Abstract on the topic of the classical law of addition of velocities. Velocity addition rule

We said that the speed of light is the maximum possible speed of signal propagation. But what happens if light is emitted by a moving source in the direction of its speed V? According to the law of addition of velocities, which follows from Galileo's transformations, the speed of light must be equal to c+V. But in the theory of relativity this is impossible. Let's see what law of velocity addition follows from the Lorentz transformations. To do this, we write them for infinitesimal quantities:

By definition of the speed of its components in the frame of reference K are found as ratios of the corresponding displacements to time intervals:

Similarly, the speed of an object in a moving frame of reference is determined K", only spatial distances and time intervals must be taken relative to this system:

Therefore, dividing the expression dx to the expression dt, we get:

Dividing the numerator and denominator by dt", we find a connection x- component of velocities in different frames of reference, which differs from the Galilean rule for adding velocities:

In addition, in contrast to classical physics, the velocity components that are orthogonal to the direction of motion also change. Similar calculations for other velocity components give:

Thus, formulas for the transformation of velocities in relativistic mechanics have been obtained. The formulas for the inverse transformation are obtained by replacing primed quantities with unprimed ones and vice versa, and by replacing V on the –V.

Now we can answer the question posed at the beginning of this section. Let at the point 0" moving reference frame K" a laser is installed that sends a pulse of light in the positive direction of the axis 0"x". What will be the momentum velocity for a stationary observer in the frame of reference To? In this case, the speed of the light pulse in the frame of reference TO" has components

Applying the law of relativistic addition of velocities, we find for the components of the momentum velocity relative to the stationary system To :

We get that the speed of the light pulse and in a fixed frame of reference, relative to which the light source moves, is equal to

The same result will be obtained for any direction of propagation of the pulse. This is natural, since the independence of the speed of light from the motion of the source and the observer is inherent in one of the postulates of the theory of relativity. The relativistic law of velocity addition is a consequence of this postulate.

Indeed, when the speed of the moving reference frame V<<c, the Lorentz transformations turn into Galilean transformations, we get the usual law of addition of velocities

In this case, the course of the flow of time and the length of the ruler will be the same in both reference systems. Thus, the laws of classical mechanics are applicable if the speed of objects is much less than the speed of light. The theory of relativity did not cross out the achievements of classical physics; it established the framework for their validity.

Example. body with speed v 0 hits a wall perpendicular to it, moving towards it with a speed v. Using the formulas for the relativistic addition of velocities, we find the speed v 1 body after bounce. The impact is absolutely elastic, the mass of the wall is much greater than the mass of the body.

Let us use the formulas expressing the relativistic law of addition of velocities.

Let's direct the axis X along the initial velocity of the body v 0 and associate the frame of reference K" with a wall. Then v x= v 0 and V= –v. In the reference frame associated with the wall, the initial velocity v" 0 body equals

Now let's go back to the laboratory frame of reference To. Substituting into the relativistic law of addition of velocities v" 1 instead v" x and considering again V = –v, we find after transformations:

And this frame of reference, in turn, moves relative to another frame), the question arises about the relationship of velocities in two frames of reference.

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Addition of velocities (kinematics) ➽ Physics Grade 10 ➽ Video lesson

Lesson 19 Velocity addition formula.

Physics. Lesson number 1. Kinematics. The law of addition of speeds

Subtitles

classical mechanics

V → a = v → r + v → e. (\displaystyle (\vec (v))_(a)=(\vec (v))_(r)+(\vec (v))_(e).)

This equality is the content of the statement of the theorem on addition velocities.

In plain language: The speed of the body relative to the fixed frame of reference is equal to the vector sum of the speed of this body relative to the moving frame of reference and the speed (relative to the fixed frame) of that point of the moving frame of reference where the body is currently located.

