulas describing natural laws must be completely invariant under Lorentz transformations. By the time SRT arose, a theory satisfying this condition had already been presented by Maxwell’s classical electrodynamics. However, all the equations of Newtonian mechanics turned out to be absolutely non-invariant with respect to other scientific postulates, and therefore SRT required a revision and clarification of mechanical laws.

As the basis for such an important revision, Einstein voiced the requirements for the feasibility of the law of conservation of momentum and internal energy, which are found in closed systems. In order for the principles of the new teaching to be carried out in all inertial concepts of reference, it turned out to be important and paramount to change the definition of the very impulse of the physical body.

If we accept and use this definition, then the law of conservation of finite momentum of interacting active particles (for example, during sudden collisions) will begin to be fulfilled in all inertial systems directly connected by Lorentz transformations. As $β → 0$, the relativistic internal impulse automatically transforms into a classical one. The mass $m$, included in the main expression for momentum, is a fundamental characteristic of the smallest particle, independent of the further choice of the reference concept, and, consequently, of the coefficient of its motion.

Relativistic impulse

Figure 3. Relativistic impulse. Author24 - online exchange of student works

The relativistic impulse is not proportional to the initial velocity of the particle, and its changes do not depend on the possible acceleration of the elements interacting in the inertial reporting system. Therefore, a force that is constant in direction and magnitude does not cause rectilinear uniformly accelerated motion. For example, in the case of one-dimensional and smooth motion along the central axis x, the acceleration of all particles under the influence of a constant force turns out to be equal to:

$a= \frac(F)(m)(1-\frac(v^2)(c^2))\frac(3)(2)$

If the speed of a certain classical particle increases indefinitely under the influence of a stable force, then the speed of relativistic matter cannot ultimately exceed the speed of light in absolute vacuum. In relativistic mechanics, just like in Newton's laws, the law of conservation of energy is fulfilled and implemented. The kinetic energy of a material body $Ek$ is determined through the external work of force necessary to communicate a given speed in the future. In order to accelerate an elementary particle of mass m from a state of rest to speed under the influence of a constant parameter $F$, this force must do work.

An extremely important and useful conclusion of relativistic mechanics is that a mass $m$ at constant rest contains an incredible amount of energy. This statement has various practical applications, including in the field of nuclear energy. If the mass of any particle or system of elements has decreased several times, then an energy equal to $\Delta E = \Delta m c^2 should be released. $

Numerous direct studies provide convincing evidence for the existence of rest energy. The first experimental proof of the correctness of Einstein's relation, which relates volume and mass, was obtained by comparing the internal energy released during instantaneous radioactive decay with the difference in the coefficients of the final products and the original nucleus.

Mass and energy in relativistic mechanics

Figure 4. Momentum and energy in relativistic mechanics. Author24 - online exchange of student works

In classical mechanics, the mass of a body does not depend on the speed of movement. And in the relativistic one it grows with increasing speed. This can be seen from the formula: $m=\frac(m_0)(√1-\frac(v^2)(c^2))$.

• $m_0$ is the mass of the material body in a calm state;
• $m$ is the mass of a physical body in that inertial reference concept relative to which it moves with speed $v$;
• $с$ is the speed of light in vacuum.

The difference in masses becomes visible only at high speeds, approaching the speed of light.

Kinetic energy at specific speeds approaching the speed of light is calculated as a certain difference between the kinetic energy of a moving body and the kinetic energy of a body at rest:

$T=\frac(mc^2)(√1-\frac(v^2)(c^2))$.

At speeds significantly lower than the speed of light, this expression turns into the formula for kinetic energy of classical mechanics: $T=\frac(1)(2mv^2)$.

The speed of light is always a limiting value. In principle, no physical body can move faster than light.

Many tasks and problems could be solved by humanity if scientists managed to develop universal devices capable of moving at speeds approaching the speed of light. For now, people can only dream of such a miracle. But someday, flying into space or to other planets at relativistic speeds will become not a fiction, but a reality.

Invariant mass is an extremely important characteristic of a group of particles, describing their scattering relative to each other. Almost no analysis of modern collider data is complete without measuring and discussing the invariant mass. However, before talking about invariant mass, let's start with one misunderstanding regarding the concept of mass.

Mass does not grow at speed!

It is a widely held belief that mass increases with speed; it is often called "relativistic mass". This belief is based on an incorrect interpretation of the relationship between energy and mass: they say, since energy increases with increasing speed, this means that mass also increases. This statement is found not only in many popular books, but also in school and even university physics textbooks.

