> Internal and external forces

Explore internal and external forces systems. Consider the impact of internal and external forces on the linear momentum of the system, elastic and inelastic collisions.

Pure external forces(which are not zero) change the total momentum of the system, and domestic- No.

Learning task

• Note the effect of external and internal forces on linear momentum and collisions.

Key Points

• External forces are created by a source located outside the system.
• Internal forces are within the system.
• To understand what is considered internal and what is external forces, a mechanical system must have clear boundaries.

Terms

• Elastic collision - elastic collision with conservation of kinetic energy.
• An inelastic collision is an inelastic collision without conservation of kinetic energy.

Linear momentum and collisions

AT isolated system, consisting of particles:

Where Newton's Second Law says that the total momentum of the entire system must be stable in the absence of net external forces. They can change the total momentum if their sum is not equal to zero. But internal ones are deprived of such influence. To analyze a mechanical system, it is necessary to clearly distinguish between internal and external forces.

Conservation of the total momentum of the system (loss due to friction is neglected)

External forces are created by a source located outside the system, while internal forces are created by internal forces. Let's simplify. You have two hockey pucks sliding across a frictionless surface. We will also remove air resistance from the calculations. They collided at t = 0.

Let's start by listing the forces present: gravity, normal (between ice and pucks), and friction during the collision.

How to define a system? Usually we are interested in the movement of the pucks. Then we will accept as a fact that we have only two washers. Beyond them everything becomes external system. Then the external forces will be gravity and normal, and the friction will be internal. The outer ones cancel each other out, so we cross them out. It turns out that total impulse two washers is the stored value.

It is worth recalling that we did not consider the nature of the impact between the pucks. Even without touching the internal forces, it was possible to determine that the total momentum of the system is a conserved quantity. It works in elastic and inelastic collision.

Do not forget: if you take into account the Earth, then gravity and normal will become internal.

mechanical system such a set of material points or bodies is called in which the position or movement of each point or body depends on the position and movement of all the others. So, for example, when studying the motion of the Earth and the Moon relative to the Sun, the combination of the Earth and the Moon is a mechanical system consisting of two material points; when a projectile breaks into fragments, we consider the fragments as a mechanical system. A mechanical system is any mechanism or machine.

If the distances between points mechanical system do not change when the system is in motion or at rest, then such a mechanical system is called immutable.

The concept of an unchanging mechanical system makes it possible to study the arbitrary motion of rigid bodies in dynamics. In this case, as in statics and kinematics, by a solid body we mean such a material body, in which the distance between each two points does not change when the body moves or is at rest. Any solid can be mentally divided into enough big number small enough parts, the totality of which can be approximately considered as a mechanical system. Since a solid body forms a continuous extension, in order to establish its exact (rather than approximate) properties, it is necessary to make a limit transition, a limit fragmentation of the body, when the dimensions of the considered parts of the body simultaneously tend to zero.

Thus, knowledge of the laws of motion of mechanical systems makes it possible to study the laws of arbitrary motions of solid bodies.

All forces acting on the points of a mechanical system are divided into external and internal forces.

External forces in relation to a given mechanical system are forces acting on the points of this system from material points or bodies that are not included in the system. Designations: -external force applied to the -th point; -main vector external forces; - the main moment of external forces relative to the pole.

Internal forces are the forces with which material points or bodies included in a given mechanical system act on points or bodies of the same system. In other words, internal forces are forces of interaction between points or bodies of a given mechanical system. Designations: - internal force applied to the -th point; - the main vector of internal forces; - the main moment of internal forces relative to the pole.

3.2 Properties of internal forces.

First property.The main vector of all internal forces of the mechanical system is equal to zero, i.e.

. (3.1)

Second property.The main moment of all internal forces of a mechanical system with respect to any pole or axis is zero, that is

, . (3.2)

Thus, for each internal force there is a directly opposite internal force and, consequently, internal forces form a certain set of pairwise opposite forces. But the geometric sum of two opposite forces is zero, so

.

As shown in statics, the geometric sum of the moments of two opposite forces about the same pole is zero, therefore

.

A similar result is obtained when calculating the main moment about the axis

.

3.3 Differential equations of motion of a mechanical system.

Consider a mechanical system consisting of material points whose masses are . For each point, we apply the basic equation of point dynamics

, ,

, (3.3)

de is the resultant of external forces applied to the -th point, and is the resultant of internal forces.

