His movements, i.e. value .
Pulse is a vector quantity coinciding in direction with the velocity vector.
The unit of momentum in the SI system: kg m/s .
The impulse of a system of bodies is equal to the vector sum of the impulses of all bodies included in the system:
Law of conservation of momentum
If additional external forces act on the system of interacting bodies, for example, then in this case the relation is valid, which is sometimes called the law of momentum change:
For a closed system (in the absence of external forces), the law of conservation of momentum is valid:
The action of the law of conservation of momentum can explain the phenomenon of recoil when shooting from a rifle or during artillery shooting. Also, the operation of the law of conservation of momentum underlies the principle of operation of all jet engines.
When solving physical problems, the law of conservation of momentum is used when knowledge of all the details of motion is not required, but the result of the interaction of bodies is important. Such problems, for example, are the problems of impact or collision of bodies. The law of conservation of momentum is used when considering the motion of bodies of variable mass, such as launch vehicles. Most of the mass of such a rocket is fuel. In the active phase of the flight, this fuel burns out, and the mass of the rocket rapidly decreases in this part of the trajectory. Also, the law of conservation of momentum is necessary in cases where the concept is inapplicable. It is difficult to imagine a situation where a motionless body acquires some speed instantly. In normal practice, bodies always accelerate and pick up speed gradually. However, during the movement of electrons and other subatomic particles, the change in their state occurs abruptly without staying in intermediate states. In such cases, the classical concept of "acceleration" cannot be applied.
Examples of problem solving
EXAMPLE 1
| Exercise | A projectile with a mass of 100 kg, flying horizontally along a railway track at a speed of 500 m/s, hits a wagon with sand of mass 10 tons and gets stuck in it. What speed will the car get if it moves at a speed of 36 km/h in the direction opposite to the projectile? |
| Solution | The wagon+projectile system is closed, so in this case the momentum conservation law can be applied. Let's make a drawing, indicating the state of the bodies before and after the interaction.
When the projectile and the car interact, an inelastic impact occurs. The law of conservation of momentum in this case will be written as: Choosing the direction of the axis coinciding with the direction of movement of the car, we write the projection of this equation onto the coordinate axis: where is the speed of the car after a projectile hits it:
We convert units to the SI system: t kg. Let's calculate: |
| Answer | After hitting the projectile, the car will move at a speed of 5 m/s. |
EXAMPLE 2
| Exercise | A projectile with mass m=10 kg had a speed v=200 m/s at the top point . At this point, it broke into two pieces. A smaller part with a mass m 1 =3 kg received a speed v 1 =400 m/s in the same direction at an angle to the horizon. With what speed and in what direction will most of the projectile fly? |
| Solution | The trajectory of the projectile is a parabola. The speed of the body is always directed tangentially to the trajectory. At the top of the trajectory, the velocity of the projectile is parallel to the axis.
Let's write the momentum conservation law: Let's pass from vectors to scalars. To do this, we square both parts of the vector equality and use the formulas for: Given that and also that , we find the speed of the second fragment: Substituting the numerical values of physical quantities into the resulting formula, we calculate: The direction of flight of most of the projectile is determined using:
Substituting numerical values into the formula, we get: |
| Answer | Most of the projectile will fly at a speed of 249 m / s down at an angle to the horizontal direction. |
EXAMPLE 3
| Exercise | The mass of the train is 3000 tons. The coefficient of friction is 0.02. What should be the size of the steam locomotive for the train to pick up a speed of 60 km / h 2 minutes after the start of movement. |
| Solution | Since an (external force) acts on the train, the system cannot be considered closed, and the law of conservation of momentum does not hold in this case. Let's use the law of momentum change: Since the friction force is always directed in the direction opposite to the movement of the body, in the projection of the equation on the coordinate axis (the direction of the axis coincides with the direction of the train movement), the friction force impulse will enter with a minus sign: |
Basic dynamic quantities: force, mass, momentum of the body, moment of force, moment of impulse.
