Change in speeds of bodies during interaction. Interaction of bodies. Force. Newton's second law. Non-system units of mass

Interaction of bodies. Experience shows that when bodies (or systems of bodies) come closer, the nature of their behavior changes. Since these changes are reciprocal in nature, they say that the bodies interact with each other . When moving bodies apart over very large distances (to infinity), all currently known interactions disappear.

Galileo was the first to give the correct answer to the question, what kind of motion is characteristic of free (i.e. non-interacting bodies). Contrary to the then existing opinion that free bodies “strive” to a state of rest (), he argued that in the absence of interaction the bodies are in a state of uniform motion (
), including peace as a special case.

Inertial reference systems. Within the framework of the formal mathematical approach implemented in kinematics, Galileo’s statement seems meaningless, since a uniform motion in one reference system may turn out to be accelerated in another, which is “no worse” than the original one. The presence of interaction allows us to identify a special class of reference systems in which free bodies move without acceleration (in these systems, most laws of nature have the simplest form). Such systems are called inertial.

All inertial systems are equivalent to each other, in any of them the laws of mechanics are manifested in the same way. This property was also noted by Galileo in his formulation principle of relativity: n and by any mechanical experience in a closed (i.e. not communicating with outside world) it is impossible for a reference system to determine whether it is at rest or moving uniformly. Any reference system that moves uniformly relative to an inertial one is also inertial.

There is a fundamental difference between inertial and non-inertial reference systems: an observer located in a closed system is able to establish the fact of movement with the acceleration of the latter, “without looking outside” (for example, when an airplane accelerates, passengers feel that they are “pressed” into their seats). It will be shown later that in non-inertial systems the geometry of space ceases to be Euclidean.

Newton's laws as the basis of classical mechanics. The three laws of motion formulated by I. Newton, in principle, make it possible to solve the main task of mechanics , i.e. Based on the known initial position and velocity of the body, determine its position and velocity in arbitrary moment time.

Newton's first law postulates the existence of inertial frames of reference.

Newton's second law States that in inertial systems, the acceleration of a body is proportional to the appliedstrength , a physical quantity that is a quantitative measure of interaction. The magnitude of the force characterizing the interaction of bodies can be determined, for example, by the deformation of an elastic body additionally introduced into the system so that the interaction with it completely compensates for the original one. The proportionality coefficient between force and acceleration is called body weight :

(1) F= m a

Under the influence of equal forces, bodies with greater mass acquire smaller accelerations. Massive bodies, when interacting, change their speeds to a lesser extent, “trying to maintain natural motion by inertia.” It is sometimes said that mass is a measure of the inertia of bodies (Fig. 4_1).

The classical properties of mass include 1) its positivity (bodies acquire acceleration in the direction of applied forces), 2) additivity (the mass of a body is equal to the sum of the masses of its parts), 3) independence of mass from the nature of movement (for example, from speed).

Third Law states that interactions both objects experience forces, and these forces are equal in magnitude and opposite in direction.

Types of fundamental interactions. Attempts to classify interactions led to the idea of identifying a minimum set fundamental interactions , with the help of which all observed phenomena can be explained. As natural science developed, this set changed. In the course of experimental research, new natural phenomena were periodically discovered that did not fit into the accepted fundamental set, which led to its expansion (for example, the discovery of the structure of the nucleus required the introduction nuclear forces). Theoretical understanding, generally striving for a unified, most economical description of the observed diversity, has repeatedly led to “great unifications” of seemingly completely dissimilar natural phenomena (Newton realized that the fall of an apple and the movement of planets around the Sun are the results of the manifestation of gravitational interactions, Einstein established the unified nature of electrical and magnetic interactions, Butlerov refuted claims about the different nature of organic and inorganic substances).

A set of four types of fundamental interactions is currently accepted:gravitational, electromagnetic, strong and weak nuclear. All the others known today can be reduced to a superposition of those listed.

Gravitational interactions are caused by the presence of mass in bodies and are the weakest of the fundamental set. They dominate at distances of cosmic scales (in the mega-world).

Electromagnetic interactions due to the specific properties of the series elementary particles called electric charge. They play a dominant role in the macro and micro worlds up to distances exceeding the characteristic dimensions of atomic nuclei.

Nuclear interactions play a dominant role in nuclear processes and appear only at distances comparable to the size of the nucleus, where the classical description is obviously inapplicable.

Nowadays, discussions about biofield , with the help of which a number of natural phenomena associated with biological objects that are not very reliably established experimentally are “explained.” Taking the concept of a biofield seriously depends on the specific meaning. Embedded in this term. If the concept of a biofield is used to describe interactions involving biological objects, reduced to four fundamental ones, this approach does not raise fundamental objections, although the introduction of a new concept to describe “old” phenomena contradicts the generally accepted tendency in natural science to minimize theoretical description. If the biofield is understood as a new type of fundamental interactions, manifested at the macroscopic level (the possibility of the existence of which a priori is obviously pointless to deny), then such far-reaching conclusions require very serious theoretical and experimental justifications, made in the language and methods of modern natural science, which are up to have not been presented at this time.