The system of differential equations (3.3) is called differential equations movement of a mechanical system in vector form.

Projecting vector equations (3.3) onto rectangular Cartesian coordinate axes, we obtain differential equations of motion of a mechanical system in coordinate form:

,

, (3.4)

,

.

These equations are a system of second order ordinary differential equations. Therefore, to find the motion of a mechanical system according to given forces and initial conditions for each point of this system, it is necessary to integrate a system of differential equations. The integration of the system of differential equations (3.4), generally speaking, involves significant, often insurmountable mathematical difficulties. However, in theoretical mechanics methods have been developed that make it possible to circumvent the main difficulties that arise when using the differential equations of motion of a mechanical system in the form (3.3) or (3.4). These include methods that give general theorems of the dynamics of a mechanical system that establish the laws of change of some total (integral) characteristics of the system as a whole, and not the laws of motion of its individual elements. These are the so-called measures of motion - the main vector of momentum; principal moment of momentum; kinetic energy. Knowing the nature of the change in these quantities, it is possible to form a partial, and sometimes complete, idea of ​​the motion of a mechanical system.

IV. BASIC (GENERAL) THEOREMS OF THE DYNAMICS OF A POINT AND A SYSTEM

4.1 The theorem on the motion of the center of mass.

4.1.1. Center of mass of the mechanical system.

Consider a mechanical system consisting of material points whose masses are .

The mass of the mechanical system, consisting of material points, we will call the sum of the masses of the points of the system:

Definition. The center of mass of a mechanical system is a geometric point, the radius vector of which is determined by the formula:

where is the radius vector of the center of mass; -radius-vectors of system points; -their masses (Fig. 18).

; ; . (4.1")

The center of mass is not a material point, but geometric. It may not coincide with any material point of the mechanical system. In a uniform gravity field, the center of mass coincides with the center of gravity. This, however, does not mean that the concepts of center of mass and center of gravity are the same. The concept of the center of mass is applicable to any mechanical systems, and the concept of the center of gravity is applicable only to mechanical systems that are under the action of gravity (that is, attraction to the Earth). So, for example, in celestial mechanics, when considering the problem of the motion of two bodies, for example, the Earth and the Moon, one can consider the center of mass of this system, but one cannot consider the center of gravity.

Thus, the concept of the center of mass is broader than the concept of the center of gravity.

4.1.2. The theorem on the motion of the center of mass of a mechanical system.

Theorem. The center of mass of a mechanical system moves as material point, the mass of which is equal to the mass of the entire system and to which all external forces acting on the system are applied, that is

. (4.2)

Here is the main vector of external forces.

Proof. Consider a mechanical system, the material points of which move under the action of external and internal forces. is the resultant of external forces applied to the -th point, and is the resultant of internal forces. According to (3.3), the equation of motion of the -th point has the form

, .

Adding the left and right sides of these equations, we get

.

Since the main vector of internal forces is equal to zero (section 3.2, first property), then

.

Let us transform the left side of this equality. From formula (4.1), which determines the radius vector of the center of mass, it follows:

.

Everywhere below, we will assume that only mechanical systems of constant composition are considered, that is, and . Let us take the second derivative with respect to time from both sides of this equality

Because , - acceleration of the center of mass of the system, then, finally,

.

Projecting both parts of this vector equality onto the coordinate axes, we get:

,

, (4.3)

,

where , , are projections of force ;

Projections of the main vector of external forces on the coordinate axes.

Equations (4.3) - differential equations of motion of the center of mass of a mechanical system in projections onto the Cartesian coordinate axes.

Equations (4.2) and (4.3) imply that It is impossible to change the nature of the movement of the center of mass of a mechanical system by internal forces alone. Internal forces can have an indirect effect on the movement of the center of mass only through external forces. For example, in a car, the internal forces developed by the engine affect the movement of the center of mass through the forces of friction between the wheels and the road.

4.1.3. Laws of conservation of motion of the center of mass

(corollaries from the theorem).

The following corollaries can be obtained from the theorem on the motion of the center of mass.

Consequence 1.If the main vector of external forces acting on the system is equal to zero, then its center of mass is at rest or moves in a straight line and uniformly.

