Momentum is a physical quantity that, under certain conditions, remains constant for a system of interacting bodies. The modulus of momentum is equal to the product of mass and velocity (p = mv). The law of conservation of momentum is formulated as follows:

In a closed system of bodies, the vector sum of the bodies’ momenta remains constant, i.e., does not change. By closed we mean a system where bodies interact only with each other. For example, if friction and gravity can be neglected. Friction can be small, and the force of gravity is balanced by the force of the normal reaction of the support.

Let's say one moving body collides with another body of the same mass, but motionless. What will happen? Firstly, a collision can be elastic or inelastic. In an inelastic collision, the bodies stick together into one whole. Let's consider just such a collision.

Since the masses of the bodies are the same, we denote their masses by the same letter without an index: m. The momentum of the first body before the collision is equal to mv 1, and the second is equal to mv 2. But since the second body is not moving, then v 2 = 0, therefore, the momentum of the second body is 0.

After an inelastic collision, the system of two bodies will continue to move in the direction where the first body was moving (the momentum vector coincides with the velocity vector), but the speed will become 2 times less. That is, the mass will increase by 2 times, and the speed will decrease by 2 times. Thus, the product of mass and speed will remain the same. The only difference is that before the collision the speed was 2 times greater, but the mass was equal to m. After the collision, the mass became 2m, and the speed was 2 times less.

Let us imagine that two bodies moving towards each other inelastically collide. The vectors of their velocities (as well as impulses) are directed in opposite directions. This means that the pulse modules must be subtracted. After the collision, the system of two bodies will continue to move in the direction in which the body with greater momentum was moving before the collision.

For example, if one body had a mass of 2 kg and moved with a speed of 3 m/s, and the other had a mass of 1 kg and a speed of 4 m/s, then the impulse of the first is 6 kg m/s, and the impulse of the second is 4 kg m /With. This means that the velocity vector after the collision will be codirectional with the velocity vector of the first body. But the speed value can be calculated like this. The total impulse before the collision was equal to 2 kg m/s, since the vectors are opposite directions, and we must subtract the values. It should remain the same after the collision. But after the collision, the body mass increased to 3 kg (1 kg + 2 kg), which means that from the formula p = mv it follows that v = p/m = 2/3 = 1.6(6) (m/s). We see that as a result of the collision the speed decreased, which is consistent with our everyday experience.

If two bodies are moving in one direction and one of them catches up with the second, pushes it, engaging with it, then how will the speed of this system of bodies change after the collision? Let's say a body weighing 1 kg moved at a speed of 2 m/s. A body weighing 0.5 kg, moving at a speed of 3 m/s, caught up with him and grappled with him.

Since the bodies move in one direction, the impulse of the system of these two bodies is equal to the sum of the impulses of each body: 1 2 = 2 (kg m/s) and 0.5 3 = 1.5 (kg m/s). The total impulse is 3.5 kg m/s. It should remain the same after the collision, but the body mass here will already be 1.5 kg (1 kg + 0.5 kg). Then the speed will be equal to 3.5/1.5 = 2.3(3) (m/s). This speed is greater than the speed of the first body and less than the speed of the second. This is understandable, the first body was pushed, and the second, one might say, encountered an obstacle.

Now imagine that two bodies are initially coupled. Some equal force pushes them in different directions. What will be the speed of the bodies? Since equal force is applied to each body, the modulus of the impulse of one must be equal to the modulus of the impulse of the other. However, the vectors are oppositely directed, so when their sum will be equal to zero. This is correct, because before the bodies moved apart, their momentum was equal to zero, because the bodies were at rest. Since momentum is equal to the product of mass and speed, in this case it is clear that the more massive the body, the lower its speed will be. The lighter the body, the greater its speed will be.

Instructions

Find the mass of the moving body and measure its motion. After its interaction with another body, the speed of the body under study will change. In this case, subtract the initial speed from the final (after interaction) and multiply the difference by the body mass Δp=m∙(v2-v1). Measure the instantaneous speed with a radar and the body mass with a scale. If, after the interaction, the body begins to move in the direction opposite to that in which it moved before the interaction, then the final speed will be negative. If it is positive, it has increased, if negative, it has decreased.