Examples

The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed that the point of the record under the fly has relative to the ground (that is, from which the record carries it due to its rotation).
If a person walks along the corridor of the car at a speed of 5 kilometers per hour relative to the car, and the car moves at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of the train, and at a speed of 50 - 5 = 45 kilometers per hour when he goes in the opposite direction. If a person in the carriage corridor moves relative to the Earth at a speed of 55 kilometers per hour, and a train at a speed of 50 kilometers per hour, then the speed of a person relative to the train is 55 - 50 = 5 kilometers per hour.
If the waves move relative to the coast at a speed of 30 kilometers per hour, and the ship also at a speed of 30 kilometers per hour, then the waves move relative to the ship at a speed of 30 - 30 = 0 kilometers per hour, that is, they become stationary relative to the ship.

Relativistic mechanics

In the 19th century, classical mechanics faced the problem of extending this rule for adding velocities to optical (electromagnetic) processes. In essence, there was a conflict between the two ideas of classical mechanics, transferred to a new field of electromagnetic processes.

For example, if we consider the example with waves on the surface of water from the previous section and try to generalize it to electromagnetic waves, then we get a contradiction with observations (see, for example, Michelson's experiment).

The classical rule for adding velocities corresponds to the transformation of coordinates from one system of axes to another system, moving relative to the first without acceleration. If, with such a transformation, we retain the concept of simultaneity, that is, we can consider two events to be simultaneous not only when they are registered in one coordinate system, but also in any other inertial frame, then the transformations are called Galilean. In addition, with Galilean transformations, the spatial distance between two points - the difference between their coordinates in one inertial frame of reference - is always equal to their distance in another inertial frame.

The second idea is the principle of relativity. Being on a ship moving uniformly and rectilinearly, it is impossible to detect its movement by some internal mechanical effects. Does this principle extend to optical effects? Is it possible to detect the absolute motion of the system by the optical or, what is the same, electrodynamic effects caused by this motion? Intuition (rather explicitly related to the classical principle of relativity) says that absolute motion cannot be detected by any kind of observation. But if light propagates at a certain speed relative to each of the moving inertial frames, then this speed will change when moving from one frame to another. This follows from the classical rule for adding velocities. Speaking mathematically, the magnitude of the speed of light will not be invariant under the Galilean transformations. This violates the principle of relativity, or rather, does not allow the principle of relativity to be extended to optical processes. Thus, electrodynamics destroyed the connection between two seemingly obvious provisions of classical physics - the rule of addition of velocities and the principle of relativity. Moreover, these two positions as applied to electrodynamics turned out to be incompatible.

The theory of relativity provides an answer to this question. It expands the concept of the principle of relativity, extending it to optical processes as well. In this case, the rule for adding velocities is not canceled at all, but is only refined for high velocities using the Lorentz transformation:

v r e l = v 1 + v 2 1 + v 1 v 2 c 2 . (\displaystyle v_(rel)=(\frac ((v)_(1)+(v)_(2))(1+(\dfrac ((v)_(1)(v)_(2)) (c^(2))))).)

It can be seen that in the case when v / c → 0 (\displaystyle v/c\rightarrow 0), the Lorentz transformations go over to the Galilean transformations . This suggests that special relativity reduces to Newtonian mechanics at speeds small compared to the speed of light. This explains how these two theories relate - the first is a generalization of the second.

Let us derive a law relating the projections of the particle velocity in the IFR K and K".

Based on the Lorentz transformations (1.3.12), for infinitely small increments of particle coordinates and time, one can write

Dividing in (1.6.1) the first three equalities by the fourth, and then the numerators and denominators of the right-hand sides of the resulting relations by dt" and taking into account that

are the projections of the particle velocities on the CO axes K and K", we arrive at the desired law:

If the particle makes a one-dimensional motion along the axes OX and O"X", then, in accordance with (1.6.2),

Example 1. ISO K" moving at a speed V relatively ISO K. at an angle 0" to the direction of travel ISO K" bullet fired at a speed v". What is this angle 0 in ISO K?

