One of the most key categories of dialectics is the category of “law”. In its most general form, a law can be defined as a connection (relationship) between phenomena and processes, which is:

a) objective, since it is inherent primarily in the real world, the sensory-objective activity of people, expresses the real relationships of things;

b) essential, concrete-universal. Being a reflection of what is essential in the movement of the universe, any law is inherent in all processes of a given class, of a certain type (species), without exception, and operates always and wherever the corresponding processes and conditions unfold:

c) necessary, because being closely connected with the essence, the law acts and is implemented with “iron necessity” in appropriate conditions;

d) internal, since it reflects the deepest connections and dependencies of a given subject area in the unity of all its moments and relationships within the framework of some integral system;

e) repeating, stable, since the law is solid (remaining) in the phenomenon, the law is identical in the phenomenon, the law is “a calm reflection of phenomena. And therefore every law is narrow, incomplete, approximate.” It is an expression of a certain constancy of a certain process, the regularity of its occurrence, the uniformity of its action under similar conditions.

The stability and invariance of laws always correlates with the specific conditions of their action, the change of which removes this invariance and gives rise to a new one, which means a change in the laws, their deepening, expansion or narrowing of the scope of their action, their modifications, etc. Any law is not something immutable, but is a concrete historical phenomenon. With changes in relevant conditions, with the development of practice and knowledge, some laws disappear from the scene, others reappear, the forms of action of laws, methods of their use, etc. change.

The most important, key task of scientific research is to “raise experience to the universal”, to find the laws of a given subject area, a certain sphere (fragment) of real reality, to express them in appropriate concepts, abstractions, theories, ideas, principles, etc. The solution to this problem can be successful only if the scientist proceeds from two main premises: the reality of the world in its integrity and development and the conformity of this world with laws, i.e. the fact that it is “permeated” by a set of objective laws. The latter regulate the entire world process, provide it with a certain order, necessity, and the principle of self-propulsion and are completely knowable.

Understanding laws is a complex, difficult and deeply contradictory process of reflecting reality. But the cognizing subject cannot reflect the entire real world, especially at once, completely and entirely. He can only forever approach this, creating various concepts and other abstractions, formulating certain laws, applying a whole range of techniques and methods in their entirety (experiment, observation, idealization, modeling, etc.). Describing the features of the laws of science, the American physicist R. Feynman wrote that, in particular, the laws of physics often do not have an obvious direct relationship to our experience, but represent its more or less abstract expression. Very often there is a huge distance between elementary laws and the main aspects of real phenomena.

Laws are first discovered in the form of assumptions and hypotheses. Further experimental material, new facts lead to the “purification of these hypotheses”, eliminating some of them, correcting others, until, finally, the law is established in its pure form. One of the most important requirements that a scientific hypothesis must satisfy is its fundamental verifiability in practice (in experience, experiment, etc.), which distinguishes a hypothesis from all kinds of speculative constructions, groundless inventions, unfounded fantasies, etc.

Since laws belong to the sphere of essence, the deepest knowledge about them is achieved not at the level of direct perception, but at the stage of theoretical research. It is here that the reduction of the random, visible only in phenomena, to actual internal movement ultimately occurs. The result of this process is the discovery of a law, or rather a set of laws inherent in a given area, which in their interconnection form the “core” of a certain scientific theory.

Revealing the mechanism for discovering new laws, R. Feynman noted that "... the search for a new law is carried out in the following way. First of all, they guess about it. Then they calculate the consequences of this guess and find out what this law will entail if it turns out that it is true Then the results of calculations are compared with what is observed in nature, with the results of special experiments or with our experience, and based on the results of such observations it is determined whether this is true or not. If the calculations disagree with the experimental data, then the law is incorrect." It should be emphasized that at all stages of the movement of knowledge, an important role is played by the philosophical attitudes that guide the researcher. Already at the beginning of the path to the law, according to R. Feynman, “it is philosophy that helps to make conjectures,” and it is difficult to make a final choice.

1 Feynman R. The nature of physical laws. M., 1987 P. 142

The discovery and formulation of a law is the most important, but not the last task of science, which must still show how the law it discovers makes its way. To do this, it is necessary, with the help of the law, relying on it, to explain all the phenomena of a given subject area (even those that seem to contradict it), to derive them all from the corresponding law through a number of intermediary links.