Since the cause of a change in the speed of any body is force, it is also the cause of a change in momentum. To calculate the change in momentum of any body, it is enough to find the momentum of the force acting on this body at some time. Using a dynamometer, measure the force that causes a body to change speed, giving it acceleration. At the same time, use a stopwatch to measure the time that this force acts on the body. If a force causes a body to move, then consider it positive, but if it slows down its movement, consider it negative. An impulse of force equal to the change in impulse will be the product of the force and the time of its action Δp=F∙Δt.

Determining instantaneous speed with a speedometer or radar If a moving body is equipped with a speedometer (), then instantaneous speed will be continuously displayed on its scale or electronic display speed at a given moment in time. When observing a body from a fixed point (), send a radar signal to it, an instantaneous signal will be displayed on its display speed bodies at a given moment in time.

Video on the topic

Force is a physical quantity acting on a body, which, in particular, imparts some acceleration to it. To find pulse strength, you need to determine the change in momentum, i.e. pulse but the body itself.

Instructions

The movement of a material point under the influence of some strength or forces that give it acceleration. Application result strength a certain amount for a certain amount is the corresponding quantity. Impulse strength the measure of its action over a certain period of time is called: Pc = Fav ∆t, where Fav is the average force acting on the body; ∆t is the time interval.

Thus, pulse strength equal to change pulse and the body: Pc = ∆Pt = m (v – v0), where v0 is the initial speed; v is the final speed of the body.

The resulting equality reflects Newton's second law in relation to the inertial reference system: the derivative of the function of a material point with respect to time is equal to the magnitude of the constant force acting on it: Fav ∆t = ∆Pt → Fav = dPt/dt.

Total pulse a system of several bodies can change only under the influence of external forces, and its value is directly proportional to their sum. This statement is a consequence of Newton's second and third laws. Let there be three interacting bodies, then it is true: Pс1 + Pc2 + Pc3 = ∆Pт1 + ∆Pт2 + ∆Pт3, where Pci – pulse strength, acting on the body i;Pтi – pulse bodies i.

This equality shows that if the sum of external forces is zero, then the total pulse closed system of bodies is always constant, despite the fact that the internal strength

In some cases, it is possible to study the interaction of bodies without using expressions for the forces acting between the bodies. This is possible due to the fact that there are physical quantities that remain unchanged (conserved) when bodies interact. In this chapter we will look at two such quantities - momentum and mechanical energy.
Let's start with momentum.

A physical quantity equal to the product of a body’s mass m and its speed is called the body’s momentum (or simply momentum):

Momentum is a vector quantity. The magnitude of the impulse is p = mv, and the direction of the impulse coincides with the direction of the body's velocity. The unit of impulse is 1 (kg * m)/s.

1. A truck weighing 3 tons is driving along a highway in the north direction at a speed of 40 km/h. In what direction and at what speed should a passenger car weighing 1 ton travel so that its momentum is equal to the impulse of the truck?

2. A ball with a mass of 400 g falls freely without an initial speed from a height of 5 m. After the impact, the ball bounces up, and the modulus of the ball’s velocity does not change as a result of the impact.
a) What is the magnitude and direction of the ball’s momentum immediately before impact?
b) What is the magnitude and direction of the ball’s momentum immediately after impact?
c) What is the change in momentum of the ball as a result of the impact and in what direction? Find the change in momentum graphically.
Clue. If the momentum of the body was equal to 1, and became equal to 2, then the change in momentum ∆ = 2 – 1.

2. Law of conservation of momentum

The most important property of momentum is that, under certain conditions, the total momentum of interacting bodies remains unchanged (conserved).

Let's put experience

Two identical carts can roll along a table along the same straight line with virtually no friction. (This experiment can be carried out with modern equipment.) The absence of friction is an important condition for our experiment!

We will install latches on the carts, thanks to which the carts move as one body after a collision. Let the right cart initially be at rest, and with the left push we impart speed 0 (Fig. 25.1, a).

After the collision, the carts move together. Measurements show that their total speed is 2 times less than the initial speed of the left cart (25.1, b).

Let us denote the mass of each cart as m and compare the total impulses of the carts before and after the collision.

We see that the total momentum of the carts remained unchanged (preserved).

Maybe this is only true when the bodies move as a single unit after interaction?

