Newton's laws of mechanics do not agree with the new space-time concepts at high speeds of motion. Only at low speeds of movement, when the classical concepts of space and time are valid, Newton’s second law does not change its form when moving from one inertial reference system to another (the principle of relativity is satisfied). But at high speeds this law in its usual (classical) form is unfair. According to Newton's second law (9.4), a constant force, acting on a body for a long time, can impart an arbitrarily high speed to the body. But in reality, the speed of light in a vacuum is limiting, and under no circumstances can a body move at a speed exceeding the speed of light in a vacuum. A very small change in the equation of motion of bodies is required for this equation to be correct at high speeds. First, let's move on to the form of writing the second law of dynamics that Newton himself used: AP - B At where p =mv is the momentum of the body. In this equation, body mass was considered independent of speed. It is striking that even at high speeds, equation (9.5) does not change its form. The changes concern only the masses. As the speed of a body increases, its mass does not remain constant; it also increases. The dependence of mass on speed can be found based on the assumption that the law of conservation of momentum is also valid under new concepts of space and time. The calculations are too complicated. We present only the final result. If m0 denotes the mass of a body at rest, then the mass m of the same body, but moving with speed v, is determined by the formula1 Figure 227 shows the dependence of the mass of a body on its speed. From the figure it can be seen that the increase in mass is greater, the closer the speed of movement of the body is to the speed of light c. At speeds of motion much lower than the speed of light, expression 2 differs extremely little from unity. Thus, at a speed of a modern space rocket of 10 km/s, we obtain It is not surprising, therefore, that we notice an increase in mass with increasing speed. In modern theoretical physics, there is a tendency to call only the rest mass m0 mass, and not to introduce the concept of relativistic mass (9.6). growth at such relatively low speeds is impossible. But elementary particles in modern charged particle accelerators reach enormous speeds. If the speed of a particle is only 90 km/s less than the speed of light, then its mass increases 40 times. Powerful electron accelerators are capable of accelerating these particles to speeds that are only 35-50 m/s less than the speed of light. In this case, the mass of the electron increases approximately 2000 times. In order for such an electron to be kept in a circular orbit, magnetic field a force must act that is 2000 times greater than one would expect without taking into account the dependence of mass on speed. It is no longer possible to use Newtonian mechanics to calculate the trajectories of fast particles. Taking into account relation (9.6), the momentum of the body is equal to: (9.7) m0v Р = The basic law of relativistic dynamics is written in the same form: bР -р At However, the momentum of the body here is determined by formula (9.7), and not just the product m0v. Thus, mass, considered constant since Newton's time, actually depends on speed. As the speed of movement increases, the mass of the body, which determines its inert properties, increases. At v-*c, the mass of the body, in accordance with equation (9.6), increases unlimitedly (/l- therefore, the acceleration tends to zero and the speed practically stops increasing, no matter how long the force acts. The need to use the relativistic equation of motion when calculating charged particle accelerators means that the theory of relativity in our time has become an engineering science. The principle of correspondence. Newton's laws of dynamics and classical concepts of space and time can be considered as a special case of relativistic laws that are valid at speeds of motion much lower than the speed of light. This is a manifestation of the so-called principle of correspondence, according to which. any theory that claims to have a deeper description of phenomena and a wider scope of applicability than the old one must include the latter as a limiting case. The principle of correspondence was first formulated by Niels Bohr in relation to the connection between quantum and classical theories. The great scientist understood the essence of the matter before anyone else. The relativistic equation of motion, which takes into account the dependence of mass on speed, is used in the design of accelerators elementary particles and other relativistic devices. 1. Write down the formula for the dependence of body mass on the speed of its movement. 2. Under what conditions can the mass of a body be considered independent of speed!

assuming that the particle mass m(v) there is a certain function of its speed, which we have to determine based on the assumption that the momentum of the particle is a conserved quantity.

For this, let us consider an inelastic collision of two identical bodies, one of which is at rest (in some laboratory frame of reference K), and the other moves towards him at a speed v. After a collision, the bodies stick together and continue moving together at a certain speed. u, which we need to find.

The law of conservation of momentum in projection onto the initial direction of movement (which we choose as the axis x) in the laboratory system reads

In this system, the first particle is at rest, and the second one attacks it with a speed - v. As a result, the resulting composite particle moves at a speed – u(since the process looks symmetrical in this system compared to the system K). Now applying the law of addition of velocities, we can relate u And v. To do this, in the formula

Regarding speed u it is there quadratic equation. Choosing from two roots the root that corresponds to a speed less than the speed of light, we obtain

In this reference frame, if we expand the picture and again make the axis x horizontal, the collision of bodies will look as shown in Fig. 5.

To determine the velocity components of bodies before and after collision in the system K"" let's use the speed conversion formulas

Likewise, since

Let us now write down the law of conservation of momentum in the system K"" in projection onto the axis x

This equality must hold for any V, including when V = 0

Resolving this equation for m(v), we arrive at the relation

Thus, we come to the already known expression for the mass of a body, depending on its speed

(43)

Along the way, we proved that if momentum is conserved (in all inertial frames of reference), then mass (depending on speed) is also conserved, or, what is the same, energy equal to the product of the body mass times the square of the speed of light.