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 recall the inertia of bodies, which, as we remember, is related to their mass. So,
One newton is such a force that changes the speed of a body with a mass of 1 kg by 1 m / s for every second.
Examples of forces are:
· Gravity- the force acting on the body as a result of gravitational interaction.
· Elastic force is the force with which a body resists an external load. Its cause is the electromagnetic interaction of body molecules.
· Strength of Archimedes- the force associated with the fact that the body displaces a certain volume of liquid or gas.
· Support reaction force- the force with which the support acts on the body located on it.
· Friction force is the force of resistance to the relative movement of the contacting surfaces of the bodies.
· The force of surface tension is the 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.
Force is measured using a special device. This device is called a dynamometer (Fig. 1). The dynamometer consists of a spring 1, the stretching of which shows us the force, an arrow 2 sliding along a scale 3, a limiter bar 4, which prevents the spring from stretching too much, and a hook 5, to which the load is suspended.
Rice. 1. Dynamometer (Source)
Many forces can act on a body. In order to correctly describe the motion of a body, it is convenient to use the concept of resultant forces.
The resultant of forces 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 the body is the vector sum of these forces.
Rice. 2. The resultant of two forces acting on the body
In addition, since we are considering the motion of a body in some coordinate system, it is usually beneficial for us to consider not the force itself, but its projection onto the axis. The projection of the force on the axis can be negative or positive, because the projection is a scalar quantity. So, Figure 3 shows the projections of forces, the projection of the force is negative, and the projection of the force is positive.
Rice. 3. Projections of forces on the axis
So, from this lesson, we have deepened our understanding of the concept of force. We remembered the units of measurement of force and the device with which force is measured. In addition, we have considered what forces exist in nature. Finally, we learned how to act if several forces act on the body.
Weight, a physical quantity, one of the main characteristics of matter, which determines its inertial and gravitational properties. Accordingly, the inertial Mass and the Gravitational Mass (heavy, gravitating) are distinguished.
The concept of Mass was introduced into mechanics by I. Newton. In classical Newtonian mechanics, mass is included in the definition of momentum (momentum) of a body: momentum R proportional to the speed of the body v, p=mv(one). The coefficient of proportionality is a constant value for a given body m- and there is the mass of the body. An 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. Defined by relations (1) and (2) Mass is called inertial mass, or inertial mass; it characterizes the dynamic properties of the body, is a measure of the inertia of the body: at 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).
Acting on different bodies with the same force and measuring their accelerations, we can determine the ratios of 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, one can find the Mass of the remaining bodies.
In Newton's theory of gravity, Mass appears in a different form - as the 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- universal gravitational constant, a m 1 and m2- Masses of attracting bodies. From formula (3) it is easy to obtain a formula for weight R bodies of mass m in the Earth's gravitational field: P = mg (4).
Here g \u003d G * M / r 2 is the 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 determines the inertia of the same body. However, experience has shown that the inertial Mass and the 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 (first formulated by him) into the basis of the general theory of relativity. The principle of equivalence has been established experimentally with very high accuracy. For the first time (1890-1906) a precision check of the equality of the inertial and gravitational Masses was made by L. Eötvös, 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 reduced the error to 10 -12 .
The principle of equivalence allows the most natural way to determine body weight by weighing.
Initially, Mass was considered (for example, by Newton) as a measure of the amount of matter. Such a definition has a clear meaning only for comparing homogeneous bodies built from the same material. It emphasizes the additivity of the Mass - the Mass of a body is equal to the sum of the Masses of its parts. The mass of a homogeneous body is proportional to its volume, so we can introduce the concept of density - Mass per unit volume of the 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 (substance), 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) with very high speeds - comparable to the speed of light with ~ 3 10 10 cm/sec. In the new mechanics - it's called relativistic mechanics - the relationship between momentum and particle velocity is given by:
(5)
At low speeds ( v << c) this relation becomes the Newtonian relation p = mv. Therefore, the value m0 is called the rest mass, and the mass of the moving particle m is defined as the speed-dependent proportionality factor between p and v:
(6)
Bearing in mind, in particular, this formula, they say that the Mass of a particle (body) increases with an increase in its speed. Such a relativistic increase in the Mass of a particle as its velocity increases must be taken into account when designing high-energy charged particle accelerators. rest mass m0(Mass in the reference frame associated with the particle) is the most important internal characteristic of the particle. All elementary particles have strictly defined values m0 inherent in this kind of particles.