Let's put experience
Let's replace the latches with an elastic spring and repeat the experiment (Fig. 25.2).

This time the left cart stopped, and the right one acquired a speed equal to the initial speed of the left cart.

3. Prove that in this case the total momentum of the carts is conserved.

Maybe this is true only when the masses of the interacting bodies are equal?

Let's put experience
Let's attach another similar cart to the right cart and repeat the experiment (Fig. 25.3).

Now, after the collision, the left cart began to move in the opposite direction (that is, to the left) at a speed equal to -/3, and the double cart began to move to the right at a speed of 2/3.

4. Prove that in this experiment the total momentum of the carts was conserved.

To determine under what conditions the total momentum of bodies is conserved, let us introduce the concept of a closed system of bodies. This is the name given to a system of bodies that interact only with each other (that is, they do not interact with bodies that are not part of this system).

Exactly closed systems of bodies do not exist in nature, if only because it is impossible to “turn off” the forces of universal gravity.

But in many cases, a system of bodies can be considered closed with good accuracy. For example, when external forces (forces acting on the bodies of the system from other bodies) balance each other or can be neglected.

This is exactly what happened in our experiments with carts: the external forces acting on them (gravity and the normal reaction force) balanced each other, and the friction force could be neglected. Therefore, the speeds of the carts changed only as a result of their interaction with each other.

The experiments described, as well as many others like them, indicate that
law of conservation of momentum: the vector sum of the momenta of the bodies that make up a closed system does not change during any interactions between the bodies of the system:
The law of conservation of momentum is satisfied only in inertial frames of reference.

Law of conservation of momentum as a consequence of Newton's laws

Let us show, using the example of a closed system of two interacting bodies, that the law of conservation of momentum is a consequence of Newton’s second and third laws.

Let us denote the masses of the bodies as m 1 and m 2, and their initial velocities as 1 and 2. Then the vector sum of the momenta of the bodies

Let the interacting bodies move with accelerations 1 and 2 during a period of time ∆t.

5. Explain why the change in the total momentum of bodies can be written in the form

Clue. Use the fact that for each body ∆ = m∆, and also the fact that ∆ = ∆t.

6. Let us denote 1 and 2 forces acting on the first and second bodies, respectively. Prove that

Clue. Take advantage of Newton's second law and the fact that the system is closed, as a result of which the accelerations of bodies are caused only by the forces with which these bodies act on each other.

7. Prove that

Clue. Use Newton's third law.

So, the change in the total momentum of the interacting bodies is zero. And if the change in a certain quantity is zero, then this means that this quantity is conserved.

8. Why does it follow from the above reasoning that the law of conservation of momentum is satisfied only in inertial frames of reference?

3. Force impulse

There is a saying: “If only I knew where you would fall, I would lay down straws.” Why do you need a “straw”? Why do athletes fall or jump on soft mats during training and competitions rather than on the hard floor? Why after a jump should you land on bent legs and not straightened ones? Why do cars need seat belts and airbags?
We can answer all these questions by becoming familiar with the concept of “force impulse”.

The impulse of a force is the product of a force and the time interval ∆t during which this force acts.

It is no coincidence that the name “impulse of force” “echoes” the concept of “impulse”. Let us consider the case when a body of mass m is acted upon by a force during a period of time ∆t.

9. Prove that the change in the momentum of the body ∆ is equal to the momentum of the force acting on this body:

Clue. Use the fact that ∆ = m∆ and Newton's second law.

Let us rewrite formula (6) in the form

This formula is another form of writing Newton's second law. (It was in this form that Newton himself formulated this law.) It follows from it that a large force acts on a body if its momentum changes significantly in a very short period of time ∆t.

This is why large forces arise during impacts and collisions: impacts and collisions are characterized by precisely a short interaction time interval.

To weaken the force of an impact or reduce the forces arising when bodies collide, it is necessary to lengthen the period of time during which the impact or collision occurs.

10. Explain the meaning of the saying given at the beginning of this section, and also answer the other questions placed in the same paragraph.

11. A ball with a mass of 400 g hit a wall and bounced off it with the same absolute speed, equal to 5 m/s. Just before impact, the ball's speed was directed horizontally. What is the average force exerted by the ball on the wall if it was in contact with the wall for 0.02 s?

