Problems in physics (ELEMENTS OF THE SPECIAL THEORY OF RELATIVITY), on the topic
Relativistic law of addition of velocities. Dependence of mass on speed. Law of relationship between mass and energy
From the manual: GDZ for the Rymkevich problem book for grades 10-11 in physics, 10th edition, 2006.

Compare the time of reception of a light signal from the same distance sent from a rocket if: a) the rocket moves away from the observer; b) the rocket approaches the observer
SOLUTION

Elementary particle The neutrino moves at the speed of light c. The observer moves towards the neutrino with speed v. What is the speed of the neutrino relative to the observer
SOLUTION

Two particles, the distance between which is L = 10 m, fly towards each other with speeds v = 0.6. After what period of time, according to the laboratory clock, will the collision occur?
SOLUTION

Two particles move away from each other, each having a speed of 0.8c, relative to an observer on earth. What is relative speed particles
SOLUTION

WITH spaceship, moving towards the Earth at a speed of 0.4c, send two signals: a light signal and a beam of fast particles having a speed relative to the ship of 0.8c. At the moment the signals were launched, the ship was at a distance of 12 Gm from the Earth. Which signal and how much earlier will be received on Earth?
SOLUTION

What is the mass of a proton traveling at a speed of 2.4 * 108 m/s? The rest mass of a proton is considered equal to 1 a. eat
SOLUTION

How many times does the mass of a particle increase when moving at a speed of 0.99c?
SOLUTION

How much will the mass of an alpha particle increase when moving at a speed of 0.9c? Assume the rest mass of the alpha particle is equal to 4 a. eat
SOLUTION

At what speed must a proton fly (m0 = 1 a.m.u.) so that its mass becomes equal to the rest mass of the particle (m0 = 4 a.m.u.)
SOLUTION

At what speed of the spacecraft will the mass of food increase by 2 times? Will the power reserve usage time double?
SOLUTION

Find the ratio of the electron charge to its mass at an electron speed of 0.8c. The ratio of an electron's charge to its rest mass is known
SOLUTION

Power total radiation Solar 3.83 x 1026 W. How much does the mass of the Sun decrease every second due to this?
SOLUTION

The crane lifted a load weighing 18 tons to a height of 5 m. How much did the weight of the load change?
SOLUTION

How much will the mass of a spring with a stiffness of 10 kN/m increase when it is stretched by 3 cm?
SOLUTION

The rest mass of the spacecraft is 9 tons. How much does the mass of the spacecraft increase when it moves at a speed of 8 km/s
SOLUTION

The electron moves at a speed of 0.8c. Determine the total and kinetic energy of the electron
SOLUTION

A kettle with 2 kg of water was heated from 10 °C to boiling. How much has the mass of water changed?
SOLUTION

How much does the mass of 1 kg of ice change when melting?
SOLUTION

Determine the momentum of a proton if its energy is equal to the rest energy of an α-particle. What accelerating potential difference must a proton pass through to acquire such momentum?

The connection between the change in mass and the change in energy obtained above does not concern the transition from one system to another, it is related to the question of the nature 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 famous 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

From point of view classical mechanics the mass of a body does not depend on its movement. If the mass of a body at rest is equal to m 0, then for a moving body this mass will remain exactly the same. The theory of relativity shows that this is not actually the case. Body mass T, moving at speed v, expressed in terms of rest mass as follows:

m = m 0 / √(1 - v 2 /c 2) (5)

Let us immediately note that the speed appearing in formula (5) can be measured in any inertial frame. In different inertial systems, a body has different speeds; in different inertial systems it will also have different mass.

Mass is the same relative quantity as speed, time, distance. We cannot talk about the magnitude of the mass until the frame of reference in which we study the body is fixed.

From what has been said, it is clear that when describing a body, one cannot simply say that its mass is such and such. For example, the sentence “the mass of the ball is 10 g” is completely indefinite from the point of view of the theory of relativity. The numerical value of the mass of the ball does not tell us anything until the inertial system in relation to which this mass is measured is indicated. Typically, the mass of a body is specified in an inertial system associated with the body itself, i.e., the rest mass is specified.

In table Figure 6 shows the dependence of body mass on its speed. It is assumed that the mass of the body at rest is 1 a. Speeds less than 6000 km/sec are not given in the table, since at such speeds the difference between the mass and the rest mass is negligible. At high speeds this difference becomes noticeable. The greater the speed of a body, the greater its mass. So, for example, when driving at a speed of 299,700 km/sec body weight increases almost 41 times. At high speeds, even a slight increase in speed significantly increases body weight. This is especially noticeable in Fig. 41, which graphically shows the dependence of mass on speed.

Rice. 41. Dependence of mass on speed (rest mass of a body is 1 g)

In classical mechanics, only slow motions are studied, for which the mass of a body differs completely insignificantly from the rest mass. When studying slow movements, we can calculate body mass equal mass peace. The mistake we make in this case is almost invisible.