It should be noted that in relativistic mechanics the definition of the Mass from the equation of motion (2) is not equivalent to the definition of the Mass as a proportionality factor between the momentum and the velocity 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 velocity.
According to the theory of relativity, the mass of a particle m associated with her energy E ratio:
(7)
The rest mass determines the internal energy of the particle - the so-called rest energy E 0 \u003d m 0 s 2. Thus, energy is always associated with Mass (and vice versa). Therefore, there is no separately (as in classical physics) the law of conservation of Mass and the law of conservation of energy - they are merged into a single law of conservation of total (ie, 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 the particle velocities are small ( v << c) and the processes of transformation of particles do not occur.
In relativistic mechanics Mass is not an additive characteristic of a body. When two particles combine to form one composite stable state, then an excess of energy (equal to the binding energy) is released D E, which corresponds to Mass D m = D E/c 2. Therefore, the Mass of a compound particle is less than the sum of the Masses of its constituent particles by the value D E/c 2(so-called mass defect). This effect is especially pronounced in nuclear reactions. For example, the mass of the deuteron ( 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), which is born during the formation of a deuteron: p + n -> d + g, E g = Dmc 2. The Mass defect, which occurs during the formation of a compound 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 the mass of the electron me, or in energy units, indicating the rest energy of the corresponding particle. So, the mass of an electron is 0.511 MeV, the mass of a proton is 1836.1 me, 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 associated with it (electromagnetic, nuclear, and others). However, the quantitative theory of Mass has not yet been created. There is also no theory explaining why the masses of elementary particles form a discrete spectrum of values, and even more so, allowing to determine this spectrum.
In astrophysics, the mass of a body that creates a gravitational field determines the so-called gravitational radius of the body R gr \u003d 2GM / s 2. Due to gravitational attraction, no radiation, including light, can go outside, beyond the surface of a body with a radius R=< R гр . Stars of this size would be invisible; hence they were called "black holes". Such celestial bodies must play an important role in the universe.
Force impulse. body momentum
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 the momentum the momentum, it is a measure of such, proportional to the speed of the 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 can be seen that the momentum is a vector quantity and its direction coincides with the direction of the body's velocity, the unit of momentum is:
= [kg m/s]
Let us consider how this physical quantity is related to the laws of motion. Let's write Newton's second law, given that acceleration is a change in speed over time:
There is a connection between the force acting on the body, more precisely, the resultant force and the change in its momentum. The magnitude of the product of a force over a period of time is called the impulse of the force. From the above formula it can be seen that the change in the momentum of the body is equal to the momentum of the force.
What effects can be described using this equation (Fig. 1)?
Rice. 1. Relation of the impulse of force with the momentum of the body (Source)
An arrow fired from a bow. The longer the contact of the bowstring with the arrow (∆t), the greater the change in the momentum of the arrow (∆ ), and therefore, the higher its final speed.
Two colliding balls. While the balls are in contact, they act on each other with equal forces, as Newton's third law teaches us. This means that the changes in their momenta must also be equal in absolute value, even if the masses of the balls are not equal.
After analyzing the formulas, two important conclusions can be drawn:
1. The same forces acting for the same period of time cause the same changes in momentum for 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 for a long period of time, or by acting for a short time 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 the body to the period of time during which this change occurred is equal to the sum of the forces acting on the body.
After analyzing 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 a 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 aircraft or space rocket, which, when moving, burn fuel, and the products of this burnt material are thrown 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 O 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 , for strength M 0 = [RF ] or in other notation M 0 = r F (rice.). Numerically M. s. is equal to the product of the modulus of force and the arm h, i.e., the length of the perpendicular dropped from O to the line of action of 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 . The side you are going to M 0 , is chosen conditionally ( M 0 - axial vector). With the right coordinate system, the vector M 0 is directed in the direction from which the turn made by the force is visible counterclockwise.