Decision. When moving, there is not only a reduction in spatial, but also a stretching of time intervals. To find tg0 = v y / v x it is necessary in (1.6.2) to divide the second formula by the first, and then the numerator and denominator of the resulting fraction - by v "x = v" cos0 " Considering that v " y / v" x = tg0 ", we find

For speeds that are small compared to the speed of light, formulas (1.6.2) turn into the well-known law of classical mechanics (1.1.4):

From the formulas for the transformation of particle velocity projections (1.6.2), it is easy to determine the velocity modulus and its direction in the IFR K through the particle velocity in the IFR K. , and in the X"0"Y" plane), and denote by 0 (0") the angle between

V (V") and the axis OX (O "X"). Then

v x = vcos0, v = vsin0, v" x = v"cos©", v* = v"sin©", v z = v" z = 0 (1.6.4) or

As for the direction of the particle velocity in CO K (angle 0), it is determined by term-by-term division in (1.6.5) of the second formula by the first one:

and substitution (1.6.4) into (1.6.2) gives

After squaring both equalities (1.6.5) and adding them, we obtain

The inverse transformation formulas are obtained by replacing primed values with unprimed ones and vice versa and replacing V with -V.

Task 2. Determine relative speed v 0TH rendezvous of two spacecraft 1 and 2 moving towards each other with speedsX And V2-

Decision. Let's connect the mobile CO K" with the spacecraft 1. Then V = Vi, and the desired relative speed v 0TH will be the speed of the craft 2 in this CO. Applying the relativistic law of velocity addition (1.6.3) to the second craft, taking into account the direction of its velocity (v "2 = -v 0TH) we have

Numerical estimates for v, = v 2 = 0.9 s give

Task 3. body with speed v0 hits a wall perpendicular to it, moving towards it with speed. Using the relativistic law of addition of velocities, find the speed v 0Tp body after rebound. The impact is absolutely elastic, the mass of the wall is much greater than the mass of the body. To find v 0Tp , if v 0 \u003d v \u003d c / 3. Analyze extreme cases.

where V is the speed of CO K "relative to CO K. Let's connect CO K" with the wall. Then V \u003d -v and in this CO the initial velocity of the body, according to the expression for v",

Let us now return back to the laboratory CO K. Substituting into

(1.6.3) v" 0Tp instead of v" and taking into account again that V = -v, after simple transformations we obtain the desired result:

Let us now analyze the limiting cases.

If the velocities of the body and the wall are small (v 0 « s, v « s), then we can neglect all the terms where these velocities and their product are divided by the speed of light. Then from the general formula obtained above we arrive at the well-known result of classical mechanics: v 0Tp = -(v 0 + 2v) -

the speed of the body after the rebound increases by twice the speed of the wall; it is directed, of course, opposite to the initial one. It is clear that in the relativistic case this result is incorrect. In particular, when v 0 =v = c/3, it follows from it that the speed of the body after the rebound will be equal to - c, which cannot be.

Let now a body moving at the speed of light hit the wall (for example, a laser beam is reflected from a moving mirror). Substituting v 0 \u003d c into the general expression for v, we get v \u003d -c.

This means that the speed of the laser beam has changed direction, but not its absolute value - in full accordance with the principle of invariance of the speed of light in vacuum.

Let us now consider the case when the wall moves with a relativistic velocity v -> with. In this case

The body after the bounce will also move at a speed close to the speed of light.

Finally, we substitute into the general formula for v 0Tp the values

v n \u003d v \u003d c / 3. Then = -s * -0.78 s. Unlike the classical

mechanics, the theory of relativity gives a value for the speed after the bounce, less than the speed of light.

In conclusion, let's see what happens if the wall moves away from the body with the same speed v = -v 0 . In this case, the general formula for v 0Tp leads to the result: v = v 0 . As in classical mechanics, the body will not catch up with the wall and, therefore, its speed will not change.

The results of the experiment were described by the formulas

where n is the refractive index of water, and V is the speed of its flow.

Prior to the creation of SRT, the results of the Fizeau experiment were considered on the basis of the hypothesis put forward by O. Fresnel, within which it was necessary to assume that moving water partially entrains the "world ether". Value

was called the drag coefficient of the ether, and formulas (1.7.1) and (1.7.2) with this approach directly follow from the classical law of addition of velocities: c/n is the speed of light in water relative to the ether, kV is the velocity of the ether relative to the pilot plant.