Relationship between energy and mass. Einstein's formula

The most important result of the special theory of relativity relates to the concept of mass. In pre-relativistic physics there were two conservation laws: the law of conservation of mass and the law of conservation of energy. Both of these fundamental laws were considered completely independent of each other. The theory of relativity combined them into one. So, if a body moving with speed v and receiving energy E 0 in the form of radiation 3 without changing its speed, it increases its energy by an amount

Consequently, the body has the same energy as a body moving with speed v and having rest mass m 0 +E 0 /c 2. Thus, we can say that if the body receives energy E 0, then its rest mass increases by the amount E 0 /c 2. So, for example, a heated body has more mass than a cold one, and if we had very accurate scales at our disposal, we would be convinced of this directly by weighing.

However, in nonrelativistic physics, energy changes E 0 that we could communicate to the body were, as a rule, not large enough to notice changes in inert body mass. Magnitude E 0 /c 2 in our everyday life is too small compared to the rest mass m 0 that the body had before the energy change. This circumstance explains the fact that the law of conservation of mass had an independent meaning in physics for so long.

The situation is completely different in relativistic physics. It is well known that with the help of accelerators we can impart enormous energy to bodies (elementary particles), sufficient for the birth of new (elementary) particles - a process that is now observed quite often in modern particle accelerators. Einstein's formula "works" in nuclear reactors nuclear power plants, where energy is released through the process of fission of the nuclei of heavy elements. The mass of the final reaction products is less than the mass of the starting substance. This mass difference divided by the square of the speed of light is the useful energy released. In the same way, our Sun provides us with heat, where, due to the thermonuclear fusion reaction, hydrogen is converted into helium and a huge amount of energy is released.

It can now be considered firmly established that the inert mass of the body is determined by the amount of energy stored in the body. This energy can be fully obtained in the process annihilation matter with antimatter, for example, an electron with a positron. As a result of this reaction, two gamma quanta are formed - photons of very high energy. This energy source may be used in the future in photonic rocket engines to achieve sub-light speeds when flying to distant galaxies.

1 Since when x<< 1

2 When such deviations are discovered, it eventually turns out that this is either an error, or, if it turns out that there is no error, it leads to the discovery of new elementary particles. The most striking example of this kind is the discovery of neutrinos.

3 Here E 0 is the energy received by the body when observed from a coordinate system moving with the body.

LECTURE 6

· Relationship between energy and momentum in relativistic mechanics.

· Doppler effect. Moment of impulse.

· Particle decay. Stellar reactions with energy conversion.

· Compton effect. Antiproton threshold.

Remember from the general physics course what Galilean transformations are. These transformations are some way to determine whether a given case is relativistic or not. The relativistic case means movement at sufficiently high speeds. The magnitude of such speeds leads to the fact that Galileo's transformations become impossible. As you know, these coordinate transformation rules are just a transition from one coordinate system, which is at rest, to another (moving).

Remember that the speed corresponding to the case of relativistic mechanics is a speed close to the speed of light. In this situation, Lorentz coordinate transformations come into force.

Relativistic impulse

Write out an expression for relativistic momentum from a physics textbook. The classical formula for momentum, as is known, is the product of the mass of a body and its speed. In the case of high speeds, a typical relativistic addition is added to the classical expression of momentum in the form of the square root of the difference between unity and the square of the ratio of the speed of the body and the speed of light. This multiplier must be in , the numerator of which is the classical representation of momentum.

Pay attention to the form of the relativistic momentum relationship. It can be divided into two parts: the first part of the work is the ratio of the classical mass of the body to the relativistic addition, the second part is the speed of the body. If we draw an analogy with the formula for the classical impulse, then the first part of the relativistic impulse can be taken as the total mass characteristic of the case of motion at high speeds.

Relativistic mass

Note that the mass of a body becomes dependent on the magnitude of its speed if the relativistic expression is taken as the general form of mass. The classical mass in the numerator of the fraction is usually called the rest mass. From its name it becomes clear that the body possesses it when its speed is zero.

If the speed of the body becomes close to the speed of light, then the denominator of the fraction of the expression for mass tends to zero, and it itself tends to infinity. Thus, as the speed of a body increases, its mass also increases. Moreover, from the form of the expression for the mass of the body, it becomes clear that changes become noticeable only when the speed of the body is sufficiently high and the ratio of the speed of movement to the speed of light is comparable to unity.

The theory of relativity - a hoax of the twentieth century Sekerin Vladimir Ilyich

6.3. Mass growth depending on speed

The representation of the dependence of mass on velocity occupies a special position in modern physics. The history of the formation of the relationship between mass and energy is outlined by V.V. Cheshev in his work, where, in particular, it is said: “The idea of ​​an increase in the mass of the electron was partly initiated by the hypothesis of the ether. In 1881, J. J. Thomson, based on theoretical considerations, pointed out that “an electrically charged body, due to the magnetic field which it produces, according to Maxwell’s theory, should behave as if its mass increased by some amount depending from its charge and shape." Subsequently, Thomson showed that the mass of a moving charge should increase with increasing its motion. Kaufman’s experiments consolidated the idea of ​​an increase in the mass of a moving electron.”

