Why is the concept of a material point introduced? What is a material point called? Material point

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

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 in 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 visible from the following 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 - a geometric point with 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 - A point with mass...

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

. Rectilinear and curvilinear movement. Body movement along...

The didactic material is intended for students of all specialties of the correspondence faculty of GUCMiZ, studying a course in mechanics according to the program for engineering and technical specialties.

The didactic material contains a brief summary of the theory on the topic being studied, adapted to the level of training of part-time students, examples of solving typical problems, questions and assignments similar to those offered to students in exams, and reference material.

The purpose of such material is to help a part-time student independently, in a short time, to master the kinematic description of translational and rotational movements using the analogy method; learn to solve numerical and qualitative problems, understand issues related to the dimension of physical quantities.

Particular attention is paid to solving qualitative problems, as one of the methods for deeper and more conscious mastery of the fundamentals of physics, necessary when studying special disciplines. They help to understand the meaning of occurring natural phenomena, to understand the essence physical laws and clarify the scope of their application.

Didactic material may be useful to full-time students.

KINEMATICS

The part of physics that studies mechanical motion is called mechanics . Mechanical movement is understood as a change over time in the relative position of bodies or their parts.

Kinematics - the first section of mechanics, it studies the laws of motion of bodies, without being interested in the reasons that cause this movement.

1. Material point. Reference system. Trajectory.

Path. Move vector

The simplest kinematics model is material point . This is a body whose dimensions can be neglected in this problem. Any body can be represented as a collection of material points.

To mathematically describe the movement of a body, it is necessary to decide on a reference system. Reference system (CO) consists of bodies of reference and related coordinate systems And hours. If there are no special instructions in the problem statement, it is considered that the coordinate system is related to the Earth's surface. The most commonly used coordinate system is Cartesian system.

Let it be necessary to describe the motion of a material point in Cartesian system coordinates XYZ(Fig. 1). At some point in time t 1 point is in position A. The position of a point in space can be characterized by a radius vector r 1 drawn from the origin to the position A, and coordinates x 1 , y 1 , z 1 . Here and below, vector quantities are indicated in bold italics. By the time t 2 = t 1 + Δ t the material point will move to position IN with radius vector r 2 and coordinates x 2 , y 2 , z 2 .

Trajectory of movement called a curve in space along which a body moves. Based on the type of trajectory, rectilinear, curvilinear and circular movements are distinguished.

Path length (or path ) - length of the section AB, measured along the trajectory of movement, is denoted by Δs (or s). Distance in the International System of Units (SI) is measured in meters (m).

Move vector material point Δ r represents the vector difference r 2 And r 1, i.e.

Δ r = r 2 - r 1.

The magnitude of this vector, called displacement, is the shortest distance between positions A And IN(start and end) moving point. Obviously, Δs ≥ Δ r, and equality holds for rectilinear motion.

When a material point moves, the value of the distance traveled, the radius vector and its coordinates change with time. Kinematic equations of motion (further equations of motion) are called their dependences on time, i.e. equations of the form

s=s( t), r=r (t), x=X(t), y=at(t), z=z(t).

If such an equation is known for a moving body, then at any moment of time you can find the speed of its movement, acceleration, etc., which we will verify later.

Any movement of a body can be represented as a set progressive And rotational movements.

2. Kinematics of translational motion

Progressive is a movement in which any straight line rigidly connected to a moving body remains parallel to itself .

Speed characterizes the speed of movement and direction of movement.

Medium speed movements in the time interval Δ t is called the quantity

(1)

where - s is the segment of path traveled by the body in time during time  t.

Instant speed movement (speed at a given time) is a quantity whose modulus is determined by the first derivative of the path with respect to time

(2)

Speed is a vector quantity. The instantaneous velocity vector is always directed along tangent to the trajectory of movement (Fig. 2). The unit of speed is m/s.

The value of the speed depends on the choice of reference system. If a person is sitting in a train carriage, he and the train move relative to the CO connected to the ground, but are at rest relative to the CO connected to the car. If a person walks along a carriage at a speed , then his speed relative to the “ground” CO  s depends on the direction of movement. Along the movement of the train  z =  trains + , against   z =  trains - .

Projections of the velocity vector on the coordinate axes υ X ,υ y ,υ z are defined as the first derivatives of the corresponding coordinates with respect to time (Fig. 2):

If the projections of velocity on the coordinate axes are known, the velocity modulus can be determined using the Pythagorean theorem:

(3)

Uniform called movement at a constant speed (υ = const). If the direction of the velocity vector does not change v, then the movement will be uniform and rectilinear.

Acceleration - physical quantity characterizing the rate of change in speed in magnitude and direction Average acceleration is defined as

(4)

where Δυ is the change in speed over a period of time Δ t.

Vector instantaneous acceleration is defined as the derivative of the velocity vector v by time:

(5)

Since during curvilinear motion the speed can change both in magnitude and in direction, it is customary to decompose the acceleration vector into two mutually perpendicular components

A = A τ + A n.