Classical mechanics uses the concept of the absolute velocity of a point. It is defined as the sum of the vectors of relative and translational velocities of this point. Such an equality contains the assertion of the theorem on the addition of velocities. It is customary to imagine that the speed of a certain body in a fixed frame of reference is equal to the vector sum of the speed of the same physical body relative to the moving frame of reference. The body itself is located in these coordinates.

Figure 1. The classical law of addition of velocities. Author24 - online exchange of student papers

Examples of the law of addition of velocities in classical mechanics

Figure 2. An example of speed addition. Author24 - online exchange of student papers

There are several basic examples of adding velocities according to established rules taken as a basis in mechanical physics. When considering physical laws, a person and any moving body in space with which there is a direct or indirect interaction can be taken as the simplest objects.

Example 1

For example, a person who moves along the corridor of a passenger train at a speed of five kilometers per hour, while the train moves at a speed of 100 kilometers per hour, then he moves at a speed of 105 kilometers per hour relative to the surrounding space. In this case, the direction of movement of a person and a vehicle must match. The same principle applies when moving in the opposite direction. In this case, a person will move relative to the earth's surface at a speed of 95 kilometers per hour.

If the speeds of two objects relative to each other coincide, then they will become stationary from the point of view of moving objects. During rotation, the speed of the object under study is equal to the sum of the speeds of the object relative to the moving surface of another object.

Galileo's principle of relativity

Scientists were able to formulate basic formulas for the acceleration of objects. It follows from it that the moving reference frame moves away relative to the other one without visible acceleration. This is natural in those cases when the acceleration of bodies occurs in the same way in different frames of reference.

Such arguments originate in the days of Galileo, when the principle of relativity was formed. It is known that, according to Newton's second law, the acceleration of bodies is of fundamental importance. The relative position of two bodies in space, the speed of physical bodies depends on this process. Then all equations can be written in the same way in any inertial frame of reference. This suggests that the classical laws of mechanics will not depend on the position in the inertial frame of reference, as is customary to act in the implementation of the study.

The observed phenomenon also does not depend on the specific choice of reference system. Such a framework is currently regarded as Galileo's principle of relativity. It enters into some contradictions with other dogmas of theoretical physicists. In particular, Albert Einstein's theory of relativity presupposes other conditions of action.

Galileo's principle of relativity is based on several basic concepts:

in two closed spaces that move in a straight line and uniformly relative to each other, the result of external influence will always have the same value;
a similar result will be valid only for any mechanical action.

In the historical context of studying the foundations of classical mechanics, such an interpretation of physical phenomena was formed largely as a result of Galileo's intuitive thinking, which was confirmed in Newton's scientific works when he presented his concept of classical mechanics. However, such requirements according to Galileo may impose some restrictions on the structure of mechanics. This affects its possible formulations, design and development.

The law of motion of the center of mass and the law of conservation of momentum

Figure 3. Law of conservation of momentum. Author24 - online exchange of student papers

One of the general theorems in dynamics was the theorem of the center of inertia. It is also called the theorem on the motion of the center of mass of the system. A similar law can be derived from Newton's general laws. According to him, the acceleration of the center of mass in a dynamic system is not a direct consequence of the internal forces that act on the bodies of the entire system. It is able to connect the acceleration process with external forces that act on such a system.

Figure 4. The law of motion of the center of mass. Author24 - online exchange of student papers

The objects referred to in the theorem are:

momentum of a material point;
phone system

These objects can be described as a physical vector quantity. It is a necessary measure of the impact of the force, while it completely depends on the time of the force.

When considering the law of conservation of momentum, it is stated that the vector sum of the impulses of all bodies, the system is completely represented as a constant value. In this case, the vector sum of external forces that act on the entire system must be equal to zero.