12. A cast iron block weighing 200 kg falls from a height of 1.25 m into sand and sinks 5 cm into it.
a) What is the momentum of the blank immediately before the impact?
b) What is the change in momentum of the blank during the impact?
c) How long did the blow last?
d) What is the average impact force?

Additional questions and tasks

13. A ball with a mass of 200 g moves at a speed of 2 m/s to the left. How should another ball of mass 100 g move so that the total momentum of the balls is zero?

14. A ball with a mass of 300 g moves uniformly in a circle of radius 50 cm at a speed of 2 m/s. What is the modulus of change in the momentum of the ball:
a) for one full circulation period?
b) for half the circulation period?
c) in 0.39 s?

15. The first board lies on the asphalt, and the second board is the same - on loose sand. Explain why it is easier to hammer a nail into the first board than into the second?

16. A bullet weighing 10 g, flying at a speed of 700 m/s, pierced the board, after which the bullet speed became equal to 300 m/s. Inside the board, the bullet moved for 40 μs.
a) What is the change in momentum of the bullet due to passing through the board?
b) What average force did the bullet exert on the board as it passed through it?

Impulse of force. Body impulse

Basic dynamic quantities: force, mass, body impulse, moment of force, angular momentum.

Force is a vector quantity, which is a measure of the action of other bodies or fields on a given body.

Strength is characterized by:

· Module

Direction

Application point

In the SI system, force is measured in newtons.

In order to understand what a force of one Newton is, we need to remember that a force applied to a body changes its speed. In addition, let us remember the inertia of bodies, which, as we remember, is associated with their mass. So,

One newton is a force that changes the speed of a body weighing 1 kg by 1 m/s every second.

Examples of forces include:

· Gravity– a force acting on a body as a result of gravitational interaction.

· Elastic force- the force with which a body resists an external load. Its cause is the electromagnetic interaction of body molecules.

· Archimedes' force- a force associated with the fact that a body displaces a certain volume of liquid or gas.

· Ground reaction force- the force with which the support acts on the body located on it.

· Friction force– the force of resistance to the relative movement of the contacting surfaces of bodies.

· Surface tension is a force that occurs at the interface between two media.

· Body weight- the force with which the body acts on a horizontal support or vertical suspension.

And other forces.

Strength is measured using a special device. This device is called a dynamometer (Fig. 1). The dynamometer consists of spring 1, the stretching of which shows us the force, arrow 2, sliding along scale 3, limiter bar 4, which prevents the spring from stretching too much, and hook 5, from which the load is suspended.

Rice. 1. Dynamometer (Source)

Many forces can act on the body. In order to correctly describe the movement of a body, it is convenient to use the concept of resultant forces.

The resultant force is a force whose action replaces the action of all forces applied to the body (Fig. 2).

Knowing the rules for working with vector quantities, it is easy to guess that the resultant of all forces applied to a body is the vector sum of these forces.

Rice. 2. Resultant of two forces acting on a body

In addition, since we are considering the movement of a body in some coordinate system, it is usually advantageous for us to consider not the force itself, but its projection onto the axis. The projection of force on the axis can be negative or positive, because the projection is a scalar quantity. So, in Figure 3 the projections of forces are shown, the projection of force is negative, and the projection of force is positive.

Rice. 3. Projections of forces onto the axis

So, from this lesson we have deepened our understanding of the concept of strength. We remembered the units of measurement of force and the device with which force is measured. In addition, we looked at what forces exist in nature. Finally, we learned how to act when several forces act on the body.

Weight, a physical quantity, one of the main characteristics of matter, determining its inertial and gravitational properties. Accordingly, a distinction is made between inertial Mass and gravitational Mass (heavy, gravitating).

The concept of Mass was introduced into mechanics by I. Newton. In classical Newtonian mechanics, Mass is included in the definition of momentum (amount of motion) of a body: momentum R proportional to the speed of the body v, p = mv(1). The proportionality coefficient is a constant value for a given body m- and is the Mass of the body. The equivalent definition of Mass is obtained from the equation of motion of classical mechanics f = ma(2). Here Mass is the coefficient of proportionality between the force acting on the body f and the acceleration of the body caused by it a. The mass defined by relations (1) and (2) is called inertial mass, or inertial mass; it characterizes the dynamic properties of a body, is a measure of the inertia of the body: with a constant force, the greater the mass of the body, the less acceleration it acquires, i.e., the slower the state of its motion changes (the greater its inertia).