Indeed, if the main vector of external forces , then from equation (4.2):

If, in particular, the initial velocity of the center of mass is , then the center of mass is at rest. If the initial speed , then the center of mass moves in a straight line and uniformly.

Consequence 2.If the projection of the main vector of external forces on any fixed axis is equal to zero, then the projection of the velocity of the center of mass of the mechanical system on this axis does not change.

This corollary follows from equations (4.3). Let, for example, then

,

from here. If at the same time at the initial moment , then:

that is, the projection of the center of mass of the mechanical system on the axis in this case will not move along the axis. If , then the projection of the center of mass on the axis moves uniformly.

4.2 The momentum of a point and system.

Theorem on the change in momentum.

4.2.1. The amount of movement of the point and system.

Definition. The momentum of a material point is a vector equal to the product of the mass of the point and its speed, that is

. (4.5)

Vector collinear to the vector and directed tangentially to the trajectory of the material point (Fig. 19).

The momentum of a point in physics is often called momentum of a material point.

The unit of momentum is in SI-kg m/s or N s.

Definition. The momentum of a mechanical system is a vector equal to vector sum the number of movements (the main vector of the number of movements) of individual points included in the system, that is

(4.6)

Projections of momentum onto rectangular Cartesian coordinate axes:

System momentum vector unlike the momentum vector, a point does not have an application point. The momentum vector of a point is applied to the moving point itself, and the vector is a free vector.

Lemma of momentum. The momentum of a mechanical system is equal to the mass of the entire system multiplied by the velocity of its center of mass, i.e.

Proof. From formula (4.1), which determines the radius vector of the center of mass, it follows:

.

Take the time derivative of both sides

, or .

From here we get , which was to be proved.

From formula (4.8) it can be seen that if the body moves in such a way that its center of mass remains stationary, then the momentum of the body is zero. For example, the momentum of a body rotating around a fixed axis passing through its center of mass (Fig. 20),

, because

If the motion of the body is plane-parallel, then the amount of motion will not characterize the rotational part of the motion around the center of mass. For example, for a wheel that is rolling (Fig. 21), regardless of how the wheel rotates around the center of mass. The amount of motion characterizes only the translational part of the motion together with the center of mass.

4.2.2. Theorem on the change in the momentum of a mechanical system

in differential form.

Theorem.The time derivative of the momentum of a mechanical system is equal to the geometric sum (principal vector) of external forces acting on this system, i.e.

. (4.9)

Proof. Consider a mechanical system consisting of material points whose masses are ; is the resultant of external forces applied to the i-th point. In accordance with the momentum lemma, formula (4.8):

Take the time derivative of both sides of this equality

.

The right part of this equality from the theorem on the motion of the center is the mass formula (4.2):

.

Finally:

and the theorem is proven .

In projections onto rectangular Cartesian coordinate axes:

; ; , (4.10)

that is the time derivative of the projection of the momentum of the mechanical system onto any coordinate axis is equal to the sum of the projections (projections of the main vector) of all external forces of the system onto the same axis.

4.2.3. Laws of conservation of momentum

(corollaries from the theorem)

Corollary 1.If the main vector of all external forces of a mechanical system is equal to zero, then the momentum of the system is constant in magnitude and direction.

Indeed, if , then from the momentum change theorem, i.e., from equality (4.9), it follows that

Consequence 2.If the projection of the main vector of all external forces of a mechanical system onto a certain fixed axis is equal to zero, then the projection of the momentum of the system onto this axis remains constant.

Let the projection of the main vector of all external forces on the axis be equal to zero: . Then from the first equality (4.10):

4.2.4. Theorem on the change in the momentum of a mechanical system

in integral form.

An elemental impulse of force is called a vector quantity equal to the product of the force vector by an elementary time interval

. (4.11)

The direction of the elementary impulse coincides with the direction of the force vector.

Impulse of force over a finite period of time is equal to a certain integral of the elementary momentum

. (4.12)

If the force is constant in magnitude and direction (), then its momentum over time equals:

Projections of the force impulse on the coordinate axes:

Let us prove the theorem on the change in the momentum of a mechanical system in integral form.

Theorem.The change in the momentum of a mechanical system over a certain period of time is equal to the geometric sum of the impulses of the external forces of the system over the same period of time, i.e.