This statement is incorrect (for greater pedantry, see the note below in small print). Weight- in the form in which this word is understood by modern physics, and especially the physics of elementary particles, - does not depend on speed. The energy of the particle and its momentum depend on the speed; at near-light speeds, the laws of dynamics and kinematics change. But the mass of a particle is a quantity that is related to the total energy E and impulse p formula

m 2 = E 2 /c 4 – p 2 /c 2 ,

remains unchanged. In popular materials, this quantity is called “rest mass” and contrasted with “relativistic mass,” but we emphasize once again: this division is made only in popular materials and in some physics courses. In modern physics there is no “relativistic mass”, there is only “mass” defined by this equation. The term “relativistic mass” is an unsuccessful technique for popularizing physics, which has long ago become divorced from real physics.

For a reader who has already heard about this problem, and perhaps even participated in debates about it, this point of view may seem somewhat “extremist”. After all formally we can introduce the concept of relativistic mass and rewrite all equations using it, rather than real mass, and we will not make any mathematical mistakes. So why is the “relativistic mass” deprived of the right to exist?

The fact is that this term is sterile from a scientific point of view and harmful from a pedagogical point of view. First, experience shows that it does not at all simplify the understanding of the theory of relativity (if by understanding we mean something more than just knowing a few words). Secondly, it confuses the “everyday intuition” of the uninitiated reader and often leads him to erroneous conclusions (for example, that a body moving at a speed sufficiently close to the speed of light will inevitably turn into a black hole due to “increased masses"). This term implicitly primes the reader's intuition to accept the conclusion that changes can occur with the particle depending on the frame of reference. And finally - let's repeat it again! - “relativistic mass” does not correspond to any real characteristics of a particle that modern physics knows; This is purely a technique for popularizing physics.

Therefore, from an educational point of view, it is much more useful not to introduce this term at all.

For more information about the origin and harm of this misconception, see the numerous publications of the outstanding physicist Lev Borisovich Okun, for example, in the article “Relativistic” circle.

Invariant mass

Let us have two particles with energies E 1 and E 2 and pulses p 1 and p 2 (bold indicates that the momentum is a vector). It could be two particles colliding or two particles flying apart, it doesn’t matter. Their masses, of course, are calculated from energies and momenta in accordance with the above formula.

We now want to know something about the property of this pair of particles as unified system. We can write total energy E 12 and full impulse p 12 of this system, E 12 = E 1 + E 2 , p 12 = p 1 + p 2, while the pulses are summed up as vectors. This means that we can calculate some mass-like size m 12 by formula

m 12 2 = E 12 2 /c 4 – p 12 2 /c 2 .

This value m 12 and is called invariant mass pairs of particles. Its most important property is precisely that it is invariant, that is, it does not depend on the frame of reference in which we carry out the calculation (although the energies and momenta do).

Let us note that the invariant mass is not at all equal to the sum of the masses of two particles! Moreover, it is easy to prove that m 12 ≥ m 1 + m 2, and equality is possible only when two particles move with the same speeds (that is, the first particle is at rest from the point of view of the second). So, for a pair of particles we have three independent characteristics that do not depend on the reference frame: m 1 , m 2 and m 12 .

If we study not two particles, but more, then the invariant masses according to these rules can be calculated not only for the entire system, but also for any pair, triple, and generally any combination of these particles. Please note that having counted these masses, we still do not state anything about the particles themselves, about their origin, about the “relationships” they have with each other. These are simply additional kinematic quantities that do not depend on the reference system.

Invariant mass as a “marker” of the origin of particles

The invariant mass characterizes how violently the particles fly apart from each other, how intense is this expansion (or their collision, if we are talking about colliding particles). To put it quite simply, if the dispersion of particles is imagined as a “micro-explosion” of a collective of particles, then the invariant mass characterizes the “energy balance” of this micro-explosion. For example in Fig. Figure 1 shows two situations in which the energies of two particles E 1 and E 2 and modules of their impulses | p 1 | and | p 2 | are the same, but the invariant masses are different.

The main benefit of invariant mass is that it helps to find out the origin of these particles: whether they were obtained from the decay of some one intermediate unstable particle or whether they were born in different processes. In the first case, their invariant mass approximately coincides with the mass of this unstable particle, and in the second case it can be arbitrary. This technique is often used in analyzing the results of collisions of elementary particles; It is with its help that we learn about the fleeting existence of unstable particles and are able to separate different types of events from each other.