It should be borne in mind that each specific law almost never appears in its “pure form”, but always in conjunction with other laws of different levels and orders. In addition, we must not forget that although objective laws act with “iron necessity”, they themselves are by no means “iron”, but very “soft”, elastic in the sense that, depending on specific conditions, the one who gains the advantage is the one who that's a different law.

The elasticity of laws (especially social ones) is also manifested in the fact that they often act as trend laws, implemented in a very confusing and approximate manner, like some never firmly established average of constant fluctuations.

The conditions under which each given law is implemented can stimulate and deepen, or vice versa - suppress and remove its effect. Thus, any law in its implementation is always modified by specific historical circumstances, which either allow the law to gain full force, or slow down, weaken its action, expressing the law in the form of a breaking through trend. In addition, the effect of a particular law is inevitably modified by the concomitant effect of other laws.

Each law has the boundaries of its action, a certain sphere of its implementation (for example, the framework of a given form of motion of matter, a specific stage of development, etc.). On the basis of laws, not only the explanation of phenomena of a given class (group) is carried out, but also prediction, foresight of new phenomena, events, processes, etc., possible paths, forms and trends in the cognitive and practical activities of people.

Open laws, known patterns can - with their skillful and correct application - be used by people so that they become masters of nature and their own social relations. Since the laws of the external world are the basis for purposeful human activity, people must consciously be guided by the requirements arising from objective laws as regulators of their activities. Otherwise, the latter will not become effective and efficient, but will be carried out, at best, by trial and error. Based on the known laws, people can truly scientifically control both natural and social processes and regulate them optimally.

Relying in his activities on the “kingdom of laws,” a person can at the same time, to a certain extent, influence the mechanism for implementing a particular law. It can promote its action in a purer form, create conditions for the development of the law to its qualitative completeness, or, on the contrary, restrain this action, localize it or even transform it.

The variety of types of relationships and interactions in reality serves as the objective basis for the existence of many forms (types) of laws, which are classified according to one or another criterion (ground). According to the forms of movement of matter, laws can be distinguished: mechanical, physical, chemical, biological, social (public); In the main spheres of reality - the laws of nature, the laws of society, the laws of thinking; according to the degree of their generality, more precisely, according to the breadth of the scope of their action: universal (dialectical), general (special), particular (specific); according to the mechanism of determination - dynamic and statistical, causal and non-causal; according to their significance and role - basic and non-basic; according to the depth of fundamentality - empirical and theoretical, etc. etc.

Let's take a closer look at two special groups of laws, dynamic and statistical, because they play a certain role in the methodology of scientific research, especially when studying causal phenomena.

Dynamic patterns are objective, necessary, essential connections and dependencies that characterize the behavior of relatively isolated objects (consisting of a small number of elements), when studying which one can abstract from many random factors. Predictions based on dynamic patterns (as opposed to statistical ones) have a precisely defined, unambiguous character.

So, for example, in classical mechanics, if the law of motion of a body is known and its coordinates and speed are given, then from them one can accurately determine the position and speed of movement of the body at any other moment in time.

A dynamic pattern is usually understood as a form of causal relationship in which a given state of the system uniquely determines all its subsequent states, due to which knowledge of the initial conditions makes it possible to accurately predict the further development of the system. The dynamic pattern operates in all autonomous systems with a relatively small number of elements, little dependent on external influences. It determines, for example, the nature of the movement of planets in the solar system.

Dynamic patterns “permeate” a number of concepts of modern science. Thus, there is the concept of a “dynamic system” - a mechanical system with a finite number of degrees of freedom, for example, a system of a finite number of material points moving according to the laws of classical mechanics. Usually the law of motion of such systems is described by systems of ordinary differential equations. The absolutization of dynamic laws is closely related to the concept of mechanistic determinism (P. Laplace and others), which was discussed above.