(6) Tangential A (or tangential) acceleration

.(7)

τ characterizes the rate of change in speed in magnitude, its modulus

Tangential acceleration is directed tangentially to the trajectory of motion along the speed during accelerated motion and against the speed during slow motion (Fig. 3). Normal A (centripetal) acceleration

(8)

n characterizes the change in speed in direction, its module Where R

- radius of curvature of the trajectory.

The normal acceleration vector is directed to the center of the circle, which can be drawn tangentially to a given point on the trajectory; it is always perpendicular to the tangential acceleration vector (Fig. 3).

. (9)

The modulus of total acceleration is determined by the Pythagorean theorem A Direction of the total acceleration vector

determined by the vector sum of the normal and tangential acceleration vectors (Fig. 3) Equally variable called movement with permanent . acceleration If the acceleration is positive, then this is uniformly accelerated motion , if it is negative - .

equally slow A When moving in a straight line A = Aם =0 and Aτ. If Aם =0 and τ = 0, the body is moving straight and even Aτ. If A; at.

τ = const movement rectilinear uniformly variable At

uniform motion s the distance traveled is calculated using the formula: t→ s d t= d t=  t+ s 0 , (10)

n characterizes the change in speed in direction, its module s= ∫d t = = ∫d

0 - starting path for υ (t 0. The last formula must be remembered. s(t Graphic dependencies

) And ) are shown in Fig. 4.  = ∫ A For t = A uniformly alternating motion t d

= ∫ dt , from here

A t=0.

+  0 , (11) s where  0 is the initial speed at t = ∫(At Distance traveled t= ∫d

s = At+  0)d t + s 0 , (12)

n characterizes the change in speed in direction, its module s. t Solving this integral, we get

2 /2 +  0 A(t), υ (t 0. The last formula must be remembered. s(t 0 - initial path (for

= 0). We recommend that you remember formulas (11), (12). Graphic dependencies ) are shown in Fig. 5. To uniform motion with acceleration free fall g ) are shown in Fig. 5.›0, acceleration when moving up ) are shown in Fig. 5.‹ 0. The speed of movement and the distance traveled in this case changes according to (11):

 =  0 + ) are shown in Fig. 5.t; (13)

h = ) are shown in Fig. 5.t+  0)d t +h 0 . (14)

Let's consider the movement of a body thrown at an angle to the horizon (ball, stone, cannon shell,...). This complex movement consists of two simple ones: horizontally along the axis OH and verticals along the axis OU(Fig. 6). Along the horizontal axis, in the absence of environmental resistance, the movement is uniform; along the vertical axis - uniformly variable: uniformly slowed down to the maximum lifting point and uniformly accelerated after it. The trajectory of movement has the form of a parabola. Let  0 be the initial speed of a body thrown at an angle α to the horizon from a point A(origin). Its components along the selected axes:

 0x =  x =  0 cos α = const; (15)

 0у =  0 sinα.

(16) According to formula (13) we have for our example at any point of the trajectory to the point

WITH ) are shown in Fig. 5. t y =  0y - =  0 sinα. - t ;

 x =  0х =  0 cos α = const. According to formula (13) we have for our example at any point of the trajectory to the point At the highest point of the trajectory, point

WITH ) are shown in Fig. 5. t y =  0y - =  0 sinα. - t = 0 → t , vertical component of speed  y = 0. From here you can find the time of movement to point C: =  0 sinα/. (17)

h Knowing this time, you can determine the maximum height of the body lifting using (14): t- ) are shown in Fig. 5.t max =  0у ) are shown in Fig. 5.– ) are shown in Fig. 5. 2 /2= 0 sinα  0 sinα/ /) are shown in Fig. 5.( 0 sinα ) are shown in Fig. 5.) (18)

) 2 /2 = ( 0 sinα) 2 /(2 IN Since the trajectory of movement is symmetrical, the total time of movement to the end point

t 1 =2 t equals ) are shown in Fig. 5.. (19)

= 2 0 sinα / AB Range of flight

AB taking into account (15) and (19) will be determined as follows: t=  x =  0 sinα/ 1 =  0 cosα 2 0 sinα/ =  0 sinα/. (20)

= 2 0 2 cosα sinα/ The total acceleration of a moving body at any point of the trajectory is equal to the acceleration of gravity g

; it can be decomposed into normal and tangential, as was shown in Fig. 3.

Definition

A material point is a macroscopic body whose dimensions, shape, rotation and internal structure can be neglected when describing its motion. The question of whether a given body can be considered as a material point depends not on the size of this body, but on the conditions of the problem being solved. For example, the radius of the Earth is much smaller than the distance from the Earth to the Sun, and its orbital motion can be well described as the movement of a material point with a mass equal to the mass of the Earth and located at its center. However, when considering the daily motion of the Earth around its own axis, replacing it with a material point does not make sense. The applicability of the material point model to a specific body depends not so much on the size of the body itself, but on the conditions of its motion. In particular, in accordance with the theorem on the motion of the center of mass of a system during translational motion, any can be considered a material point whose position coincides with the center of mass of the body.