Let's take a now famous example: the search for the Higgs boson at the Large Hadron Collider through its decay into two photons. If a Higgs boson is produced in a collision, it can decay into two photons (Figure 2, left). But the same pair of photons can be obtained by itself, without any intermediate particles, simply due to the emission of photons by quarks (Fig. 2, right). In both cases, the detector will see a pair of photons and will not be able to say what caused them to appear. Simply by detecting photons, we cannot prove that we actually sometimes experience the birth and decay of the Higgs boson.

Studying the invariant mass of two photons comes to the rescue mγγ. In each specific event with two photons, we need to calculate this invariant mass, and then count how many events with what invariant mass we got, and build a graph: the number of events depending on mγγ. If the Higgs boson is not in the data (or is not yet visible), this dependence will be smooth - after all, the energies and momenta of two photons are not related, so the invariant mass can turn out to be anything. If there is a Higgs boson, a bump should appear on the graph. This bump is those additional events that resulted precisely from the birth of the Higgs boson and its decay into two photons. The position of the bump will indicate the mass of the boson, and its height will indicate the intensity of this process.

In Fig. Figure 3 shows data from the ATLAS detector based on the results of 2011 and 2012 in the region of the invariant mass of two photons from 100 to 160 GeV. A more or less smooth background is visible, decreasing with growth mγγ and caused precisely by the independent production of two photons. And against this background, the desired bump in the region of 125 GeV is clearly visible. It is not very strong, but due to small errors it has great statistical significance, which means that the existence of a new particle decaying into two photons can be considered experimentally proven.

Additional literature:

• G. I. Kopylov. “Just Cinematics”, vol. eleven

> Relativistic energy and mass

Explore mass and energy of a relativistic particle in the special theory of relativity. Consider the role of the speed of light, the formula for relativistic mass and energy.

In special relativity, if an object's motion approaches the speed of light, then energy and momentum increase without limit.

Learning Objective

• Describe the possibility of an object moving at the speed of light.

Main points

Terms

• Special Theory of Relativity: The speed of light remains the same in all frames of reference.
• Rest mass is the mass of a body when it is not moving relative to the observer.
• Lorentz coefficient - used to determine the degree of time dilation, length contraction and relativistic mass.

Relativistic energy and mass

In Einstein's theory of special relativity, if an object has mass, it cannot reach the speed of light. As it approaches the mark, its energy and momentum will increase without limitation. Relativistic corrections for energy and mass are needed because the speed of light in a vacuum remains stable in all reference frames.

Conservation of mass and energy are generally accepted physical laws. For them to work, the special theory of relativity must function. If the speed of the object is lower than light, then the expressions for the values ​​of relativistic energy and mass will approximately converge with the Newtonian options.

This shows the relationship between relativistic and Newtonian kinetic energy and the speed of an object. The relativistic will increase to infinity as the object approaches the speed of light. But Newton's exponent will continue to increase as the object's speed increases

Relativistic mass

In 1934, the mass of a relativistic particle was determined by Richard K. Tolman. For a particle with zero rest mass, the Lorentz coefficient appears (v – relative speed between inertial reference systems, c – light speed).

Richard K. Tolman and Albert Einstein (1932)

If the relative velocity is equal to zero, then it reaches 1, and the relativistic mass is reduced to rest mass. As the speed of light increases, the denominator on the right side tends to zero, that is, to infinity.

In the equation for momentum, the mass will be relativistic. That is, it is a constant of proportionality between speed and momentum.

It is worth noting that despite the validity of Newton's second law, the derivative form will be invalid because it is not constant.

Relativistic energy

Relativistic energy related to rest mass through the formula:

This is the square of the Euclidean shape for the various momentum vectors in the system.

In the modern world, predictions of relativistic energy and mass are regularly confirmed in experiments with particle accelerators. Not only can the growth of relativistic momentum and energy be accurately determined, but they are also used to understand the behavior of cyclotrons and synchrotrons.

After Einstein proposed the principle of equivalence of mass and energy, it became obvious that the concept of mass can be used in two ways. On the one hand, this is the mass that appears in classical physics; on the other hand, one can introduce the so-called relativistic mass as a measure of the total (including kinetic) energy of a body. These two masses are related to each other by the relationship:

where is the relativistic mass, m- “classical” mass (equal to the mass of a body at rest), v- body speed. The relativistic mass introduced in this way is a coefficient of proportionality between the momentum and velocity of the body:

A similar relationship holds for classical momentum and mass, which is also given as an argument in favor of introducing the concept of relativistic mass. The relativistic mass introduced in this way subsequently led to the thesis that the mass of a body depends on the speed of its movement.