Statistical patterns are a form of manifestation of the interconnection of phenomena in which a given state of the system determines all its subsequent states not unambiguously, but only with a certain probability, which is an objective measure of the possibility of realizing the trends of change inherent in the past. This (probabilistic) nature of predictions is due to the action of many random factors. The necessity manifested in statistical laws arises as a result of mutual compensation and balancing of many contingencies. These patterns are interconnected with dynamic ones, but cannot be reduced to them.

Many random factors usually occur in “statistical groups” or mass events (for example, a large number of molecules in a gas, people in social groups, etc.). The actions of many random factors are characterized by a stable frequency. This makes it possible to reveal the necessity that “breaks through” through the combined action of many accidents.

A statistical pattern arises as a result of the interaction of a large number of elements that make up a team, and therefore characterizes not so much the behavior of an individual element, but rather the behavior of the team as a whole. The necessity manifested in statistical laws arises as a result of mutual compensation and balancing of many random factors. “Although statistical patterns can lead to statements whose degree of probability is so high that it borders on certainty, nevertheless, in principle, exceptions are always possible.”

1 Heisenberg W. Steps beyond the horizon. M., 1987. P. 125.

Statistical laws, although they do not give unambiguous and reliable predictions, are nevertheless the only possible ones in the study of mass phenomena of a random nature. Behind the combined action of various factors of a random nature, which are practically impossible to cover, statistical laws reveal something stable, necessary, and repeating.

Statistical laws confirm the dialectic of transforming the random into the necessary. Dynamic laws turn out to be the limiting case of statistical ones, when probability becomes practical certainty.

It should also be said that statistical patterns are fundamentally irreducible to dynamic patterns (although they are interrelated). This is due to the following main circumstances: 1. the inexhaustibility of matter and the openness of systems; 2. the impossibility of implementing many development trends inherent in past states of systems; 3. the emergence in the process of development of opportunities and trends of qualitatively new states.

When characterizing statistical methods, concepts such as “statistics” and “probability” are important. In general, the concept of “statistics” is used in two main aspects: a) obtaining and processing information that characterizes the quantitative laws of life (technical, economic, social, political phenomena, culture) in inextricable connection with their qualitative content - a broad meaning; b) a set of data about any phenomenon or process. In the natural sciences, the concept of “statistics” means the analysis of mass phenomena based on the application of methods of probability theory - a narrow meaning.

Statistics develops special methods of research and processing of material: mass statistical observations, the grouping method, the method of average values, the index method, the balance method, the method of graphical images, etc. It is important to pay attention to the fact that statistical probability does not directly characterize an individual event, but a certain class events.

Probability is a concept that expresses the degree, the “measure of possibility,” and provides a quantitative characteristic of the feasibility of a possibility under a given set of specific conditions. If the probability is equal to one, then this is already reality; if it is equal to zero, it is an impossibility. Typically, there are three concepts of probability in scientific knowledge - classical, statistical and logical (inductive), which is widely used in probabilistic and inductive logic. The concept of "probability" is the starting point for the development of probabilistic statistical methods. The latter are based on taking into account the action of many random factors (which are characterized by a stable frequency), through which necessity and regularity “break through”. One of the main tasks of probability theory, as a science of mass random phenomena, is to clarify the patterns that arise during the interaction of a large number of random factors.

Probabilistic methods are based on the theory of probability, which is often called the science of randomness, and in the minds of many scientists, probability and randomness are practically inseparable. Moreover, it was on the basis of the analysis of statistical data that this theory was largely developed. Like statistics, probability theory is the science of patterns that characterize mass phenomena, but not mass phenomena in general, but a certain class of them, the specificity of which is expressed through ideas about randomness. There is even an idea that today chance appears as “an independent beginning of the world, its structure and evolution.”

The categories of necessity and chance are by no means outdated; on the contrary, their role in modern science has increased immeasurably. As the history of knowledge has shown, we, according to I. Prigogine, are only now beginning to appreciate the importance of the entire range of problems associated with necessity and chance.

