OGE
What is a point in physics. Dimensionless material point and different reference systems. Thin lenses, thin lens formula

What is a point in physics. Dimensionless material point and different reference systems. Thin lenses, thin lens formula

Material point

Material point(particle) - the simplest physical model in mechanics - an ideal body whose dimensions are equal to zero; the dimensions of the body can also be considered infinitesimal compared to other sizes or distances within the assumptions of the problem under study. Position material point in space is defined as the position of a geometric point.

In practice, a material point is understood as a body with mass, the size and shape of which can be neglected when solving this problem.

At straight motion a body only needs one coordinate axis to determine its position.

Peculiarities

The mass, position and speed of a material point at each specific moment in time completely determine its behavior and physical properties.

Consequences

Mechanical energy can be stored by a material point only in the form of the kinetic energy of its movement in space, and (or) the potential energy of interaction with the field. This automatically means that a material point is incapable of deformation (only an absolutely rigid body can be called a material point) and rotation around its own axis and changes in the direction of this axis in space. At the same time, the model of the motion of a body described by a material point, which consists of changing its distance from some instantaneous center of rotation and two Euler angles, which specify the direction of the line connecting this point to the center, is extremely widely used in many branches of mechanics.

Restrictions

The limited application of the concept of a material point is clear from this example: in a rarefied gas at high temperature, the size of each molecule is very small compared to the typical distance between molecules. It would seem that they can be neglected and the molecule can be considered a material point. However, this is not always the case: vibrations and rotations of a molecule are an important reservoir of the “internal energy” of the molecule, the “capacity” of which is determined by the size of the molecule, its structure and chemical properties. To a good approximation, a monatomic molecule (inert gases, metal vapors, etc.) can sometimes be considered as a material point, but even in such molecules, at a sufficiently high temperature, excitation of electron shells is observed due to collisions of molecules, followed by emission.

Notes

Wikimedia Foundation.

2010.

See what a “material point” is in other dictionaries: A point with mass. In mechanics, the concept of a material point is used in cases where the size and shape of a body do not play a role in the study of its motion, and only mass is important. Almost any body can be considered as a material point if... ...

Big Encyclopedic Dictionary A concept introduced in mechanics to designate an object that is considered as a point with mass. The position of the M. t. in law is defined as the position of the geom. points, which greatly simplifies the solution of mechanics problems. Practically, the body can be considered... ...

Physical encyclopedia material point - A point with mass. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics theoretical mechanics EN particle DE materialle Punkt FR point matériel …

Technical Translator's Guide

Modern encyclopedia In mechanics: infinitesimal body. Dictionary foreign words , included in the Russian language. Chudinov A.N., 1910 ...

Dictionary of foreign words of the Russian language Material point - MATERIAL POINT, a concept introduced in mechanics to designate a body whose dimensions and shape can be neglected. The position of a material point in space is defined as the position of a geometric point. The body can be considered material... ...

Illustrated Encyclopedic Dictionary A concept introduced in mechanics for an object of infinitesimal size that has mass. The position of a material point in space is defined as the position of a geometric point, which simplifies the solution of mechanics problems. Almost any body can... ...

Dictionary of foreign words of the Russian language - encyclopedic Dictionary geometric point , having mass; material point is an abstract image of a material body that has mass and has no dimensions...

Physical encyclopedia The beginnings of modern natural science

Physical encyclopedia- materialusis taškas statusas T sritis fizika atitikmenys: engl. mass point; material point vok. Massenpunkt, m; materieller Punkt, m rus. material point, f; point mass, f pranc. point mass, m; point matériel, m … Fizikos terminų žodynas Polytechnic terminological explanatory dictionary

Books

Set of tables. Physics. 9th grade (20 tables), . Educational album of 20 sheets. Material point. Coordinates of a moving body. Acceleration. Newton's laws. Law universal gravity. Straightforward and curvilinear movement. Body movement along...

A material point is an infinitesimal body with mass whose shape can be neglected. This is the simplest, idealized body, the geometric dimensions of which are small, and to determine it in space you need only 3 coordinates. The rotation of a material point is also neglected. They believe that there are no forces inside a material point. It does not compress, does not stretch, but is absolutely elastic. The mass of a material point is constant in time and does not depend on any other conditions.