M. s. about the z-axis rev. scalar Mz, equal to the projection on the axis z vector M. s. about any center O taken on this axis; value Mz can also be defined as a projection onto a plane hu, perpendicular to the z-axis, the area of the triangle OAB or as a moment of projection Fxy strength F to the plane hu, 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 of the force Fxy visible from positive end of the z-axis counterclockwise (in the right coordinate system). M. s. relative to the coordinate axes Oxyz can also be calculated by analytical. f-lam:
where F x , F y , F z- force projections F on the coordinate axes x, y, z- point coordinates BUT application of force. Quantities M x , M y , M z are equal to the projections of the vector M 0 on the coordinate axes.
1. As you know, the result of a force depends on its modulus, point of application, and direction. Indeed, the greater the force acting on the body, the greater the acceleration it acquires. The direction of acceleration also depends on the direction of the force. So, by applying a small force to the handle, we easily open the door, if the same force is applied near the hinges on which the door hangs, then it may not be opened.
Experiments and observations show that the result of the action of a force (interaction) depends not only on the modulus of the force, but also on the time of its action. Let's do an experiment. We will hang a load on a tripod on a thread, to which another thread is tied from below (Fig. 59). If you sharply pull the lower thread, it will break, and the load will remain hanging on the upper thread. If now slowly pull the lower thread, the upper thread will break.
The impulse of force is called a vector physical quantity equal to the product of the force and the time of its action F t .
Unit of momentum of force in SI - newton second (1 N s): [ft] = 1 N s.
The force impulse vector coincides in direction with the force vector.
2. You also know that the result of a force depends on the mass of the body on which the force acts. So, the greater the mass of the body, the less acceleration it acquires under the action of the same force.
Let's consider an example. Imagine that there is a loaded platform on the rails. A wagon moving at a certain speed collides with it. As a result of the collision, the platform will acquire acceleration and move a certain distance. If a wagon moving at the same speed collides with a light wagon, then as a result of the interaction it will move a significantly greater distance than a loaded platform.
Another example. Let's assume that a bullet flies up to the target with a speed of 2 m/s. The bullet will most likely bounce off the target, leaving only a small dent on it. If the bullet flies at a speed of 100 m / s, then it will pierce the target.
Thus, the result of the interaction of bodies depends on their mass and speed.
The momentum of a body is a vector physical quantity equal to the product of the mass of the body and its speed.
|
p = m v. |
Unit of momentum of a body in SI - kilogram meter per second(1 kg m/s): [ p] = [m][v] = 1 kg 1m/s = 1 kg m/s.
The direction of the body's momentum coincides with the direction of its velocity.
Impulse is a relative quantity, its value depends on the choice of reference system. This is understandable, since speed is a relative value.
3. Let us find out how the momentum of the force and the momentum of the body are related.
According to Newton's second law:
F = ma.
Substituting in this formula the expression for acceleration a= , we get:
F= , or
ft = mv – mv 0 .
On the left side of the equality is the impulse of force; on the right side of the equality - the difference between the final and initial momenta of the body, i.e. e. change in momentum of the body.
In this way,
the momentum of the force is equal to the change in the momentum of the body.
|
F t =D( m v). |
This is a different formulation of Newton's second law. This is how Newton put it.
4. Let's assume that two balls moving on the table collide. Any interacting bodies, in this case balls, form system. Forces act between the bodies of the system: the force of action F 1 and counter force F 2. At the same time, the force of action F 1 according to Newton's third law is equal to the reaction force F 2 and is directed opposite to it: F 1 = –F 2 .
The forces with which the bodies of the system interact with each other are called internal forces.
In addition to internal forces, external forces act on the bodies of the system. So, the interacting balls are attracted to the Earth, they are affected by the reaction force of the support. These forces are in this case external forces. During the movement, the air resistance force and the friction force act on the balls. They are also external forces in relation to the system, which in this case consists of two balls.