Some scientists (N. Wiener, M. Bunge, Yu. Sachkov, etc.) believe that the main concept of probability theory is “probability distribution”. Thus, N. Wiener quite definitely states that “statistics is the science of distribution.” The concept of “probability distribution” means that a mass random phenomenon (a system of independent entities) is divided (disintegrates) into subsystems, the relative “weight” of which, the relative number of elements in each subsystem, is very stable. The presence of this stability is correlated with the concept of probability. Each of the elements is characterized by some common property, the values ​​of which change chaotically when moving from one element to another, but the relative number of elements with a certain given value, we emphasize again, is very stable.

It should be noted that the concept of “distribution” is central not only to probability theory, but also to statistics. This is how it is in mathematical statistics as a basic science that studies arrays of statistical data. The application of statistical ideas and methods in real knowledge is based on the recognition of the fundamental nature of the concept of “distribution”. Only on the basis of ideas about distributions is it possible to set problems and formulate basic dependencies in the corresponding scientific theories. Statistical patterns express the dependencies between the distributions of various quantities of the systems under study, as well as the nature of changes in these distributions over time.

Today, among those who recognize the fundamental importance of the probability-theoretic style of thinking and its greater generality compared to the approach based on the principle of rigid determination, there is a widespread belief that thinking that does not include the idea of ​​chance in its orbit is primitive (M. Bunge ). By analogy, we can say that those studies (natural sciences and social and humanities) that do not involve the analysis of statistical data in their orbit should also now be considered as quite primitive.

Probabilistic-statistical methods are widely used in the study of mass phenomena - especially in such scientific disciplines as mathematical statistics, statistical physics, quantum mechanics, chemistry, biology, cybernetics, synergetics, etc. In practical terms, the statistical method of generalization plays the greatest role both in scientific research and in decision-making in other areas of activity.

Recent research has shown that statistical generalization does not simply postulate that the conclusion is plausible, but quantifies (in percentage) the degree of probability of the conclusion based on the study of the sample. For scientific and practical forecasts, such a quantitative characteristic is especially important when one has to act in conditions of uncertainty and instability. Statistical laws are laws of averages that operate in the field of mass phenomena, in particular in atomic physics and in the social and human sciences.

Probabilistic ideas and research methods are important to the social sciences. Probability is included primarily in statistics as the science of quantitative relationships in mass social phenomena. Without the processing of statistical data, the development of social sciences is simply impossible.

It would not be an exaggeration to say that the entry of probability into real knowledge marks a great scientific, or more precisely, a methodological revolution, thanks to which people began to talk about a probabilistic style of thinking. It is within the framework of the latter that only adequate knowledge of complex, self-organizing, developing integral systems is possible.

The law of motion is given by the vector equation

L E C T I O N No. 1. K I N E M A T I C A

Kinematics is a branch of mechanics that studies the movement of bodies without considering the reasons causing the movement.

The movement of a body is the change in its position relative to another body in space over time.

The bodies relative to which the motion being studied is considered are called bodies of reference (for example, the walls of a laboratory, the Earth...).

Usually a coordinate system is associated with these bodies. We will use a right-handed rectangular coordinate system X, Y, Z.

A reference system is a coordinate system equipped with a clock and rigidly connected to an absolutely rigid body.

An absolutely rigid body is a body whose distance between any two points always remains the same.

Kinematics of a material point. Path, displacement, speed and acceleration

With the vector method, the position of point A, Fig. 1, at time t determined by its radius vector , drawn from the origin to the moving point.

The law of motion is given by the vector equation

With the coordinate method, the position of point A is determined by the coordinates x, y, z, and the law of motion is given by three equations:

at the same time , (3)

where are the modulo unit and mutually perpendicular vector vectors of the coordinate system.

Moving a point over a period of time - a vector connecting the position of a point at moments t And t+ . From Fig. 2 it can be seen that the displacement vector

Speed

The instantaneous speed of a material point is determined by the relation

, (5)

those. instantaneous speed is the derivative of the radius vector with respect to time. It is directed tangentially to the trajectory of the moving point.

In physics, it is customary to denote time derivatives not with a prime, but with an (×) above the letter.

From Fig. 2 it is clear that when , so the speed module

You can describe movement through trajectory parameters. To do this, we take a certain point on the trajectory as the initial one, then any other point is characterized by the distance S(t) from her. The radius vector becomes a complex function of the form , therefore from (5) it follows:

– unit vector tangent to the trajectory; – speed module.