Newton's laws and the main task of mechanics. To solve the main problem of mechanics (determining the position of a body at an arbitrary moment in time from a known initial position and speed), it is enough to find the acceleration of the body as a function of time a(t). This problem is solved by Newton's laws (1) under the condition of known forces. In general, forces can depend on time, position and speed of the body:

(2) F=F(r,v, t) ,

those. To find the acceleration of a body, you need to know its position and speed. The described situation in mathematics is called second order differential equation :

(3)
,

(4)

Mathematics shows that problem (3-4) in the presence of two initial conditions (position and speed at the initial moment of time) always has a solution and, moreover, a unique one. That. The main problem of mechanics, in principle, always has a solution, but finding it is often very difficult.

Laplace determinism. The German mathematician Laplace applied a similar theorem on the existence and uniqueness of a solution to a problem of type (3-4) for a system of a finite number of equations to describe the motion of all particles of the real world interacting with each other and came to the conclusion that it is fundamentally possible to calculate the position of all bodies at any time . Obviously, this meant the possibility of an unambiguously predicted future (at least in principle) and complete determinism (predetermination) of our world. The statement made, which is more of a philosophical rather than a naturally scientific nature, was called Laplace determinism . If desired, one could draw from it very far-reaching philosophical and social conclusions about the impossibility of influencing the predetermined course of events. The fallacy of this doctrine was that atoms or elementary particles (“material points” from which real bodies are composed) do not actually obey the classical law of motion (3), which is true only for macroscopic objects (i.e. those with sufficiently large masses and sizes). A correct description from the point of view of today's physics of the movement in time of microscopic objects, such as the atoms and molecules that make up macroscopic bodies, is given by the equations quantum mechanics, , which make it possible to determine only the probability of finding a particle at a given point, but fundamentally does not make it possible to calculate trajectories of motion for subsequent moments of time.

What is the reason for the movement of bodies? The answer to this question is provided by a branch of mechanics called dynamics.
How can you change the speed of a body, make it move faster or slower? Only when interacting with other bodies. When interacting, bodies can change not only speed, but also direction of movement and deform, thereby changing shape and volume. In dynamics, for a quantitative measure of the interaction of bodies on each other, a quantity called force is introduced. And the change in speed during the action of the force is characterized by acceleration. Force is the cause of acceleration.

Concept of strength

Force is a vector physical quantity that characterizes the action of one body on another, manifested in the deformation of the body or a change in its movement relative to other bodies.

Force is denoted by the letter F. The SI unit of measurement is the Newton (N), which is equal to the force under the influence of which a body weighing one kilogram receives an acceleration of one meter per second squared. The force F is completely defined if its magnitude, direction in space and point of application are given.
To measure forces, a special device called a dynamometer is used.

How many forces are there in nature?

Forces can be divided into two types:

They act through direct interaction, contact (elastic forces, friction forces);
Act at a distance, long-range (force of attraction, gravity, magnetic, electrical).

During direct interaction, for example, a shot from a toy pistol, bodies experience a change in shape and volume compared to the original state, that is, compression, stretching, and bending deformation. The pistol spring is compressed before firing, and the bullet is deformed when it hits the spring. IN in this case forces act at the moment of deformation and disappear along with it. Such forces are called elastic. Friction forces arise from the direct interaction of bodies when they roll and slide relative to each other.

An example of forces acting at a distance is a stone thrown upward, due to gravity it will fall to the Earth, ebbs and flows that occur on the ocean coasts. As the distance increases, such forces decrease.
Depending on the physical nature of the interaction, forces can be divided into four groups:

weak;
strong;
gravitational;
electromagnetic.

We encounter all types of these forces in nature.
Gravitational or universal forces are the most universal; everything that has mass is capable of experiencing these interactions. They are omnipresent and pervasive, but very weak, so we do not notice them, especially at great distances. Long-range gravitational forces bind all bodies in the Universe.

Electromagnetic interactions occur between charged bodies or particles through the action of an electromagnetic field. Electromagnetic forces allow us to see objects, since light is a form of electromagnetic interactions.

Weak and strong interactions became known through the study of the structure of the atom and the atomic nucleus. Strong interactions arise between particles in nuclei. Weak ones characterize the mutual transformations of elementary particles into each other; they act during thermonuclear fusion reactions and radioactive decays of nuclei.

What if several forces act on a body?

When several forces act on a body, this action is simultaneously replaced by one force equal to their geometric sum. The force obtained in this case is called the resultant force. It imparts to the body the same acceleration as the forces simultaneously acting on the body. This is the so-called principle of superposition of forces.

In classical mechanics it is believed that:

a) The mass of a material point does not depend on the state of motion of the point, being its constant characteristic.

b) Mass is an additive quantity, i.e. the mass of the system (for example, a body) is equal to the sum of the masses of the milestones material points included in this system.

c) The mass of a closed system remains unchanged during any processes occurring in this system (law of conservation of mass).