When determining the speed in classical mechanics, the dynamics of the rotational motion of a rigid body and the angular momentum are also used. The angular momentum has all the characteristic features of the amount of rotational motion. Researchers use this concept as a quantity that depends on the amount of rotating mass, as well as how it is distributed over the surface relative to the axis of rotation. In this case, the speed of rotation matters.

Rotation can also be understood not only from the point of view of the classical representation of the rotation of a body around an axis. When a body moves rectilinearly past some unknown imaginary point that does not lie on the line of motion, the body can also have an angular momentum. When describing the rotational motion, the angular momentum plays the most significant role. This is very important when setting and solving various problems related to mechanics in the classical sense.

In classical mechanics, the law of conservation of momentum is a consequence of Newtonian mechanics. It clearly shows that when moving in empty space, momentum is conserved in time. If there is an interaction, then the rate of its change is determined by the sum of the applied forces.

If a person walks along the corridor of the car at a speed of 5 kilometers per hour relative to the car, and the car moves at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of the train, and at a speed of 50 - 5 = 45 kilometers per hour when he goes in the opposite direction.

The second idea is the principle of relativity. Being on a ship moving uniformly and rectilinearly, it is impossible to detect its movement by some internal mechanical effects. Does this principle extend to optical effects? Is it possible to detect the absolute motion of the system by the optical or, what is the same, electrodynamic effects caused by this motion? Intuition (fairly explicitly related to the classical principle of relativity) says that absolute motion cannot be detected by any kind of observation. But if light propagates at a certain speed relative to each of the moving inertial frames, then this speed will change when moving from one frame to another. This follows from the classical rule for adding velocities. Speaking mathematically, the magnitude of the speed of light will not be invariant under the Galilean transformations. This violates the principle of relativity, or rather, does not allow the principle of relativity to be extended to optical processes. Thus, electrodynamics destroyed the connection between two seemingly obvious provisions of classical physics - the rule of addition of velocities and the principle of relativity. Moreover, these two positions as applied to electrodynamics turned out to be incompatible.

Literature

B. G. Kuznetsov Einstein. Life, death, immortality. - M.: Nauka, 1972.
Chetaev N. G. Theoretical mechanics. - M.: Nauka, 1987.

See what the "Velocity Addition Rule" is in other dictionaries:

Addition of speeds- When considering a complex movement (that is, when a point or body moves in one frame of reference, and it moves relative to another), the question arises about the relationship of velocities in 2 frames of reference. Contents 1 Classical mechanics 1.1 Examples ... Wikipedia

Mechanics- [from 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 that occur during this. Mechanical movement is understood as a change over time ... ... Great Soviet Encyclopedia

VECTOR- In physics and mathematics, a vector is a quantity that is characterized by its numerical value and direction. In physics, there are many important quantities that are vectors, for example, force, position, speed, acceleration, torque, ... ... Collier's Encyclopedia

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RELATIVITY THEORY- a physical theory that considers the spatio-temporal properties of the physical. processes. These properties are common to all physical. processes, so they are often called. just properties of space-time. The properties of space-time depend on ... Encyclopedia of Mathematics

Velocity addition rule

classical mechanics

The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed with which it is carried by the record due to its rotation.

Relativistic mechanics

The classical rule for adding velocities corresponds to the transformation of coordinates from one system of axes to another system, moving relative to the first without acceleration. If, with such a transformation, we retain the concept of simultaneity, that is, we can consider two events to be simultaneous not only when they are registered in one coordinate system, but also in any other inertial system, then the transformations are called Galilean. In addition, with Galilean transformations, the spatial distance between two points - the difference between their coordinates in one inertial frame of reference - is always equal to their distance in another inertial frame.

It can be seen that in the case when , Lorentz transformations turn into Galilean transformations. The same happens when . This suggests that special relativity coincides with Newtonian mechanics either in a world with an infinite speed of light, or at speeds small compared to the speed of light. The latter explains how these two theories are combined - the first is a refinement of the second.