In SI, speed is measured in meters per second (m/s).

Taking into account formula (3) from (5) we obtain

are the velocity components, they are equal to the derivatives of the corresponding coordinates with respect to time.

In Fig. 2, denotes the unit tangent vector, it coincides with the direction of velocity, therefore

1.1.2. Acceleration

To characterize the rate of change in speed, a vector physical quantity called acceleration is introduced . It is defined similarly to speed:

Taking into account formulas (7) and (8) from (10) we find

(11)

are the acceleration components, they are equal to the second derivatives of the corresponding coordinates with respect to time.

Taking into account formula (9) from (10) we obtain

It can be shown that

, (14)

Where R – radius of curvature at a given point of the trajectory, and is the unit vector of the normal to the trajectory at the point where the body was at the moment of time t. Moreover, they are mutually perpendicular (see Fig. 3).

Each point on the curve can be associated with a circle that merges with the trajectory in an infinitesimal portion of it. The radius of this circle R., (see Fig. 3), characterizes the curvature of the line at the point under consideration and is called the radius of curvature.

Dynamics – this is a branch of mechanics that studies the movement of bodies under the influence of forces applied to them.

Biomechanics also considers the interaction between the human body and the external environment, between parts of the body, and between two people (for example, in martial arts). As a result, forces arise, which are a quantitative measure of these interactions.

When studying quantities that are characterized not only by magnitude, but also by direction (for example, speed, acceleration, force, etc.), their vector image is used.

Vector – directed straight line segment(arrow) fig. 1.

Two vectors are considered equal only if they have the same lengths and directions (that is, they are parallel and oriented in the same direction). With a change in orientation, the sign of the vector changes (in Fig. 1 b = a; c = - a).

The rules of vector algebra reflect the physical properties of vector quantities. So, in accordance with the fact that the resultant of two forces is found according to the parallelogram rule, the sum of two vectors (a and b) determines a new vector (c = a + b), depicted by the diagonal of a parallelogram, the sides of which are vector-commands, Fig. 2.

Subtraction is defined as the inverse of addition. In addition to the vector, biomechanics also uses a term called “scalar” (scalar quantities).

Scalar – a quantity, each value of which (unlike a vector) can be expressed as a single number, as a result of which the set of values ​​can be depicted on a linear scale(rock – hence the name). Scalar quantities are: length, area, temperature, etc.

The scalar product (a۰b) of two vectors (a and b) is a number (scalar) equal to the product of the lengths of these vectors and the cosine of the angle formed by their directions, that is, |a| ۰ |b| ۰ cos φ, see fig. 3.

The straight line along which the force is directed is called the line of action of the force. A force is completely defined if its magnitude, direction, and point of application are given. If several forces (F1, F2, ...Fn) act on the elements of the biomechanical system of the human body, then they can be replaced by one force equal to their vector sum: FR = Σ Fi. This force is called the resultant force.

For example, a long jumper is subject to the force of gravity (mg) and the force of air resistance (Fc), Fig. 4. Acceleration (negative) is created by their resultant force (Fр).

The movements of the biomechanical system of the human body are subject to Newtonian mechanics. Consequently, the three basic laws of this mechanics determine the nature of movement, since despite the biological nature of the energy supply of movement, the body is a mechanical system and obeys all the laws that are associated with the movement of material objects on Earth.

Newton's first law(law of inertia). Any material body maintains a state of rest or uniform rectilinear motion until an external influence changes this state.

Rectilinear uniform motion of a material body is called inertial (or motion by inertia). Inertia – this is the property of a material body to resist changes in the speed of its movement(both in magnitude and direction). Inertia – inherent property of matter. Such resistance is possible only because bodies have a certain mass, which is considered a quantitative measure of inertia.

Weight – quantitative measure of body inertia. The SI unit of mass is called the kilogram (kg).

Newton's first law is a fairly idealized idea of ​​motion, since a body can move rectilinearly and uniformly only in the absence of any forces. In reality, a moving body is always influenced by various forces (air resistance forces, friction forces, etc.), whose influence leads to the moving body eventually stopping. This does not mean that Newton's first law is incorrect: simply movement, if the action of forces is not excluded, leads to a change in the state of the body and, in particular, to its transition to a state of rest.