External forces are called forces that act on the bodies of the system from other bodies.
We will consider such a system of bodies, which is not affected by external forces.
A closed system is a system of bodies interacting with each other and not interacting with other bodies.
In a closed system, only internal forces act.
5. Consider the interaction of two bodies that make up a closed system. Mass of the first body m 1 , its speed before interaction v 01 , after interaction v one . Mass of the second body m 2 , its speed before interaction v 02 , after interaction v 2 .
The forces with which bodies interact, according to the third law: F 1 = –F 2. The time of action of the forces is the same, therefore
F 1 t = –F 2 t.
For each body, we write Newton's second law:
F 1 t = m 1 v 1 – m 1 v 01 , F 2 t = m 2 v 2 – m 2 v 02 .
Since the left parts of the equalities are equal, their right parts are also equal, i.e.
m 1 v 1 – m 1 v 01 = –(m 2 v 2 – m 2 v 02).
Transforming this equality, we get:
|
m 1 v 01 + m 1 v 02 = m 2 v 1 + m 2 v 2 . |
On the left side of the equality is the sum of the momenta of the bodies before the interaction, on the right - the sum of the momentum of the bodies after the interaction. As can be seen from this equality, the momentum of each body changed during the interaction, while the sum of the momenta remained unchanged.
The geometric sum of the impulses of the bodies that make up a closed system remains constant for any interactions of the bodies of this system.
This is what law of conservation of momentum.
6. A closed system of bodies is a model of a real system. There are no systems in nature that would not be affected by external forces. However, in a number of cases, systems of interacting bodies can be considered as closed ones. This is possible in the following cases: the internal forces are much greater than the external forces, the interaction time is short, and the external forces compensate each other. In addition, the projection of external forces on any direction may be equal to zero, and then the momentum conservation law is satisfied for the projections of the momentums of the interacting bodies on this direction.
7. Problem solution example
Two railway platforms are moving towards each other with speeds of 0.3 and 0.2 m/s. The weights of the platforms are respectively 16 and 48 tons. At what speed and in what direction will the platforms move after the automatic coupling?
|
Given: |
SI |
Solution |
|
v 01 = 0.3 m/s v 02 = 0.2 m/s m 1 = 16 t m 2 = 48 t v 1 = v 2 = v |
v 02 = v 02 = 1.6104kg 4.8104kg |
Let us depict in the figure the direction of movement of the platforms before and after the interaction (Fig. 60). The forces of gravity acting on the platforms and the reaction forces of the support compensate each other. The system of two platforms can be considered closed |
|
vx? |
and apply the law of conservation of momentum to it.
m 1 v 01 + m 2 v 02 = (m 1 + m 2)v.
In projections on the axis X can be written:
m 1 v 01x + m 2 v 02x = (m 1 + m 2)v x.
Because v 01x = v 01 ; v 02x = –v 02 ; v x = - v, then m 1 v 01 – m 2 v 02 = –(m 1 + m 2)v.
Where v = – .
v= – = 0.75 m/s.
After coupling, the platforms will move in the direction in which the platform with a larger mass moved before the interaction.
Answer: v= 0.75 m/s; directed in the direction of movement of the cart with a larger mass.
Questions for self-examination
1. What is called the momentum of the body?
2. What is called the impulse of force?
3. How are the momentum of a force and the change in the momentum of a body related?
4. What system of bodies is called closed?
5. Formulate the law of conservation of momentum.
6. What are the limits of applicability of the law of conservation of momentum?
Task 17
1. What is the momentum of a body of mass 5 kg moving at a speed of 20 m/s?
2. Determine the change in the momentum of a body of mass 3 kg in 5 s under the action of a force of 20 N.
3. Determine the momentum of a car with a mass of 1.5 tons moving at a speed of 20 m/s in a frame of reference associated with: a) a car that is stationary relative to the Earth; b) with a car moving in the same direction at the same speed; c) with a car moving at the same speed but in the opposite direction.