(4.14)

Proof. Let at the moment of time the amount of motion of the mechanical system be , and at the moment of time - ; is the momentum of the external force acting on the th point in time .

We use the theorem on the change in momentum in differential form - equality (4.9):

.

Multiplying both parts of this equality by and integrating within the limits from to , we obtain

, , .

The theorem on the change in momentum in integral form is proven.

In projections on the coordinate axes, according to (4.14):

,

, (4.15)

.

4.3. Theorem on the change of the kinetic moment.

4.3.1. momentum points and systems.

In statics, the concepts of moments of force relative to the pole and axis were introduced and widely used. Since the momentum of a material point is a vector, it is possible to determine its moments relative to the pole and axis in the same way as the moments of force are determined.

Definition. relative to the pole is called the moment of its momentum vector relative to the same pole, i.e.

. (4.16)

The angular momentum of a material point relative to the pole is a vector (Fig. 22) directed perpendicular to the plane containing the vector and the pole in the direction from which the vector is relative to the pole seen counter-clockwise rotation. Vector modulus

equal to the product of the module and the arm - the length of the perpendicular dropped from the pole to the line of action of the vector:

The momentum relative to the pole can be represented as a vector product: the kinetic moment of a material point relative to the pole is equal to the vector product of the radius of the vector drawn from the pole to the point by the momentum vector:

(4.17)

Definition. The kinetic moment of a material point relatively axis is called the moment of its vector of momentum relative to the same axis, i.e.

. (4.18)

The angular momentum of a material point about the axis (Fig. 23) is equal to the product of the vector projection onto the plane perpendicular to the axis, taken with a plus or minus sign , on the shoulder of this projection:

where the shoulder is the length of the perpendicular dropped from the point axis intersection with the plane on the line of action of the projection , while , if, looking towards the axis , you can see the projection about the point counter-clockwise direction, and otherwise.

The unit of angular momentum is in SI-kg m 2 /s, or N m s.

Definition. The angular momentum or the main moment of momentum of a mechanical system relative to a pole is a vector equal to the geometric sum of the angular momentum of all material points of the system relative to this pole:

. (4.19)

Definition. The angular momentum or the main moment of momentum of a mechanical system relative to an axis is the algebraic sum of the kinetic moments of all material points of the system relative to this axis:

. (4.20)

The kinetic moments of the mechanical system relative to the pole and the axis passing through this pole are connected by the same dependence as the main moments of the system of forces relative to the pole and the axis:

-projection of the kinetic moment of the mechanical system relative to the pole onto the axis ,passing through this pole is equal to the angular momentum of the system about this axis, i.e.

. (4.21)

4.3.2. Theorems on the change in the kinetic moment of a mechanical system.

Consider a mechanical system consisting of material points whose masses are . Let's prove a theorem on the change in the kinetic moment of a mechanical system with respect to the pole.

Theorem.The time derivative of the angular momentum of a mechanical system with respect to a fixed pole is equal to the main moment of the external forces of the system with respect to the same pole, i.e.

. (4.22)

Proof. We choose some fixed pole . The angular momentum of a mechanical system relative to this pole is, by definition, equality (4.19):

.

Let's differentiate this expression with respect to time:

Consider the right side of this expression. Calculating the derivative of the product:

, (4.24)

It is taken into account here that . Vectors and have the same direction, their vector product is equal to zero, hence the first sum in equality (4.24).

System of material points (or tel) any set of them that we have distinguished is called. Each body of the system can interact both with bodies that belong to this system and with bodies that are not included in it. The forces acting between the bodies of the system are called internal forces. Forces acting on the bodies of the system from bodies that are not included in this system, are called external forces. The system is called closed (or isolated) if it includes all interacting bodies. Thus, only internal forces act in a closed system.

Strictly speaking, closed systems do not exist in nature. However, it is almost always possible to formulate the problem in such a way that external forces can be neglected (because of their smallness or compensated ™, i.e., mutual annihilation) in comparison with internal ones. The choice of an imaginary surface that limits the system is the prerogative (free will) of the subject, i.e. should be carried out by the researcher on the basis of an analysis of internal and external forces. The same system of bodies can be considered closed or open in various conditions depending on the formulation of the problem and on the given accuracy of its solution.