A vector quantity equal to the product of body mass and acceleration and directed in the direction opposite to the acceleration in magnitude or direction of a given body under the influence of external forces is called the force of inertia: Fi = - m ac.

The change in the speed of a body is due to the influence of other bodies on it. The greater the acceleration it creates, the more intense the impact. On the other hand, a body with more mass has less acceleration (that is, its speed is more difficult to change). Therefore, it is customary to measure the impact on a body from all other bodies by multiplying the mass of the body by the acceleration imparted to it. This measure of influence is called force.

If the formula F = m a is transformed:

then we get Newton's second law.

The acceleration with which a body moves is directly proportional to the force acting on it, inversely proportional to the mass of the body and the direction coincides with the direction of the force

The relationship between the resultant of all external forces and the acceleration that it imparts to it can be transformed into a form that turns out to be useful in solving many problems in biomechanics:

Newton's third law. The forces with which material bodies act on each other are equal in magnitude, opposite in direction and directed in a straight line passing through these bodies.

This law shows that interaction is the action of one body on the second and the equal action of the second body on the first. Consequently, the source of force for the first body is the second, and since the forces of action and reaction are applied to different bodies, they cannot be added, and the acting forces cannot be replaced by the resultant.

A person, performing motor actions, participates in a complex movement, which consists of simpler ones - translational and rotational. Each of them has different characteristics.

Everyone paid attention to the variety of types of movement that he encounters in his life. However, any mechanical movement of the body comes down to one of two types: linear or rotational. Let us consider in the article the basic laws of motion of bodies.

What types of movement will we talk about?

As noted in the introduction, all types of body motion that are considered in classical physics are associated with either a rectilinear trajectory or a circular one. Any other trajectories can be obtained through a combination of these two. Further in the article the following laws of body motion will be considered:

1. Uniform in a straight line.
2. Uniformly accelerated (uniformly decelerated) in a straight line.
3. Uniform around the circumference.
4. Uniformly accelerated around the circle.
5. Movement along an elliptical path.

Uniform motion, or state of rest

Galileo first became interested in this movement from a scientific point of view at the end of the 16th - beginning of the 17th century. Studying the inertial properties of a body, as well as introducing the concept of a reference system, he guessed that the state of rest and uniform motion are one and the same (it all depends on the choice of the object relative to which the speed is calculated).

Subsequently, Isaac Newton formulated his first law of motion of a body, according to which the speed of the body is a constant value whenever there are no external forces that change the characteristics of motion.

Uniform rectilinear movement of a body in space is described by the following formula:

Where s is the distance that the body will cover in time t, moving at speed v. This simple expression is also written in the following forms (it all depends on the quantities that are known):

Moving in a straight line with acceleration

According to Newton's second law, the presence of an external force acting on a body inevitably leads to the appearance of acceleration in the latter. From (rate of change of speed) the expression follows:

a = v / t or v = a * t

If the external force acting on the body remains constant (does not change its magnitude or direction), then the acceleration will also not change. This type of motion is called uniformly accelerated, where acceleration acts as a coefficient of proportionality between speed and time (speed grows linearly).

For this motion, the distance traveled is calculated by integrating the speed over time. The law of body motion for a path with uniformly accelerated movement takes the form:

The most common example of this movement is the fall of any object from a height, in which the force of gravity imparts to it an acceleration g = 9.81 m/s 2 .

Rectilinear accelerated (slow) motion with initial speed

In fact, we are talking about a combination of two types of movement discussed in the previous paragraphs. Let's imagine a simple situation: a car was driving at a certain speed v 0, then the driver pressed the brakes, and the vehicle stopped after some time. How to describe the movement in this case? For the function of speed versus time, the expression is valid:

Here v 0 is the initial speed (before the car brakes). The minus sign indicates that the external force (sliding friction) is directed against the speed v 0 .

As in the previous paragraph, if we take the time integral of v(t), we obtain the formula for the path:

s = v 0 * t - a * t 2 / 2

Note that this formula only calculates the braking distance. To find out the distance traveled by the car during the entire time of its movement, you should find the sum of two paths: for uniform and for uniformly slow motion.