Figure 1 - replacing a body with a material point.

The concept of a material point is introduced into mechanics to simplify the description of the movement of material bodies. A body of arbitrary shape, which also has elasticity, can perform both translational and rotational motion. It may also become deformed. That is, individual points of the body, in addition to moving with the body, also move relative to it. In the general case, the movement of a body of arbitrary shape is quite complex and difficult to describe.

It is precisely to simplify the description of such motion that the concept of a material point is introduced. It is believed that it has the mass of the described body, but infinitesimal dimensions. In this case, it only performs forward movement. The material point is used to determine the center of mass. This is precisely the point that has mass distributed throughout the volume of the body.

Figure 2 - material point.

It is clear that you cannot simply take and replace the body of a complex handicap with an extremely simplified model. To do this, certain conditions must be met. The main one is: the size of the body must be many times smaller than the distance it travels. Also an important factor influencing the possibility of replacing a real body with a simplified model are the experimental conditions and the expected result.

Let us assume that, according to the conditions of the experiment, it is necessary to determine the time during which the train will travel the distance from point A to point B, knowing its speed. In this case, we don’t care what shape the train has or how many cars the train consists of. Since we know its speed. It can be represented as a material point. But if we need to determine the air resistance exerted on the train when moving at high speeds. It makes no sense to imagine it as a material point. Since the result of this experiment depends on the shape of the train.

But what to do in the case when the body cannot be represented in the form of a material point. Due to the fact that it has complex shape. And its individual parts move not only with linear, but also with angular speed. Then the body is represented as a sum of individual material points. Which will only make forward movement.

When solving a whole set of problems, you can ignore the shape and size of the body and consider it as a material point.

Definition

Material point in physics they call a body that has mass, but whose dimensions, in comparison with the distances to other bodies, can be neglected in the problem under consideration.

The concept of “material point”

The concept of “material point” is an abstraction. In nature, material points do not exist. But the formulation of some problems in mechanics makes it possible to use this abstraction.

When we talk about a point in kinematics, it can be considered as a mathematical point. In kinematics, a point is understood as a small mark on a body or the body itself, if its dimensions are small in comparison with the distances the body travels.

In such a section of mechanics as dynamics, we must already talk about a material point, as a point that has mass. Basic laws classical mechanics refer to a material point, a body that does not have geometric dimensions, but has mass.

In dynamics, the size and shape of a body in many cases does not affect the nature of movement; in this case, the body can be considered as a material point. But under other conditions, this same body cannot be considered a point, since its shape and size are decisive in describing the movement of the body.

So, if a person is interested in how much time a car needs to get from Moscow to Tyumen, then it is not at all necessary to know how each of the wheels of the car moves. But, if a motorist is trying to squeeze his car into a narrow parking space, the car cannot be taken as a material point, since the size of the car matters. We can take the Earth as a material point if we consider the movement of our planet around the Sun, but this cannot be done when studying its movement around its own axis, if we are trying to establish the reasons why day follows night. Thus, the same body under some conditions can be considered as a material point, but under other conditions this cannot be done.

There are some types of motion in which the body can be safely taken as a material point. So, for example, during the translational motion of a rigid body, all its parts move equally, therefore in such motion the body is usually considered as a point with a mass that is equal to the mass of the body. But if the same body rotates around its axis, then it cannot be taken as a material point.

And so, a material point is the simplest model of a body. If a body can be likened to a material point, then this significantly simplifies the solution to the problem of studying its motion.

Different types of point movement are distinguished, first of all, by the type of trajectory. If the trajectory of a point’s movement is a straight line, then the movement is called rectilinear. In relation to the movement of a macroscopic body, it makes sense to talk about the rectilinear or curvilinear movement of the body only when, when describing the movement, we can limit ourselves to considering the movement of one point of this body. In a body, in general, different points can perform different types movements.

Material point system

If a body cannot be taken as a material point, then it can be represented as a system of material points. In this case, the body is mentally divided into infinitesimal elements, each of which can be taken as a material point.