In the process of creating the theory of relativity, the concepts of longitudinal and transverse mass of a particle were discussed. Let the force acting on the particle be equal to the rate of change of the relativistic momentum. Then the relationship between force and acceleration changes significantly compared to classical mechanics:

If the speed is perpendicular to the force, then and if it is parallel, then where - relativistic factor. Therefore, it is called longitudinal mass, and - transverse mass.

The statement that mass depends on speed has been included in many educational courses and, due to its paradoxical nature, has become widely known among non-specialists. However, in modern physics they avoid using the term “relativistic mass”, using instead the concept of energy, and by the term “mass” understanding rest mass. In particular, the following disadvantages of introducing the term “relativistic mass” are highlighted:

§ non-invariance of the relativistic mass under Lorentz transformations;

§ synonymy of the concepts energy and relativistic mass, and, as a consequence, the redundancy of introducing a new term;

§ the presence of longitudinal and transverse relativistic masses of different sizes and the impossibility of uniformly writing the analogue of Newton’s second law in the form

§ methodological difficulties in teaching the special theory of relativity, the presence of special rules when and how to use the concept of “relativistic mass” in order to avoid mistakes;

§ confusion in the terms “mass”, “rest mass” and “relativistic mass”: some sources simply call one thing mass, some - another.

Despite these shortcomings, the concept of relativistic mass is used in both educational and scientific literature. It should, however, be noted that in scientific articles the concept of relativistic mass is used for the most part only in qualitative reasoning as a synonym for increasing the inertia of a particle moving at near-light speed.

17. Laws of conservation of energy and momentum in SRT.

18. Oscillations in mechanics. Elastic and quasi-elastic forces. Own vibrations.

Oscillations- a process of changing the states of a system around the equilibrium point that is repeated to one degree or another over time. For example, when a pendulum oscillates, its deviations in one direction or another from the vertical position are repeated; When oscillations occur in the electrical oscillatory circuit, the magnitude and direction of the current flowing through the coil are repeated.

Oscillations are almost always associated with the alternating transformation of the energy of one form of manifestation into another form.

Oscillations of various physical natures have many common patterns and are closely interrelated with waves. Therefore, the study of these patterns is carried out by the generalized theory of oscillations and waves. The fundamental difference from waves: during oscillations there is no transfer of energy; these are, so to speak, “local” energy transformations.

Classification

The identification of different types of oscillations depends on the emphasized properties of oscillating systems (oscillators)

[edit]By physical nature

§ Mechanical(sound, vibration)

§ Electromagnetic(light, radio waves, heat)

§ Mixed type- combinations of the above

[edit]By the nature of interaction with the environment

§ Forced- oscillations occurring in the system under the influence of external periodic influence. Examples: leaves on trees, raising and lowering a hand. With forced oscillations, the phenomenon of resonance may occur: a sharp increase in the amplitude of oscillations when the natural frequency of the oscillator coincides with the frequency of the external influence.

§ Free (or own)- these are oscillations in a system under the influence of internal forces, after the system is brought out of equilibrium (in real conditions, free oscillations are always damped). The simplest examples of free oscillations are the oscillations of a weight attached to a spring, or a weight suspended on a thread.

§ Self-oscillations- oscillations in which the system has a reserve of potential energy that is spent on oscillations (an example of such a system is a mechanical watch). A characteristic difference between self-oscillations and free oscillations is that their amplitude is determined by the properties of the system itself, and not by the initial conditions.

§ Parametric- oscillations that occur when any parameter of the oscillatory system changes as a result of external influence.

§ Random- oscillations in which the external or parametric load is a random process.

Characteristics

§ Amplitude- the maximum deviation of a fluctuating quantity from some average value for the system, (m)

§ Period- the period of time after which any indicators of the state of the system are repeated (the system makes one complete oscillation), (With)

§ Frequency- number of oscillations per unit time, ( Hz, s −1).

The oscillation period and frequency are reciprocal quantities;

In circular or cyclic processes, instead of the “frequency” characteristic, the concept is used circular (cyclic) frequency (rad/s, Hz, s −1), showing the number of oscillations per unit of time:

§ Bias- deviation of the body from the equilibrium position. Designation X, Unit of measurement meter.