By acting on different bodies with the same force and measuring their accelerations, we can determine the relationship between the mass of these bodies: m 1: m 2: m 3 ... = a 1: a 2: a 3 ...; if one of the Masses is taken as a unit of measurement, the Mass of the remaining bodies can be found.

In Newton's theory of gravity, Mass appears in a different form - as a source of the gravitational field. Each body creates a gravitational field proportional to the Mass of the body (and is affected by the gravitational field created by other bodies, the strength of which is also proportional to the Mass of the bodies). This field causes the attraction of any other body to this body with a force determined by Newton’s law of gravity:

(3)

Where r- distance between bodies, G is the universal gravitational constant, a m 1 And m 2- Masses of attracting bodies. From formula (3) it is easy to obtain the formula for weight R body mass m in the Earth's gravitational field: P = mg (4).

Here g = G*M/r 2- acceleration of free fall in the gravitational field of the Earth, and r » R- the radius of the Earth. The mass determined by relations (3) and (4) is called the gravitational mass of the body.

In principle, it does not follow from anywhere that the Mass that creates the gravitational field also determines the inertia of the same body. However, experience has shown that inertial Mass and gravitational Mass are proportional to each other (and with the usual choice of units of measurement, they are numerically equal). This fundamental law of nature is called the principle of equivalence. Its discovery is associated with the name of G. Galileo, who established that all bodies on Earth fall with the same acceleration. A. Einstein put this principle (formulated by him for the first time) into the basis of the general theory of relativity. The equivalence principle has been established experimentally with very high accuracy. For the first time (1890-1906), a precision test of the equality of inertial and gravitational Masses was carried out by L. Eotvos, who found that the Masses coincide with an error of ~ 10 -8. In 1959-64, American physicists R. Dicke, R. Krotkov and P. Roll reduced the error to 10 -11, and in 1971, Soviet physicists V.B. Braginsky and V.I. Panov - to 10 -12.

The principle of equivalence allows us to most naturally determine body weight by weighing.

Initially, Mass was considered (for example, by Newton) as a measure of the amount of matter. This definition has a clear meaning only for comparing homogeneous bodies built from the same material. It emphasizes the additivity of Mass - the Mass of a body is equal to the sum of the Mass of its parts. The mass of a homogeneous body is proportional to its volume, so we can introduce the concept of density - Mass of a unit volume of a body.

In classical physics it was believed that the mass of a body does not change in any processes. This corresponded to the law of conservation of Mass (matter), discovered by M.V. Lomonosov and A.L. Lavoisier. In particular, this law stated that in any chemical reaction the sum of the Masses of the initial components is equal to the sum of the Masses of the final components.

The concept of Mass acquired a deeper meaning in the mechanics of A. Einstein’s special theory of relativity, which considers the movement of bodies (or particles) at very high speeds - comparable to the speed of light with ~ 3 10 10 cm/sec. In new mechanics - it is called relativistic mechanics - the relationship between momentum and velocity of a particle is given by the relation:

(5)

At low speeds ( v << c) this relation goes into the Newtonian relation p = mv. Therefore the value m 0 is called rest mass, and the mass of a moving particle m is defined as the speed-dependent proportionality coefficient between p And v:

(6)

Bearing in mind, in particular, this formula, they say that the mass of a particle (body) grows with an increase in its speed. Such a relativistic increase in the mass of a particle as its speed increases must be taken into account when designing accelerators of high-energy charged particles. Rest mass m 0(Mass in the reference frame associated with the particle) is the most important internal characteristic of the particle. All elementary particles have strictly defined meanings m 0, inherent in a given type of particle.

It should be noted that in relativistic mechanics, the definition of Mass from the equation of motion (2) is not equivalent to the definition of Mass as a coefficient of proportionality between the momentum and the speed of the particle, since the acceleration ceases to be parallel to the force that caused it and the Mass turns out to depend on the direction of the particle’s speed.