4. A boy of mass 50 kg jumped off a stationary boat of mass 100 kg, located in the water near the shore. With what speed did the boat move away from the shore if the boy's speed is horizontal and equal to 1 m/s?
5. A 5 kg projectile flying horizontally exploded into two fragments. What is the speed of the projectile if a fragment with a mass of 2 kg acquired a speed of 50 m/s upon breaking, and a fragment with a mass of 3 kg acquired a speed of 40 m/s? The fragment velocities are directed horizontally.
In everyday life, in order to characterize a person who commits spontaneous acts, the epithet "impulsive" is sometimes used. At the same time, some people do not even remember, and a significant part does not even know what physical quantity this word is associated with. What is hidden under the concept of “body momentum” and what properties does it have? The answers to these questions were sought by such great scientists as Rene Descartes and Isaac Newton.
Like any science, physics operates with clearly formulated concepts. At the moment, the following definition has been adopted for a quantity called the momentum of a body: it is a vector quantity, which is a measure (quantity) of the mechanical movement of a body.
Let us assume that the issue is considered within the framework of classical mechanics, i.e., it is considered that the body moves with ordinary, and not with relativistic speed, which means that it is at least an order of magnitude less than the speed of light in vacuum. Then the momentum modulus of the body is calculated by formula 1 (see photo below).
Thus, by definition, this quantity is equal to the product of the mass of the body and its speed, with which its vector is codirected.
The unit of momentum in SI (International System of Units) is 1 kg/m/s.
Where did the term "impulse" come from?
Several centuries before the concept of the amount of mechanical motion of a body appeared in physics, it was believed that the cause of any movement in space is a special force - impetus.
In the 14th century, Jean Buridan made adjustments to this concept. He suggested that a flying boulder has an impetus directly proportional to its speed, which would be the same if there were no air resistance. At the same time, according to this philosopher, bodies with more weight had the ability to "accommodate" more of this driving force.
The concept, later called impulse, was further developed by Rene Descartes, who designated it with the words “quantity of motion”. However, he did not take into account that speed has a direction. That is why the theory put forward by him in some cases contradicted experience and did not find recognition.
The fact that the amount of motion must also have a direction was the first to guess the English scientist John Vallis. It happened in 1668. However, it took another couple of years for him to formulate the well-known law of conservation of momentum. The theoretical proof of this fact, established empirically, was given by Isaac Newton, who used the third and second laws of classical mechanics discovered by him, named after him.
Momentum of the system of material points
Let us first consider the case when we are talking about velocities much smaller than the speed of light. Then, according to the laws of classical mechanics, the total momentum of the system of material points is a vector quantity. It is equal to the sum of the products of their masses at speed (see formula 2 in the picture above).
In this case, the momentum of one material point is taken as a vector quantity (formula 3), which is co-directed with the velocity of the particle.
If we are talking about a body of finite size, then first it is mentally divided into small parts. Thus, the system of material points is again considered, however, its momentum is calculated not by the usual summation, but by integration (see formula 4).
As you can see, there is no time dependence, so the momentum of a system that is not affected by external forces (or their influence is mutually compensated) remains unchanged in time.
Proof of the conservation law
Let us continue to consider a body of finite size as a system of material points. For each of them, Newton's Second Law is formulated according to formula 5.
Note that the system is closed. Then, summing over all points and applying Newton's Third Law, we obtain expression 6.
Thus, the momentum of a closed system is a constant.
The conservation law is also valid in those cases when the total sum of the forces that act on the system from the outside is equal to zero. From this follows one important particular assertion. It states that the momentum of a body is constant if there is no external influence or the influence of several forces is compensated. For example, in the absence of friction after a hit with a club, the puck must maintain its momentum. Such a situation will be observed even despite the fact that this body is affected by the force of gravity and the reactions of the support (ice), since, although they are equal in absolute value, they are directed in opposite directions, i.e. they compensate each other.
Properties
The momentum of a body or material point is an additive quantity. What does it mean? Everything is simple: the momentum of the mechanical system of material points is the sum of the impulses of all the material points included in the system.