Mass, position, speed and some other physical properties of a material point at any given moment in time completely determine its behavior.

The position of a material point in space is defined as the position of a geometric point. In classical mechanics, the mass of a material point is assumed to be constant in time and independent of any features of its movement and interaction with other bodies. In the axiomatic approach to the construction of classical mechanics, the following is accepted as one of the axioms:

Axiom

A material point is a geometric point that is associated with a scalar called mass: $(r,m)$, where $r$ is a vector in Euclidean space related to some Cartesian coordinate system. The mass is assumed to be constant, independent of the position of the point in space and time.

Mechanical energy can be stored by a material point only in the form of kinetic energy of its movement in space and (or) 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 in 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.

The method of studying the laws of motion of real bodies by studying the motion of an ideal model - a material point - is fundamental in mechanics. Any macroscopic body can be represented as a collection of interacting material points g, with masses equal to the masses of its parts. The study of the movement of these parts comes down to the study of the movement of material points.

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.

Exercise 1

a) a car entering the garage;

b) a car on the Voronezh - Rostov highway?

a) a car entering a garage cannot be taken as a material point, since in these conditions the dimensions of the car are significant;

b) a car on the Voronezh-Rostov highway can be taken as a material point, since the size of the car is much smaller than the distance between cities.

Is it possible to take this as a material point:

a) a boy who walks 1 km on his way home from school;

b) a boy doing exercises.

a) When a boy, returning from school, walks a distance of 1 km to home, then the boy in this movement can be considered as a material point, because his size is small compared to the distance he covers.

b) when the same boy performs morning exercises, then he cannot be considered a material point.

Material 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... ...

Physical encyclopedia The beginnings of modern natural science

Polytechnic terminological explanatory dictionary

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...

By a material point we mean a macroscopic body, the properties of which (mass, rotation, shape, etc.) can be neglected if there is a need to describe its movement. You will learn about what a material point is from this article.

If we talk about whether this body can be considered as such a point, then everything here is determined not by the size of the body, but by the conditions set in the problem. As an example, the radius of our planet is an order of magnitude smaller than the distance between the Sun and the Earth, and orbital motion can be described precisely in the form of the movement of a material point that has a mass similar to the Earth and is located in its center. However, if we consider the daily motion of the planet around its own axis, then it makes no sense to replace it with a material point. The model of a point of the type under consideration for a specific body is determined not by the dimensions of the body itself, but to a greater extent by the conditions of its movement. As an example, according to the theorem on the movement of the center of mass of a system during translational movement, each rigid body can be considered as a material point, the position of which is similar to the center of mass of the body.

Physical properties of a point such as mass, speed, position and others determine its behavior at each moment of time.

The position in space of the point under consideration is determined in the form of the position of a geometric point. In mechanics, a material point has a mass that is constant in time and independent of any factors of its movement and interaction with other bodies. If we use an approach to constructing mechanics based on axioms, then the following is taken as one of them:

Axiom

A body is called a material point - geometric point, which corresponds to a scalar called mass: (r and m), where r is a vector in Euclidean space that refers to one or another Cartesian coordinate system. Mass is constant and independent of the position of a point in time and space.

Material point stores mechanical energy solely as the kinetic energy of its movement in space, or as potential energy that interacts with the field. This suggests that given point cannot be deformed, cannot rotate around its own axis, and it does not react to its changes in space. In parallel with this, the material point moves with a change in its distance from a pair of Euler angles and some instantaneous center of rotation, which give the line direction, and it, in turn, connects this point with the center. This method is very common in mechanics.

The technique by which the laws of motion of real objects are studied by studying the movement of an ideal model is the basis of mechanics. Each macroscopic body can be represented in the form of material points interacting with each other, having masses corresponding to the masses of its parts. The study of the movement of these parts comes down to studying the movement of the points in question.

The term itself is somewhat limited in application. As an example, a rarefied gas at high temperatures is characterized by a small size of molecules relative to the typical distance between them. And although this can be neglected in some cases and the molecule can be taken for a material point, in general this is not the case. Internal energy a molecule is determined by vibrations and rotations, and its capacity depends on the size, structure and properties of the particle. In some cases, monatomic molecules can be considered as examples of a material point, but even in them, at high temperatures, electron shells are excited due to collisions of molecules with further emission.

First task

a) a car entering the garage;
b) a car on the Moscow - Rostov highway?

a) a car entering a garage cannot be considered such an object, since the difference in size between the car and the garage is relatively small;
b) a car on the Moscow - Rostov highway can be considered as such a point, since the dimensions vehicle orders of magnitude less distance.

Second task

a) a boy walking home from school (path 1 km);
b) a boy doing physical exercise?
a) Since the path from school to home is a kilometer, the boy can be considered as such a point, since he is very small in size relative to the distance traveled.
b) when the same child does morning exercises, he cannot be mistaken for a material point.