According to the theory of relativity, Particle mass m connected to her energy E ratio:

(7)

The rest mass determines the internal energy of the particle - the so-called rest energy E 0 = m 0 s 2. Thus, energy is always associated with Mass (and vice versa). Therefore, there is no separate law (as in classical physics) of the conservation of Mass and the law of conservation of energy - they are merged into a single law of conservation of total (i.e., including the rest energy of particles) energy. An approximate division into the law of conservation of energy and the law of conservation of mass is possible only in classical physics, when particle velocities are small ( v << c) and particle transformation processes do not occur.

In relativistic mechanics, Mass is not an additive characteristic of a body. When two particles combine to form one compound stable state, an excess of energy (equal to the binding energy) is released D E, which corresponds to Mass D m = D E/s 2. Therefore, the Mass of a composite particle is less than the sum of the Masses of the particles forming it by the amount D E/s 2(the so-called mass defect). This effect is especially pronounced in nuclear reactions. For example, deuteron mass ( d) is less than the sum of proton masses ( p) and neutron ( n); defect Mass D m associated with energy E g gamma quantum ( g), born during the formation of a deuteron: p + n -> d + g, E g = Dmc 2. The Mass defect that occurs during the formation of a composite particle reflects the organic connection between Mass and energy.

The unit of mass in the CGS system of units is gram, and in International System of Units SI - kilogram. The mass of atoms and molecules is usually measured in atomic mass units. The mass of elementary particles is usually expressed either in units of electron mass m e, or in energy units, indicating the rest energy of the corresponding particle. Thus, the mass of an electron is 0.511 MeV, the mass of a proton is 1836.1 m e, or 938.2 MeV, etc.

The nature of Mass is one of the most important unsolved problems of modern physics. It is generally accepted that the mass of an elementary particle is determined by the fields that are associated with it (electromagnetic, nuclear and others). However, a quantitative theory of Mass has not yet been created. There is also no theory that explains why the mass of elementary particles forms a discrete spectrum of values, much less allows us to determine this spectrum.

In astrophysics, the mass of a body creating a gravitational field determines the so-called gravitational radius of the body R gr = 2GM/s 2. Due to gravitational attraction, no radiation, including light, can escape beyond the surface of a body with a radius R=< R гр . Stars of this size will be invisible; That's why they were called "black holes". Such celestial bodies must play an important role in the Universe.

Impulse of force. Body impulse

The concept of momentum was introduced in the first half of the 17th century by Rene Descartes, and then refined by Isaac Newton. According to Newton, who called momentum the quantity of motion, this is a measure of it, proportional to the speed of a body and its mass. Modern definition: The momentum of a body is a physical quantity equal to the product of the mass of the body and its speed:

First of all, from the above formula it is clear that impulse is a vector quantity and its direction coincides with the direction of the body’s speed; the unit of measurement for impulse is:

= [kg m/s]

Let us consider how this physical quantity is related to the laws of motion. Let's write down Newton's second law, taking into account that acceleration is the change in speed over time:

There is a connection between the force acting on the body, or more precisely, the resultant force, and the change in its momentum. The magnitude of the product of a force and a period of time is called the impulse of force. From the above formula it is clear that the change in the momentum of the body is equal to the impulse of the force.

What effects can be described using this equation (Fig. 1)?

Rice. 1. Relationship between force impulse and body impulse (Source)

An arrow fired from a bow. The longer the contact of the string with the arrow continues (∆t), the greater the change in the arrow's momentum (∆), and therefore, the higher its final speed.

Two colliding balls. While the balls are in contact, they act on each other with forces equal in magnitude, as Newton’s third law teaches us. This means that the changes in their momenta must also be equal in magnitude, even if the masses of the balls are not equal.

After analyzing the formulas, two important conclusions can be drawn:

1. Identical forces acting for the same period of time cause the same changes in momentum in different bodies, regardless of the mass of the latter.

2. The same change in the momentum of a body can be achieved either by acting with a small force over a long period of time, or by acting briefly with a large force on the same body.

According to Newton's second law, we can write:

∆t = ∆ = ∆ / ∆t

The ratio of the change in the momentum of a body to the period of time during which this change occurred is equal to the sum of the forces acting on the body.

Having analyzed this equation, we see that Newton's second law allows us to expand the class of problems to be solved and include problems in which the mass of bodies changes over time.