In a closed system of bodies, all phenomena are described using simple and general laws, therefore, if the conditions of the problem allow, then one should neglect the small action of external forces and consider the system as closed. This is what is often called physical model objective reality.

A special case of an ideal mechanical system is an absolutely rigid body that can neither deform nor change in volume, much less collapse (it is obvious that there are no such bodies in nature): the distance between individual material points that form such a system remains constant for all types of interaction.

Now let us introduce a very important concept in mechanics of the center of mass (center of inertia) of a system of material points. Let's take a system consisting of N material points. Center of mass of a mechanical system point C is called, the radius-vector of the position of which in an arbitrarily chosen reference system is given by the relation:

where /u, is the mass of a material point; /; - radius vector drawn from the origin of the reference system to the point where t,.

If we place the origin at point C, then Rc= 0 and then

which leads to another definition of the center of mass: center of mass of the mechanical system - this is such a point for which the sum of the products of the masses of all material points that form a mechanical system and their radius vectors drawn from this point, as the beginning of the coor

dinat, are equal to zero. Figure 1.

Rice. 1.11.

1 this is illustrated by the example of a system consisting of two bodies (for example, a diatomic molecule).

Radius vector Rc of this system, the MT in the Cartesian coordinate system has the coordinates X c, Y c , Z c(general three-dimensional case). In this case, the position of the center of mass can be determined by the following equations:

where M- the total mass of the mechanical system MT,

So far, we have operated with the set N discrete material points. And what about the definition of the center of mass of an extended body, the mass of which is distributed continuously in space? It is natural to pass in this case from summation in (1.68)-(1.70) to integration. In this case, in vector form, we get

For bodies with a plane of symmetry (as in the example) the center of mass is located in this plane. If the body has an axis of symmetry (axis X in our example), then the center of mass must certainly lie on this axis, if the body has a center of symmetry (for example, as in the case of a homogeneous ball), then this center must coincide with the position of the center of mass.

In order to determine how the center of mass of the system moves, we write expressions (1.70) in the form

=MZ C and differentiate them twice with respect to time (all masses

we assume constant)

Comparing the resulting equalities with expressions (1.51), we obtain

or (in vector form)

These equations are called differential equations of motion of the center of mass, coincide in structure with the differential equations of motion of a material point. This allows us to formulate a theorem on the motion of the center of mass: the center of mass of a mechanical system moves as a material point, the mass of which is equal to the mass of the entire system and to which all external forces acting on the system are applied.

If the system is not affected by external forces i.e. the action of external forces is compensated), then

those. the speed of the center of mass closed system always remains constant (preserved). Internal forces have no effect on the motion of the center of mass of the system. If, in particular, in this inertial system coordinates, the center of mass of a closed system is at rest at one of the instants of time, this means that it will always be at rest.

Many problems in mechanics are solved most simply in a coordinate system associated with the center of mass.

• With the coordinate system chosen in the example, Zc = 0 (flat one-dimensional case).

External forces are called forces acting on the body from points or bodies that are not included in this body or system. Forces with which the points of a given body act on each other are called internal forces.

Destruction or even simple failure of a structural element is possible only with an increase in internal forces and when they pass through a certain limiting barrier. It is convenient to count the height of this barrier from the level corresponding to the absence of external forces. In essence, it is necessary to take into account only additional internal forces that arise only in the presence of external forces. These additional internal forces are called in mechanics simply internal forces in a narrow, mechanical sense.

Internal forces are determined using the “section method”, which is based on a fairly obvious statement: if the body as a whole is in equilibrium, then any part separated from it is also in this state

Figure 2.1.5

Consider a rod that is in equilibrium under the action of a system of external forces, Fig. 2.1.5, a. With the section AB, let's mentally divide it into two parts, fig. 2.1.5, b. To each of the sections AB of the left and right parts, we apply a system of forces corresponding to the internal forces acting in a real body, Fig. 1.7, c. Thus, using the method of sections, internal forces are converted into external forces with respect to each of the cut off parts of the body, which makes it possible to determine them from the equilibrium conditions for each of these parts separately.

The section AB can be oriented in any way, but the cross section perpendicular to the longitudinal axis of the rod turns out to be more convenient for further reasoning.