RELATIVITY THEORY- a physical theory that considers spatio-temporal patterns that are valid for any physical. processes. The universality of spatio-temporal sv, considered by O. t., allows us to speak of them simply as s. s of space ... ... Physical Encyclopedia

law- a; m. 1. A normative act, a decision of the highest body of state power, adopted in the prescribed manner and having legal force. Labor Code. Z. on social security. Z. on military duty. Z. about the securities market. ... ... Encyclopedic Dictionary

When considering a complex movement (that is, when a point or body moves in one frame of reference, and it moves relative to another), the question arises about the relationship of velocities in 2 frames of reference.

In plain language: The speed of a body relative to a fixed frame of reference is equal to the vector sum of the speed of this body relative to a moving frame of reference and the speed of the most mobile frame of reference relative to a fixed frame.

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Parallelogram of speeds- a geometric construction expressing the law of addition of velocities. Rule P. s. consists in the fact that with complex motion (see Relative motion), the absolute speed of a point is represented as a diagonal of a parallelogram built on ... ... Great Soviet Encyclopedia

Special theory of relativity- Postage stamp with the formula E = mc2, dedicated to Albert Einstein, one of the creators of SRT. Special theory ... Wikipedia

Poincare, Henri- Henri Poincaré Henri Poincaré Date of birth: April 29, 1854 (1854 04 29) Place of birth: Nancy ... Wikipedia

The law of addition of velocities in classical mechanics

Main article: Velocity addition theorem

In classical mechanics, the absolute velocity of a point is equal to the vector sum of its relative and translational velocities:

This equality is the content of the statement of the theorem on the addition of velocities.

1. The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed that the point of the record under the fly has relative to the ground (that is, from which the record carries it due to its rotation).

2. If a person walks along the corridor of the car at a speed of 5 kilometers per hour relative to the car, and the car moves at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of travel train, and at a speed of 50 - 5 = 45 kilometers per hour when he goes in the opposite direction. If a person in the carriage corridor moves relative to the Earth at a speed of 55 kilometers per hour, and a train at a speed of 50 kilometers per hour, then the speed of a person relative to the train is 55 - 50 = 5 kilometers per hour.

3. If the waves move relative to the coast at a speed of 30 kilometers per hour, and the ship also at a speed of 30 kilometers per hour, then the waves move relative to the ship at a speed of 30 - 30 = 0 kilometers per hour, that is, they become motionless relative to the ship.

It follows from the formula for accelerations that if the moving reference frame moves relative to the first one without acceleration, that is, then the acceleration of the body relative to both reference frames is the same.

Since in Newtonian dynamics it is acceleration that plays the role of kinematic quantities (see Newton's second law), then if it is quite natural to assume that forces depend only on the relative position and velocities of physical bodies (and not their position relative to the abstract reference point), it turns out that that all the equations of mechanics will be written in the same way in any inertial frame of reference - in other words, the laws of mechanics do not depend on which of the inertial frames of reference we study them in, do not depend on the choice of any particular inertial frame of reference as a working one.

Also - therefore - the observed motion of bodies does not depend on such a choice of reference system (taking into account, of course, the initial velocities). This statement is known as Galileo's principle of relativity, as opposed to Einstein's principle of relativity

Otherwise, this principle is formulated (following Galileo) as follows:

If in two closed laboratories, one of which moves uniformly in a straight line (and translationally) relative to the other, the same mechanical experiment is carried out, the result will be the same.

The requirement (postulate) of the principle of relativity, together with the transformations of Galileo, which seem intuitively obvious enough, largely follows the form and structure of Newtonian mechanics (and historically they also had a significant impact on its formulation). Speaking somewhat more formally, they impose restrictions on the structure of mechanics, which significantly affect its possible formulations, which historically greatly contributed to its formation.

The center of mass of the system of material points

The position of the center of mass (center of inertia) of a system of material points in classical mechanics is determined as follows:

where is the radius vector of the center of mass, is the radius vector i-th point of the system, - mass i-th point.

For the case of continuous mass distribution:

where is the total mass of the system, is the volume, is the density. The center of mass thus characterizes the distribution of mass over a body or a system of particles.