If the speed of a body approaches the speed of light, then the mass grows unlimitedly or, as they say, the mass of the body becomes infinite. Only in one single case can a body acquire a speed equal to the speed of light.
From formula (5) it is clear that if the body moves at the speed of light, i.e. if v = With and √(1 - v 2 /c 2), then the value must also be equal to zero m 0 .

If this were not the case, then formula (5) would lose all meaning, since dividing a finite number by zero is an unacceptable operation. A finite number divided by zero equals infinity - a result that has no definite physical meaning. However, we can make sense of the expression “zero divided by zero.” It follows that only objects whose rest mass is zero can move exactly at the speed of light. Such objects cannot be called bodies in the usual sense.

The equality of the rest mass to zero means that a body with such a mass cannot be at rest at all, but must always move with speed c. An object with zero rest mass is light, more precisely, photons (light quanta). Photons can never be at rest in any inertial frame; they always move at speed With. Bodies with a rest mass different from zero can be at rest or move at different speeds, but at lower speeds of light. They can never reach the speed of light.

In an experiment to measure the mass of an electron using a mass spectrograph, only one stripe is detected on a photographic plate. Since the charge of each electron is equal to one elementary charge, we come to the conclusion that all electrons have the same mass.

The mass, however, turns out to be unstable. It increases with increasing potential difference accelerating electrons in the mass spectrograph (Fig. 351), Since kinetic energy electron is directly proportional to the accelerating potential difference, it follows that the mass of the electron increases with its kinetic energy. Experiments lead to the following dependence of mass on energy:

, (199.1)

where is the mass of the electron having kinetic energy, - constant, - speed of light in vacuum . From formula (199.1) it follows that the mass of an electron at rest (i.e., an electron with kinetic energy) is equal to . The quantity is therefore called the rest mass of the electron.

Measurements with different sources of electrons (gas discharge, thermionic emission, photoelectron emission, etc.) lead to coinciding values ​​of the electron rest mass. This mass turns out to be extremely small:

Thus, an electron (at rest or slowly moving) is almost two thousand times lighter than an atom of the lightest substance - hydrogen.

The value in formula (199.1) represents the additional mass of the electron due to its motion. While this addition is small, when calculating the kinetic energy, we can approximately replace by , and set . Then This shows that our assumption that the additional mass is small compared to the rest mass is equivalent to the condition that the electron speed is much less than the speed of light. On the contrary, when the speed of the electron approaches the speed of light, the additional mass becomes large.

Albert Einstein (1879-1955) theoretically substantiated the relation (199.1) in the theory of relativity (1905). He proved that it is applicable not only to electrons, but also to any particles or bodies without exception, and by this one must be understood the rest mass of the particle or body in question. Einstein's conclusions were further tested in various experiments and were completely confirmed. Theoretical formula Einstein, expressing the dependence of mass on speed, has the form

(199.2)

Thus, the mass of any body increases with an increase in its kinetic energy or speed. However, as with the electron, the additional mass due to motion is only noticeable when the speed of motion approaches the speed of light. Comparing expressions (199.1) and (199.2), we obtain a formula for the kinetic energy of a moving body, taking into account the dependence of mass on speed:

(199.3)

In relativistic mechanics, (i.e. mechanics based on the theory of relativity), as well as in classical mechanics, the momentum of a body is defined as the product of its mass and speed. However, now the mass itself depends on the speed (see (196.2)), and the relativistic expression for momentum has the form

(199.4)

In Newtonian mechanics, the mass of a body is considered a constant quantity, independent of its motion. This means that Newtonian mechanics (more precisely, Newton's 2nd law) is applicable only to the movements of bodies with velocities very small compared to the speed of light. The speed of light is colossal; when moving earthly or celestial bodies the condition is always satisfied, and the mass of the body is practically indistinguishable from its rest mass. Expressions for kinetic energy and momentum (199.3) and (199.4) at transform into the corresponding formulas for classical mechanics (see Exercise 11 at the end of the chapter).

In view of this, when considering the motion of such bodies, one can and should use Newtonian mechanics.

The situation is different in the world of the smallest particles of matter - electrons, atoms. Here we often have to deal with fast movements, when the speed of the particle is no longer small compared to the speed of light. In these cases, Newtonian mechanics is not applicable and it is necessary to use the more accurate, but also more complex mechanics of Einstein; the dependence of the mass of a particle on its speed (energy) is one of the important conclusions of this new mechanics.

Another characteristic conclusion of Einstein’s relativistic mechanics is the conclusion that it is impossible for bodies to move at a speed greater than the speed of light in a vacuum. The speed of light is the maximum speed of movement of bodies.

Existence top speed the motion of bodies can be considered as a consequence of the increase in mass with speed: the greater the speed, the heavier the body and the more difficult it is to further increase the speed (since acceleration decreases with increasing mass).

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 speed 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 magnetic field which it causes, according to Maxwell’s theory, should behave as if its mass were increased by some amount depending on 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 electric field tension E (Fig. 20) created by 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 a field with a charged particle in a vacuum occurs with a speed c, the speed of propagation electromagnetic field, while maintaining the equality of the impulse of force to the angular momentum. 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 accelerating field configuration occurs and own field accelerated particle, 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 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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