§ Oscillation phase- determines the displacement at any time, that is, determines the state of the oscillatory system.

QUASIELASTIC FORCE- force directed towards the center O. the modulus is proportional to the distance r from the center O to the point of application of force ( F=-cr), Where With- constant coefficient, numerically equal to the force acting per unit distance. K. s. is a central and potential force with force function U = -0,5cr 2. Examples of K. s. The forces of elasticity that arise during small deformations of elastic bodies are used (hence the term “CS”). Approximately K. s. can also be considered the tangential component of gravity acting on the mat. pendulum with small deviations from the vertical. For a material point under the influence of a cosmic system, the center O is the position of its stable equilibrium. The point removed from this position will be depending on the start. conditions or perform about O rectilinear harmonious. vibrations, or describe an ellipse (in particular, a circle).

Elastic force- a force that arises during deformation of a body and counteracts this deformation.

In the case of elastic deformations, it is potential. The elastic force is of an electromagnetic nature, being a macroscopic manifestation of intermolecular interaction. In the simplest case of tension/compression of a body, the elastic force is directed opposite to the displacement of the particles of the body, perpendicular to the surface.

The force vector is opposite to the direction of deformation of the body (displacement of its molecules).

[edit]Hooke's Law

Main article:Hooke's law

In the simplest case of one-dimensional small elastic deformations, the formula for the elastic force has the form:

where is the rigidity of the body, is the magnitude of the deformation.

In its verbal formulation, Hooke's law sounds like this:

The elastic force that arises during deformation of a body is directly proportional to the elongation of the body and is directed opposite to the direction of movement of particles of the body relative to other particles during deformation.

[edit]Nonlinear deformations

As the amount of deformation increases, Hooke's law ceases to apply, and the elastic force begins to depend in a complex way on the amount of stretching or compression.

Natural vibrations, free vibrations, vibrations in a mechanical, electrical or any other physical system, occurring in the absence of external influence due to the initially accumulated energy (due to the presence of an initial displacement or initial speed). The nature of natural vibrations is determined mainly by the system’s own parameters (mass, inductance, capacitance, elasticity). In real systems, due to energy dissipation, natural oscillations are always damped, and with large losses they become aperiodic.

19. Equations of motion of the simplest mechanical oscillatory systems without friction.

Oscillatory system- a physical system in which free vibrations can exist

20. Energy of the oscillatory system.

21. Free vibrations. Equation of motion of oscillatory systems with liquid friction.

22. Attenuation coefficient. Logarithmic decrement. Good quality.

Let us find the ratio of the amplitude values ​​of damped oscillations at moments of time t and (Fig. 3.1):

where β is the attenuation coefficient.

The natural logarithm of the ratio of amplitudes following each other through a period T is called logarithmic damping decrement χ:

Let's find out physical meaningχiβ.

Relaxation time τ – time during which amplitude A decreases by e times.

Hence, damping coefficient β is a physical quantity,inverse time,during which the amplitude decreases by a factor of e.

Let N the number of oscillations after which the amplitude decreases by e once. Then

Hence, logarithmic damping decrement χ is a physical quantity reciprocal to the number of oscillations, after which the amplitude A decreases by e times.

If χ = 0.01, then N = 100.

With a large damping coefficient, not only does the amplitude rapidly decrease, but the oscillation period also increases noticeably. When the resistance becomes equal critical , then the circular frequency becomes zero (w=0), and (t-), the oscillations stop. This process is called aperiodic (Fig. 3.2).

The differences are as follows. When a body oscillates and returns to its equilibrium position, has a reserve of kinetic energy. When aperiodic motion When returning to the equilibrium position, the energy of the body is spent on overcoming the forces of resistance and friction.

Good quality- a characteristic of an oscillatory system that determines the resonance band and shows how many times the energy reserves in the system are greater than the energy losses during one oscillation period.

The quality factor is inversely proportional to the rate of decay of natural oscillations in the system. That is, the higher the quality factor of the oscillatory system, the less energy loss for each period and the slower the oscillations decay.

The general formula for the quality factor of any oscillatory system:

,

§ - resonant vibration frequency

§ - energy stored in the oscillatory system

§ - power dissipation.

23. Forced vibrations. Resonance.

Forced vibrations- vibrations that occur under the influence of external forces that change over time.

Self-oscillations differ from forced oscillations in that the latter are caused by periodic external influence and occur with the frequency of this influence, while the occurrence of self-oscillations and their frequency are determined by the internal properties of the self-oscillatory system itself.