Let us introduce the notation:

principal vectors and principal moments of external and internal forces applied to the left cut-off part. Taking into account the introduced notation, the equilibrium conditions for this body can be written as:

0, + =0 (2.1.1)

Similar expressions can be made for the right cut off part of the rod. After simple transformations, you can get:

=- , =- (2.1.1)

which can be interpreted as a consequence of the well-known law of mechanics: an action is always accompanied by an equal and oppositely directed reaction.

In the case of solving the problem of dynamic action on the rod, one can refer to the well-known d'Alembert principle, according to which inertial forces are added to external forces, which again reduces the problem to equilibrium equations. Therefore, the section method procedure remains

The values ​​and do not depend on the orientation of the section AB (see Fig. 2.1.5). However, in practical calculations, the most convenient is the use of the cross section. In this case, the normal to the section coincides with the longitudinal axis of the rod. Further, the main vector and the main moment of internal forces are usually represented as their projections onto orthogonal coordinate axes, and one of the axes (for example, the x axis) is aligned with the mentioned normal, see Fig. 2.1.6.

Figure 2.1.6

Let us expand the vectors , , , along the coordinate axes, Fig. 2.1.6, a-d. The principal vector and principal moment components have common names. The force N x normal to the plane of the section is called the normal (longitudinal) force, and Q x and Q y are called transverse (cutting) forces. Moments relative to axes at and z, i.e. M y and M z will be bending and the moment about the longitudinal axis X, i.e. M x - twisting.

The components of the main moment of internal forces in the resistance of materials are most often displayed as shown in Fig. 2.1.6, e and f.

Vector equations equilibrium can be represented as a projection onto the coordinate axes:

Thus, each component of the main vector for the main moment of internal forces is calculated as the sum of the projections of all external forces on the corresponding axis or as the sum of the moments of all external forces about this axis (taking into account the accepted sign rule) located on one side of the section.

The projection of a vector onto the coordinate axis, being a scalar quantity, can be either positive or negative. It depends on whether the projection direction coincides with the positive or negative direction of the axis, respectively. For internal forces, this rule is observed only for the case when the normal X is external, as was the case for the left cut-off part in Fig. 2.1.6. In a situation where normal X is internal, see the right cut-off part in fig. 2.1.6, the sign of the internal force is taken positive when its direction coincides with the negative direction of the axis. On fig. 2.1.6 all projections of internal forces N x , Q x , Q y , M x , M y and M z (both related to the left and related to the right cut-off parts) are shown positive.

Introduce strong man easy enough. powerful physique, big muscles, confident look. But do these signs always prove true strength? And what is this inner strength that one hears about so often? Does it match the imposing appearance? Can physically less developed person to be stronger than his superior opponent? In what cases is the inner strength of a person manifested? Is it possible to develop it, or is it an innate quality that is inherited? Let's try to understand this issue.

What is inner strength?

Inner strength is the strength of the spirit, a set of strong-willed qualities that make it possible to overcome various life difficulties. Accordingly, it manifests itself in stressful cases, when a person, feeling that he cannot control the situation, still continues to act “in character”.

This quality literally endows people with superhuman abilities, allowing them to pass where even two-meter bouncers will break. Inner strength does not depend on age, gender or other parameters of a person.

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It can manifest itself in anyone, the main thing is not to suppress it. The main factors that suppress the development of internal strength can be considered harmful, complexes, stresses, fears, experiences, and.

How does inner strength come about?

The inner strength of a person does not depend on his external power, but does not exclude it either. After all, for any power, there is always a greater power. And in the event of a collision with it, it is precisely the inner strength that manifests itself.

Of course, it is easier to defeat a weaker opponent. But we all know examples when a small but “spiritual” person emerges victorious from a skirmish with someone who is clearly superior in size. Why is this happening? Apparently he is more and this confidence is transferred to the enemy, literally disarming him. According to the principle of the textbook Moska, which strikes terror into all local elephants.

There are five main components that make up the inner strength of a person:

• The strength of the spirit is the core of the personality;
• Life energy is all that is necessary for life;
• Willpower is an internal reserve that opens up during times of difficulty;
• Self-control - the ability to control your body and thoughts;
• Psychic energy - emotional and mental stability.

Their interaction determines how strong a person will be in a given situation, therefore it is very important to pay attention to the development of each of these components.