In the example described above, if the driver pressed the gas pedal rather than the brake pedal, then the “-” sign in the presented formulas would change to “+”.

Circular movement

Any movement in a circle cannot occur without acceleration, since even if the magnitude of the velocity is maintained, its direction changes. The acceleration that is associated with this change is called centripetal (it is this that bends the trajectory of the body, turning it into a circle). The module of this acceleration is calculated as follows:

a c = v 2 / r, r - radius

In this expression, the speed can depend on time, as happens in the case of uniformly accelerated motion in a circle. In the latter case, a c will increase rapidly (quadratic dependence).

Centripetal acceleration determines the force that must be applied to keep a body in a circular orbit. An example is hammer throwing competitions, where athletes exert significant force to spin the projectile before throwing it.

Rotation around an axis at constant speed

This type of motion is identical to the previous one, only it is customary to describe it not using linear physical quantities, but using angular characteristics. The law of rotational motion of a body, when the angular velocity does not change, is written in scalar form as follows:

Here L and I are the moments of momentum and inertia, respectively, ω is the angular velocity, which is related to the linear velocity by the equality:

The value ω shows how many radians the body will rotate per second. The quantities L and I have the same meaning as momentum and mass for linear motion. Accordingly, the angle θ through which the body will rotate in time t is calculated as follows:

An example of this type of motion is the rotation of a flywheel located on the crankshaft in a car engine. The flywheel is a massive disk, which is very difficult to give any acceleration. Thanks to this, it ensures a smooth change in torque, which is transmitted from the engine to the wheels.

Rotation around an axis with acceleration

If an external force is applied to a system that is capable of rotating, it will begin to increase its angular velocity. This situation is described by the following law of body motion around:

Here F is the external force that is applied to the system at a distance d from the axis of rotation. The product on the left side of the equality is called the moment of force.

For uniformly accelerated motion in a circle, we find that ω depends on time as follows:

ω = α * t, where α = F * d / I - angular acceleration

In this case, the angle of rotation over time t can be determined by integrating ω over time, that is:

If the body was already rotating at a certain speed ω 0, and then the external moment of force F*d began to act, then by analogy with the linear case the following expressions can be written:

ω = ω 0 + α * t;

θ = ω 0 * t + α * t 2 / 2

Thus, the appearance of an external moment of force is the reason for the presence of acceleration in a system with an axis of rotation.

For completeness of information, we note that the rotation speed ω can be changed not only with the help of an external moment of force, but also by changing the internal characteristics of the system, in particular its moment of inertia. This situation was seen by every person who watched the skaters spin on the ice. When grouping, athletes increase ω by decreasing I, according to the simple law of body movement:

Movement along an elliptical trajectory using the example of the planets of the solar system

As you know, our Earth and other planets of the solar system rotate around their star not in a circle, but along an elliptical trajectory. For the first time, mathematical laws to describe this rotation were formulated by the famous German scientist Johannes Kepler at the beginning of the 17th century. Using the results of his teacher Tycho Brahe's observations of the motion of the planets, Kepler came to the formulation of his three laws. They are formulated as follows:

1. The planets of the Solar System move in elliptical orbits, with the Sun located at one of the foci of the ellipse.
2. The radius vector that connects the Sun and the planet describes equal areas in equal periods of time. This fact follows from the conservation of angular momentum.
3. If we divide the square of the orbital period by the cube of the semimajor axis of the elliptical orbit of a planet, we obtain a certain constant that is the same for all planets in our system. Mathematically it is written like this:

T 2 / a 3 = C = const

Subsequently, Isaac Newton, using these laws of motion of bodies (planets), formulated his famous law of universal gravitation, or gravitation. Using it, we can show that the constant C in 3 is equal to:

C = 4 * pi 2 / (G * M)

Where G is the gravitational universal constant, and M is the mass of the Sun.

Note that movement along an elliptical orbit in the case of the action of a central force (gravity) leads to the fact that the linear speed v is constantly changing. It is maximum when the planet is closest to the star, and minimum away from it.