The second property of this quantity is that it remains unchanged during interactions that change only the mechanical characteristics of the system.
In addition, momentum is invariant with respect to any rotation of the frame of reference.
Relativistic case
Let us assume that we are talking about non-interacting material points having velocities of the order of 10 to the 8th power or slightly less in the SI system. The three-dimensional momentum is calculated by formula 7, where c is understood as the speed of light in vacuum.
In the case when it is closed, the law of conservation of momentum is true. At the same time, the three-dimensional momentum is not a relativistically invariant quantity, since there is its dependence on the reference frame. There is also a 4D version. For one material point, it is determined by formula 8.
Momentum and energy
These quantities, as well as the mass, are closely related to each other. In practical problems, relations (9) and (10) are usually used.
Definition via de Broglie waves
In 1924, a hypothesis was put forward that not only photons, but also any other particles (protons, electrons, atoms) have wave-particle duality. Its author was the French scientist Louis de Broglie. If we translate this hypothesis into the language of mathematics, then it can be argued that any particle with energy and momentum is associated with a wave with a frequency and length expressed by formulas 11 and 12, respectively (h is Planck's constant).
From the last relationship, we obtain that the pulse modulus and the wavelength, denoted by the letter "lambda", are inversely proportional to each other (13).
If a particle with a relatively low energy is considered, which moves at a speed incommensurable with the speed of light, then the momentum modulus is calculated in the same way as in classical mechanics (see formula 1). Consequently, the wavelength is calculated according to expression 14. In other words, it is inversely proportional to the product of the mass and velocity of the particle, i.e., its momentum.
Now you know that the momentum of a body is a measure of mechanical movement, and you have become familiar with its properties. Among them, in practical terms, the Law of Conservation is especially important. Even people who are far from physics observe it in everyday life. For example, everyone knows that firearms and artillery pieces recoil when fired. The law of conservation of momentum is also clearly demonstrated by playing billiards. It can be used to predict the direction of expansion of the balls after the impact.
The law has found application in the calculations necessary to study the consequences of possible explosions, in the field of creating jet vehicles, in the design of firearms, and in many other areas of life.
3.2. Pulse
3.2.1. body momentum, body system momentum
Only moving bodies have momentum.
The momentum of the body is calculated by the formula
P → = m v → ,
where m - body weight; v → - body speed.
In the International System of Units, the momentum of a body is measured in kilograms times a meter divided by a second (1 kg m/s).
Impulse of the body system(Fig. 3.1) is the vector sum of the impulses of the bodies included in this system:
P→=P→1+P→2+...+P→N=
M 1 v → 1 + m 2 v → 2 + ... + m N v → N ,
where P → 1 = m 1 v → 1 is the momentum of the first body (m 1 is the mass of the first body; v → 1 is the speed of the first body); P → 2 \u003d m 2 v → 2 - momentum of the second body (m 2 - mass of the second body; v → 2 - speed of the second body), etc.
Rice. 3.1
To calculate the momentum of a system of bodies, it is advisable to use the following algorithm:
1) choose a coordinate system and find the projections of the impulses of each body on the coordinate axes:
P 1 x , P 2 x , ..., P Nx ;
P 1 y , P 2 y , ..., P Ny ,
where P 1 x , ..., P Nx ; P 1 y , ..., P Ny - projections of body impulses on coordinate axes;
P x = P 1 x + P 2 x + ... + P Nx ;
P y = P 1 y + P 2 y + ... + P Ny ;
3) calculate the momentum modulus of the system using the formula
P \u003d P x 2 + P y 2.
Example 1. A body rests on a horizontal surface. A force of 30 N, directed parallel to the surface, begins to act on it. Calculate the momentum modulus of the body 5.0 s after the start of motion if the friction force is 10 N.
Solution. The momentum modulus of the body depends on time and is determined by the product
P(t) = mv,
where m - body weight; v is the modulus of the body's velocity at time t 0 = 5.0 s.