In mechanics, each body can be represented as a system of material points. Having the laws of motion of a point, we can consider that we have a method for describing any body.

In mechanics significant role plays the concept of an absolutely rigid body, which is defined as a system of material points, the distances between which are constant, for any interactions of this body.

Examples of problems with solutions

Example 1

Exercise. In what case can a body be considered a material point:

An athlete throws a shot at a competition. Can the core be considered a material point?

The ball rotates around its axis. Is a ball a material point?

A gymnast performs an exercise on the uneven bars.

The runner covers the distance.

Example 2

Exercise. Under what conditions can a stone moving upward be considered a material point? See Fig.1 and Fig.2.

Solution: In Fig. 1 the size of the stone cannot be considered small in comparison with the distance to it. In this case, the stone cannot be considered a material point

In Fig. 2, the stone rotates, therefore, it cannot be considered a material point.

Answer. A stone thrown upward can be considered a material point if its dimensions are small in comparison with the distance to it, and it moves translationally (there will be no rotation).

Material point

Material point(particle) - the simplest physical model in mechanics - an ideal body whose dimensions are equal to zero; the dimensions of the body can also be considered infinitesimal compared to other sizes or distances within the assumptions of the problem under study. The position of a material point in space is defined as the position of a geometric point.

In practice, a material point is understood as a body with mass, the size and shape of which can be neglected when solving this problem.

When a body moves in a straight line, one coordinate axis is sufficient to determine its position.

Peculiarities

The mass, position and speed of a material point at each specific moment in time completely determine its behavior and physical properties.

Consequences

Restrictions

Notes

Wikimedia Foundation.

Mechanical movement
Absolutely solid body

2010.

MATERIAL POINT- a point with mass. In mechanics, the concept of a material point is used in cases where the size and shape of a body do not play a role in the study of its motion, and only mass is important. Almost any body can be considered as a material point if... ... A point with mass. In mechanics, the concept of a material point is used in cases where the size and shape of a body do not play a role in the study of its motion, and only mass is important. Almost any body can be considered as a material point if... ...

MATERIAL POINT- a concept introduced in mechanics to designate an object, which is considered as a point with mass. The position of the M. t. in law is defined as the position of the geom. points, which greatly simplifies the solution of mechanics problems. Practically, the body can be considered... ... A concept introduced in mechanics to designate an object that is considered as a point with mass. The position of the M. t. in law is defined as the position of the geom. points, which greatly simplifies the solution of mechanics problems. Practically, the body can be considered... ...

Physical encyclopedia- A point with mass. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics theoretical mechanics EN particle DE materialle Punkt FR point matériel ... EN particle DE materialle Punkt FR point matériel …

MATERIAL POINT Technical Translator's Guide

MATERIAL POINT- In mechanics: an infinitesimal body. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910 ... , included in the Russian language. Chudinov A.N., 1910 ...

Physical encyclopedia- a concept introduced in mechanics for an object of infinitesimal size that has mass. The position of a material point in space is defined as the position of a geometric point, which simplifies the solution of mechanics problems. Almost any body can... ... A concept introduced in mechanics for an object of infinitesimal size that has mass. The position of a material point in space is defined as the position of a geometric point, which simplifies the solution of mechanics problems. Almost any body can... ...

Dictionary of foreign words of the Russian language- a geometric point with mass; material point is an abstract image of a material body that has mass and has no dimensions... , having mass; material point is an abstract image of a material body that has mass and has no dimensions...

Physical encyclopedia The beginnings of modern natural science

Books

Set of tables. Physics. 9th grade (20 tables), . Educational album of 20 sheets. Material point. Coordinates of a moving body. Acceleration. Newton's laws. The law of universal gravitation. Rectilinear and curvilinear movement. Body movement along...

MATERIAL POINT– a model concept (abstraction) of classical mechanics, denoting a body of vanishingly small dimensions, but having a certain mass.

On the one hand, a material point is the simplest object of mechanics, since its position in space is determined by only three numbers. For example, three Cartesian coordinates that point in space in which our material point is located.