Thomson's initial, uncertain assumption about the observed “as if” increase in mass has now grown into the confidence of the equivalence between mass and energy, enshrined in the well-known formula E = mc 2, where E is energy, m is mass. For our case, the following remark from the cited work is significant: “The results of Kaufman’s experiments suggest that the effect exerted by the field on a moving charge differs from its effect on a charge at rest.”

This phenomenon seems to manifest itself during the operation of charged particle accelerators. But in accelerators of charged particles, what is observed is not a change in the mass of particles depending on the speed (this is impossible to observe), but a change in the acceleration of charged particles under controlled electric and magnetic fields, which is inexplicable in modern physical concepts.

From Newton's second law a = F/m, where a is acceleration, F is force, m is mass, it is clear that acceleration depends on both force and mass. Therefore, it seems more logical to explain the observed acceleration not by an increase in mass, but by the result of a change in the forces of interaction of electric and magnetic fields with charged particles moving in these fields.

The change in interaction forces is determined by the finite speed of propagation of the disturbance (change) in field strength. The constancy of interaction forces during the movement of interacting bodies is possible only if the speed of propagation of the disturbance is infinite.

Rice. 20

No matter how quickly the charge q is moved to point K of the electric field of intensity E (Fig. 20), created by the charged plates B and D, the position shown in Fig. 21, can only take place after a finite time interval, determined by the speed of propagation of the disturbance in the field E.

Rice. 21

We believe that the interaction of the field with a charged particle in a vacuum occurs with a speed c, the speed of propagation of the electromagnetic field, while the equality of the momentum of the force to the angular momentum is maintained. Then the interaction force F (v) of the electric field of intensity E and a particle having a charge q and moving in this field with a speed v will be equal to:

Where? - the angle between the vectors of tension E and velocity v.

Under the influence of an accelerating field, the speed increases, and with it the kinetic energy of the particle. In this case, a certain change in the configuration of the accelerating field and the own field of the accelerated particle occurs, which leads to an increase in its potential energy, i.e., the transition of the potential energy of the accelerating field into kinetic energy and potential energy of the accelerated charge. The total energy of particle A, equal to qU (U is the potential difference passed), is composed of its kinetic energy - E k and potential energy - E p

The kinetic energy of an accelerated particle is limited by the limit

The potential energy of an accelerated particle may have no limit, it is not yet visible. Therefore, the total energy of the accelerated particle, despite the speed limit, continues to grow and is determined only by the potential difference passed through. This process is reversible; when an accelerated particle interacts with a decelerating field, the stored energy is released.

The Lorentz force - F (v), acting on a charge moving in a magnetic field, is determined in a similar way:

where B is induction, ? - the angle between the directions of speed and induction. The Lorentz force is directed perpendicular to the plane in which the vectors B and v lie.

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The connection obtained above between a change in mass and a change in energy does not concern the transition from one system to another, it is related to the question of the nature of electromagnetic radiation. But the possibility of changing body weight will entail corresponding changes in dynamics. Let's see this using the example of calculating kinetic energy.

Let the body have mass m has speed u . The energy of its movement can be calculated from the work done by external forces:

If we use Newton's second law, then

Integrating equation (5.42) will lead to the well-known expression for kinetic energy.

The situation will be completely different if we question the constancy of the mass, the assumption of which is tacitly contained in (5.42): the mass is taken out of the differential sign and remains constant when energy is imparted to the system. In the light of new ideas, this is not at all the case.

Indeed, if the mass can change, then it also needs to be differentiated. Then

Replacing the change in energy through the change in mass according to the law obtained above (5.40), we obtain:

The last equality contains two variables and when integrating they should be separated:

Where m 0 – mass in the system where the body is at rest. This system, as a rule, is associated directly with the moving particle itself. m – the mass of a particle in the system relative to which it moves. As a result of integration we obtain:

The dependence of mass on speed (5.46) is similar to that for the duration of an event (5.17): the time of the event is minimal in the system where this event occurs. Likewise, mass is minimal in the system where the body is at rest.

Equation (5.46) can be verified experimentally where particles move at speeds close to the speed of light, that is, in the microcosm. An increase in mass with increasing speed was first noticed in cyclotrons, the first generation accelerators. This effect led to the fact that further acceleration of particles became impossible. As a result, the design of the cyclotron had to be changed and accelerators created that take into account the increase in particle mass with increasing speed.

It is appropriate to note here that there is a particle that can only move at the speed of light; when the speed decreases - braking - it ceases to exist, transferring its energy and momentum to other bodies (or turns into other particles). This particle is called photon- a particle of light. For him it is zero. Therefore, if for the remaining particles integration (5.40) in the range from to m gives