With uniformly accelerated motion with zero initial speed (v 0 \u003d 0), the speed of the body depends on time according to the law
v(t) = at,
where a is the acceleration module; t - time.
Substituting the dependence v (t) into the formula for determining the momentum modulus gives the expression
P(t) = mat.
Thus, the solution of the problem is reduced to finding the product ma .
To do this, we write the basic law of dynamics (Newton's second law) in the form:
F → + F → tr + N → + m g → = m a → ,
or in projections on the coordinate axes
O x: F − F tr = m a ; O y: N − m g = 0, )
where F is the modulus of force applied to the body in the horizontal direction; F tr - modulus of friction force; N is the modulus of the force of the normal reaction of the support; mg is the modulus of gravity; g - free fall acceleration modulus.
The forces acting on the body and the coordinate axes are shown in the figure.
It follows from the first equation of the system that the desired product is determined by the difference
ma = F − F tr.
Therefore, the dependence of the momentum of the body on time is determined by the expression
P (t ) = (F − F tr)t ,
and its value at the specified time t 0 = 5 c - by the expression
P (t) \u003d (F - F tr) t 0 \u003d (30 - 10) ⋅ 5.0 \u003d 100 kg ⋅ m / s.
Example 2. A body moves in the xOy plane along a trajectory of the form x 2 + y 2 \u003d 64 under the action of a centripetal force, the value of which is 18 N. The mass of the body is 3.0 kg. Assuming that the x and y coordinates are given in meters, find the momentum of the body.
Solution. The trajectory of the body movement is a circle with a radius of 8.0 m. According to the condition of the problem, only one force acts on the body, directed towards the center of this circle.
The modulus of this force is a constant value, so the body has only normal (centripetal) acceleration. The presence of constant centripetal acceleration does not affect the magnitude of the body's velocity; therefore, the motion of the body in a circle occurs at a constant speed.
The figure illustrates this circumstance.
The magnitude of the centripetal force is determined by the formula
F c. c \u003d m v 2 R,
where m - body weight; v is the modulus of the body's velocity; R is the radius of the circle along which the body moves.
Let us express the modulus of the body's velocity from here:
v = F c. with R m
and substitute the resulting expression into the formula that determines the magnitude of the momentum:
P = m v = m F c. with R m = F c. with R m .
Let's do the calculation:
P = 18 ⋅ 8.0 ⋅ 3.0 ≈ 21 kg ⋅ m/s.
Example 3. Two bodies move in mutually perpendicular directions. The mass of the first body is 3.0 kg, and its velocity is 2.0 m/s. The mass of the second body is 2.0 kg, and its velocity is 3.0 m/s. Find the momentum module of the system tel.
Solution. The bodies moving in mutually perpendicular directions will be depicted in the coordinate system, as shown in the figure:
- direct the velocity vector of the first body along the positive direction of the axis Ox ;
- let us direct the velocity vector of the second body along the positive direction of the axis Oy .
To calculate the momentum modulus of a system of bodies, we use the algorithm:
1) write down the projections of the impulses of the first P → 1 and the second P → 2 bodies on the coordinate axes:
P 1 x \u003d m 1 v 1; P2x=0;
P 1 y \u003d 0, P 2 y \u003d m 2 v 2,
where m 1 is the mass of the first body; v 1 - the value of the speed of the first body; m 2 - mass of the second body; v 2 - the value of the speed of the second body;
2) find the projections of the momentum of the system on the coordinate axes, summing up the corresponding projections of each of the bodies:
P x \u003d P 1 x + P 2 x \u003d P 1 x \u003d m 1 v 1;
P y \u003d P 1 y + P 2 y \u003d P 2 y \u003d m 2 v 2;
3) calculate the magnitude of the momentum of the system of bodies according to the formula
P = P x 2 + P y 2 = (m 1 v 1) 2 + (m 2 v 2) 2 =
= (3.0 ⋅ 2.0) 2 + (2.0 ⋅ 3.0) 2 ≈ 8.5 kg ⋅ m/s.