On the other hand, a material point is the main supporting object of mechanics, since it is for it that the basic laws of mechanics are formulated. All other objects of mechanics - material bodies and environments - can be represented in the form of one or another set of material points. For example, any body can be “cut” into small parts and each of them can be taken as a material point with the corresponding mass.

When it is possible to “replace” a real body with a material point when posing a problem about the motion of a body, it depends on the questions that must be answered by the solution of the formulated problem.

Various approaches to the question of using the material point model are possible.

One of them is empirical in nature. It is believed that the material point model is applicable when the sizes of moving bodies are negligible compared to the magnitude of the relative movements of these bodies. As an illustration we can cite solar system. If we assume that the Sun is a stationary material point and assume that it acts on another material point-planet according to the law of universal gravitation, then the problem of the movement of a point-planet has a known solution. Among the possible trajectories of a point’s motion, there are also those on which Kepler’s laws, empirically established for the planets of the solar system, are satisfied.

Thus, when describing the orbital motions of planets, the material point model is quite satisfactory. (However, the construction mathematical model such phenomena as solar and lunar eclipses requires taking into account the actual sizes of the Sun, Earth and Moon, although these phenomena are obviously associated with orbital movements.)

The ratio of the diameter of the Sun to the diameter of the orbit of the nearest planet - Mercury - is ~ 1·10 -2, and the ratio of the diameters of the planets closest to the Sun to the diameters of their orbits is ~ 1 ÷ 2·10 -4. Can these numbers serve as a formal criterion for neglecting the size of a body in other problems and, therefore, for the acceptability of the point model? Practice shows that no.

For example, a small bullet size l= 1 ÷ 2 cm distance flies L= 1 ÷ 2 km, i.e. ratio, however, the flight trajectory (and range) significantly depends not only on the mass of the bullet, but also on its shape, and on whether it rotates. Therefore, even a small bullet, strictly speaking, cannot be considered a material point. If in problems of external ballistics the thrown body is often considered a material point, then this is accompanied by a number of reservations additional conditions, as a rule, empirically taking into account the real characteristics of the body.

If we turn to astronautics, then when spacecraft(SC) is launched into a working orbit; in further calculations of its flight trajectory, it is considered a material point, since no changes in the shape of the spacecraft have any noticeable effect on the trajectory. Only sometimes, when making trajectory corrections, does it become necessary to ensure precise orientation of jet engines in space.

When the descent compartment approaches the Earth’s surface at a distance of ~100 km, it immediately “turns” into a body, since the “side” it enters the dense layers of the atmosphere determines whether the compartment will deliver the cosmonauts and the returned materials to the desired point on the Earth .

The model of a material point turned out to be practically unacceptable for describing the movements of such physical objects of the microworld as elementary particles, atomic nuclei, electron, etc.

Another approach to the question of using the material point model is rational. According to the law of change in the momentum of a system, applied to an individual body, the center of mass C of the body has the same acceleration as some (let’s call it equivalent) material point, which is acted upon by the same forces as on the body, i.e.

Generally speaking, the resulting force can be represented as a sum , where depends only on and (radius vector and speed of point C), and - and on angular velocity body and its orientation.

If F 2 = 0, then the above relation turns into the equation of motion of an equivalent material point.

In this case, they say that the movement of the center of mass of the body does not depend on the rotational motion of the body. Thus, the possibility of using a material point model receives a rigorous mathematical (and not just empirical) justification.

Naturally, in practice the condition F 2 = 0 is performed rarely and usually F 2 No. 0, however, it may turn out that F 2 is in some ways small compared to F 1 . Then we can say that the model of an equivalent material point is some approximation in describing the motion of a body. An estimate of the accuracy of such an approximation can be obtained mathematically, and if this estimate turns out to be acceptable for the “consumer,” then replacing the body with an equivalent material point is acceptable, otherwise such a replacement will lead to significant errors.

This can also occur when the body moves translationally and, from the point of view of kinematics, it can be “replaced” by some equivalent point.

Naturally, the model of a material point is not suitable for answering questions such as “why does the Moon face the Earth with only one side?” Similar phenomena are associated with rotational movement bodies.

Vitaly Samsonov