If we try to solve problems with variable mass of bodies using the usual formulation of Newton’s second law:

then attempting such a solution would lead to an error.

An example of this is the already mentioned jet plane or space rocket, which burn fuel while moving, and the products of this combustion are released into the surrounding space. Naturally, the mass of an aircraft or rocket decreases as fuel is consumed.

MOMENT OF POWER- quantity characterizing the rotational effect of the force; has the dimension of the product of length and force. Distinguish moment of power relative to the center (point) and relative to the axis.

M. s. relative to the center ABOUT called vector quantity M 0 equal to the vector product of the radius vector r , carried out from O to the point of application of force F , to strength M 0 = [rF ] or in other notations M 0 = r F (rice.). Numerically M. s. equal to the product of the modulus of force and the arm h, i.e. by the length of the perpendicular lowered from ABOUT on the line of action of the force, or twice the area

triangle built on the center O and strength:

Directed vector M 0 perpendicular to the plane passing through O And F . Side to which it is heading M 0, selected conditionally ( M 0 - axial vector). With a right-handed coordinate system, the vector M 0 is directed in the direction from which the rotation made by the force is visible counterclockwise.

M. s. relative to the z-axis called scalar quantity Mz, equal to the projection onto the axis z vector M. s. relative to any center ABOUT, taken on this axis; size Mz can also be defined as a projection onto a plane xy, perpendicular to the z axis, the area of ​​the triangle OAB or as a moment of projection Fxy strength F to the plane xy, taken relative to the point of intersection of the z axis with this plane. T. o.,

In the last two expressions of M. s. is considered positive when the rotation force Fxy visible from positive the end of the z axis counterclockwise (in the right coordinate system). M. s. relative to coordinate axes Oxyz can also be calculated analytically. f-lam:

Where Fx, Fy, Fz- force projections F on the coordinate axes, x, y, z- point coordinates A application of force. Quantities M x , M y , M z are equal to the projections of the vector M 0 on the coordinate axes.

The momentum of a body is a vector physical quantity, which is equal to the product of the speed of the body and its mass. Also, the momentum of a body has a second name - momentum. The direction of the body's momentum coincides with the direction of the velocity vector. The momentum of a body in the C system does not have its own unit of measurement. Therefore, it is measured in the units included in its composition: kilogrammometer per second kgm/s.

Formula 1 - Body impulse.

m is body weight.

v is the speed of the body.

The momentum of a body is, in fact, a new interpretation of Newton's second law. In which the acceleration was simply expanded. In this case, the value Ft was called the impulse of force, and mv was called the impulse of the body.

The impulse of a force is a physical quantity of a vector nature that determines the degree of action of a force over the period of time during which it acts.

Formula 2 - Newton's second law, body momentum.

m is body weight.

v1 is the initial speed of the body.

v2 is the final speed of the body.

a is the acceleration of the body.

p is the momentum of the body.

t1 - start time

t2 is the final time.

This was done so that it was possible to calculate problems associated with the movement of bodies of variable mass and at speeds comparable to the speed of light.

The new interpretation of Newton's second law should be understood as follows. As a result of the action of force F during time t on a body of mass m, its speed will become equal to V.

In a closed system, the magnitude of the momentum is constant, this is the law of conservation of momentum. Let us recall that a closed system is a system that is not affected by external forces. An example of such a system would be two dissimilar balls moving along a straight path towards each other, at the same speed. The balls have the same diameter. There are no friction forces during movement. Since the balls are made of different materials, they have different masses. But at the same time, the material ensures absolute elasticity of bodies.

As a result of the collision of the balls, the lighter one will bounce off at a higher speed. And the heavier one will roll back more slowly. Since the impulse of the body imparted by a heavier ball to a lighter one is greater than the impulse given by a light ball to a heavy one.

Figure 1 - Law of conservation of momentum.

Thanks to the law of conservation of momentum, reactive motion can be described. Unlike other types of motion, reactive motion does not require interaction with other bodies. For example, a car moves due to the force of friction, which pushes it away from the surface of the earth. During jet motion, interaction with other bodies does not occur. Its cause is the separation of part of its mass from the body at a certain speed. That is, part of the fuel is separated from the engine in the form of expanding gases, while they move at enormous speed. Accordingly, the engine itself acquires a certain impulse, which imparts speed to it.