Density ρ body at a given point M called mass ratio dm small body element including a point M, to the value dV volume of this element:

The dimensions of the element under consideration must be so small that by changing the density within its limits many times greater intermolecular distances can be achieved.

The body is called homogeneous , if the density is the same at all its points. The mass of a homogeneous body is equal to the product of its density and volume:

Mass of a heterogeneous body:

dV,

where ρ is a function of coordinates, and integration is carried out over the entire volume of the body. Medium density (ρ) of an inhomogeneous body is called the ratio of its mass to volume: (ρ)=m/V.

Center of mass of the system material points is called point C, radius vector

which is equal to: and – mass and radius vector i th material point, n – total number material points in the system, and m= is the mass of the entire system.

Center of mass speed:

Vector quantity

, equal to the product of the mass of a material point and its speed, is called impulse, or amount of movement , this material point. Impulse of the system of material points is called a vector p, equal to the geometric sum of the momenta of all material points of the system:

The momentum of the system is equal to the product of the mass of the entire system and the speed of its center of mass:

Newton's second law

The basic law of the dynamics of a material point is Newton’s second law, which talks about how the mechanical motion of a material point changes under the influence of forces applied to it. Newton's second law reads: rate of change of momentum ρ material point is equal to the force acting on it F, i.e.

, or

where m and v are the mass and speed of the material point.

If several forces simultaneously act on a material point, then under the force F in Newton's second law, you need to understand the geometric sum of all acting forces - both active and reaction reactions, i.e. resultant force.

Vector quantity Fdt called elementary impulse strength F in a short time dt her actions. Impulse force F for a finite period of time from

to is equal to a definite integral:

Where F, in general, depends on time t.

According to Newton's second law, the change in momentum of a material point is equal to the momentum of the force acting on it:

d p= F dt And

, is the value of the momentum of the material point at the end ( ) and at the beginning ( ) of the time period under consideration.

Since in Newtonian mechanics the mass m material point does not depend on the state of motion of the point, then

Therefore, the mathematical expression of Newton's second law can also be represented in the form

– acceleration of a material point, r is its radius vector. Accordingly, the wording Newton's second law states: the acceleration of a material point coincides in direction with the force acting on it and is equal to the ratio of this force to the mass of the material point.

The tangential and normal acceleration of a material are determined by the corresponding components of the force F

, is the magnitude of the velocity vector of the material point, and R– radius of curvature of its trajectory. The force imparting normal acceleration to a material point is directed towards the center of curvature of the point’s trajectory and is therefore called centripetal force.

If several forces simultaneously act on a material point

, then its acceleration. Consequently, each of the forces simultaneously acting on a material point imparts to it the same acceleration as if there were no other forces (the principle of independence of the action of forces).

Differential equation of motion of a material point called the equation

In projections onto the axes of a rectangular Cartesian coordinate system, this equation has the form

, ,

where x, y and z are the coordinates of the moving point.

Newton's third law. Movement of the center of mass

The mechanical action of bodies on each other is manifested in the form of their interaction. This is what he says Newton's third law: two material points act on each other with forces that are numerically equal and directed in opposite directions along the straight line connecting these points.

– force acting on i- yu material point from the side k- th, a – force acting on kth material point from i-th side, then, according to Newton’s third law, they are applied to different material points and can be mutually balanced only in those cases when these points belong to the same absolutely rigid body.

Newton's third law is an essential addition to the first and second laws. It allows you to move from the dynamics of a single material point to the dynamics of an arbitrary mechanical system (system of material points). From Newton’s third law it follows that in any mechanical system the geometric sum of all internal forces is equal to zero: where

– the resultant of external forces applied to i th material point.

From Newton's second and third laws it follows that the first derivative with respect to time t from impulse p mechanical system is equal to the main vector of all external forces applied to the system,

This equation expresses law of change in the momentum of the system.

4.1. Interaction of bodies– the action of bodies on each other, i.e. The action of bodies on each other is always a two-way action.

Examples:

The interaction is shown by arrows:

∙ cube acts on surface - surface on cube,

∙ ball on thread – thread on ball,

∙ the traction force of the engine through the wheels acts forward - the friction force of the road acts backward through the wheels,

4.2. The consequence of the interaction is – disturbance of body rest, change in its speed or deformation, i.e. change in body shape.

An illustrative example:

Conclusion from experience:

The more mass the more inert the body is.

The more the speed of a body changes during interaction, the stronger the body resists disturbance of rest and change in speed.

Example from practical life:

With the same impact force, it is more difficult to change the speed of a massive body, i.e. by the train.

4.3. Inertia of the physical body– this is the property of a physical body to maintain peace or speed.

Examples:(See in 4.2.)

4.4. Body mass– a physical quantity that is a measure of the inertia of a body: the greater the mass of the body, the more inert the body.

Units of mass: 1kg (SI)– equal to the mass of the international prototype kilogram, which was obtained by comparison with the mass of 1 liter of water under certain conditions.

Comment: the 1kg prototype is stored in Sevres near Paris, in the International Chamber of Weights and Measures.