It can be shown that if the system does not consist of material points, but of extended bodies with masses , then the radius vector of the center of mass of such a system is related to the radius vectors of the centers of mass of the bodies by the relation:

In other words, in the case of extended bodies, a formula is valid, which in its structure coincides with that used for material points.

Law of motion of the center of mass

Theorem on the motion of the center of mass (center of inertia) of the system- one of the general theorems of dynamics, is a consequence of Newton's laws. Asserts that the acceleration of the center of mass of a mechanical system does not depend on the internal forces acting on the bodies of the system, and relates this acceleration to the external forces acting on the system.

The objects referred to in the theorem may, in particular, be the following:

The impulse of a material point and a system of bodies is a physical vector quantity, which is a measure of the action of a force, and depends on the time of the force.

Law of conservation of momentum (proof)

Law of conservation of momentum(The law of conservation of momentum) states that the vector sum of the impulses of all bodies of the system is a constant value if the vector sum of the external forces acting on the system is equal to zero.

In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. From Newton's laws, it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces.

Like any of the fundamental conservation laws, the law of conservation of momentum is associated, according to Noether's theorem, with one of the fundamental symmetries, - homogeneity of space.

According to Newton's second law for a system of N particles:

where is the momentum of the system

a is the resultant of all forces acting on the particles of the system

For systems from N particles in which the sum of all external forces is zero

or for systems whose particles are not affected by external forces (for all k from 1 to n), we have

As you know, if the derivative of some expression is equal to zero, then this expression is a constant relative to the differentiation variable, which means:

(constant vector).

That is, the total momentum of the system from N particles, where N Any integer is a constant value. For N=1 we obtain an expression for one particle.

The law of conservation of momentum is satisfied not only for systems that are not affected by external forces, but also for systems where the sum of all external forces is zero. Equality to zero of all external forces is sufficient, but not necessary for the fulfillment of the law of conservation of momentum.

If the projection of the sum of external forces on any direction or coordinate axis is equal to zero, then in this case one speaks of the law of conservation of the projection of momentum on a given direction or coordinate axis.

Dynamics of rotational motion of a rigid body

The basic law of the dynamics of a MATERIAL POINT during rotational motion can be formulated as follows:

“The product of the moment of inertia and the angular acceleration is equal to the resulting moment of forces acting on a material point: “M = I e.

The basic law of the dynamics of rotational motion of a RIGID BODY relative to a fixed point can be formulated as follows:

“The product of the moment of inertia of a body and its angular acceleration is equal to the total moment of external forces acting on the body. The moments of forces and inertia are taken relative to the axis (z), around which the rotation occurs: "

Basic concepts: moment of force, moment of inertia, moment of impulse

Moment of power (synonyms: torque, torque, torque, torque) is a vector physical quantity equal to the vector product of the radius vector (drawn from the axis of rotation to the point of application of the force - by definition) by the vector of this force. Characterizes the rotational action of force on a rigid body.

The concepts of “rotating” and “torque” moments are generally not identical, since in technology the concept of “rotating” moment is considered as an external force applied to an object, and “torque” is an internal force that occurs in an object under the action of applied loads (this the concept is used in the resistance of materials).

Moment of inertia- a scalar (in the general case - tensor) physical quantity, a measure of inertia in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion. It is characterized by the distribution of masses in the body: the moment of inertia is equal to the sum of the products of elementary masses and the square of their distances to the base set (point, line or plane).

Unit of measure in the International System of Units (SI): kg m².

angular momentum(kinetic moment, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed about the axis of rotation, and how fast the rotation occurs.

It should be noted that rotation here is understood in a broad sense, not only as a regular rotation around an axis. For example, even with a rectilinear motion of a body past an arbitrary imaginary point that does not lie on the line of motion, it also has an angular momentum. Perhaps the greatest role is played by the angular momentum in describing the actual rotational motion. However, it is extremely important for a much wider class of problems (especially if the problem has central or axial symmetry, but not only in these cases).

Comment: angular momentum about a point is a pseudovector, and angular momentum about an axis is a pseudoscalar.

The angular momentum of a closed system is conserved.