What body can be taken for the material point. Material point

Material point

Material point (Particle) - the simplest physical model in the mechanics is the perfect body whose sizes are zero, you can also count the sizes of the body are infinitely small compared to other sizes or distances within the assuming tasks under study. The position of the material point in space is defined as the position of the geometric point.

Practically under the material point understands the body weight, size and form of which can be neglected when solving this task.

With the straight-line movement of the body, one coordinate axis is enough to determine its position.

Features

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

Corollary

Mechanical energy can be stacked with 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 the inability of the material point to deformities (the material point can only be called an absolutely solid body) and the rotation around its own axis and changes in the direction of this axis in space. At the same time, the model of the body moved, described by the material point, which consists in changing its distance from some instantaneous center of rotation and two Euler angles, which set the direction of the line connecting this point with the center, is extremely widely used in many sections of mechanics.

Restrictions

The limited application of the concept of the material point is visible 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 considered a material point molecule. However, this is not always the case: oscillations and rotation of the molecule - an important tank of "internal energy" of the molecule, the "capacity" of which is determined by the dimensions of the molecule, its structure and chemical properties. In a good approximation, as a material point, it is sometimes possible to consider a monoomic molecule (inert gases, pairs of metals, etc.), but even in such molecules at a sufficiently high temperature there is an excitation of electron shells due to collisions of molecules, followed by highlighting.

Notes

Wikimedia Foundation. 2010.

Mechanical movement
Absolutely solid body

Watch what is a "material point" in other dictionaries:

MATERIAL POINT - point having a lot. In the mechanics, the material point is used in cases where the sizes and shape of the body do not play roles when studying its movement, but only mass is important. Almost any body can be viewed as a material point if ... ... Big Encyclopedic Dictionary

MATERIAL POINT - The concept administered in the mechanics to designate the object, is considered as a point having a mass. The position of M. t. In Pré is defined as the position of the geom. Points that significantly simplifies the solution of the problems of mechanics. Practically body can be considered ... ... Physical encyclopedia

material point - point that has a mass. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee of Scientific Technical Terminology. 1984] Themes Theoretical Mechanics En Particle De Materialle Punkt FR Point Matériel ... Technical translator directory

MATERIAL POINT Modern encyclopedia

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

Material point - The material point, the concept administered in the mechanics to designate the body, the dimensions and the form of which can be neglected. The position of the material point in space is defined as the position of the geometric point. The body can be considered material ... ... Illustrated Encyclopedic Dictionary

material point - The concept administered in the mechanics for the object of infinitely small sizes having a mass. The position of the material point in space is defined as the position of the geometric point, which simplifies the solution of the mechanics problems. Virtually any body can ... ... encyclopedic Dictionary

Material point - geometric point with mass; Material dot abstract image of a material body, having a mass and having no size ... The start of modern natural science

material point - Materialusis Taškas Statusas T Sritis Fizika Atitikmenys: Angl. Mass Point; Material Point Vok. Massenpunkt, m; Materieller Punkt, M Rus. material point, f; Point mass, F pranc. Point Masse, m; Point Matériel, M ... Fizikos Terminų žodynas

material point - point having a lot of ... Polytechnic Terminology Dictionary

Books

Set of tables. Physics. Grade 9 (20 tables) ,. An academic album of 20 sheets. Material point. Coordinates of a moving body. Acceleration. Newton's laws. The law of global gravity. Straight and curvilinear movement. Body Movement

Under the material point, a macroscopic body is meant, the properties of which (mass, rotation, shape, etc.) can be neglected if there is a need to describe its movement. What is the material point, you will learn from this article.

If we talk about whether this body is considered as such a point, then everything is determined by non-sizes of the body, but from the conditions set in the task. As an example, the radius of our planet is an order of magnitude less than the distance between the Sun and the Earth, and the orbital movement can be described just in the form of a material point, which has similar land mass and is located in its center. However, if we consider the daily movement of the planet around your own axis, then it is meaningless to replace it on the material point. The model of the point of the considered type to a particular body is not determined by the sizes of the body itself, but more the conditions for its movement. As an example, according to the theorem on the movement of the center of the system of the system when moving the progressive type, each solid body can be considered as a material point, the position of which is similar to the center of mass body.

Such physical properties of the point as a mass, speed, position and other determine its behavior at each time.

The position in the space of the considered point is determined as a position of the geometric point. In the mechanics, the material point has a lot of time constant and independent of any factors of its movement and interaction with other bodies. If you use an approach to the construction of mechanics based on axioms, then the following is taken for one of them:

Axiom

The material point is called the body - the geometric point, which corresponds to the scalar, called the mass: (R and M), where R is a vector in the Euclidean space, which refers to a particular Cartesian coordinate system. Mass are constant and independent of the position of the point in time and space.

The material point spares mechanical energy exclusively as the kinetic energy of its movement in space, or as a potential energy that enters into interaction with the field. This suggests that this point cannot be deformed, rotate around its own axis, and it does not respond to its changes in space. In parallel with this, the material point moves with the change in its distance from the pair of the angles of Euler and any instantaneous center of rotation, the direction of the direction of the direction, and in turn connects this point with the center. This method is very common in mechanics.

The technique in which the laws of movement of real objects are being studied by studying the movement of the ideal model - this 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 is reduced to the study of the movement of the points under consideration.

The term itself is somewhat limited in use. As an example, the rarefied gas at high temperature mode 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 take the molecule for the material point, mostly all is not so. The internal energy of the molecule is determined by oscillations and rotations, and its capacity depends on the size, structure and properties of the particle. In some cases, monohydomic molecules can be considered as examples of a material point, but even at high temperature regime, electronic shells are excited due to collisions of molecules with further highlighting.

First task

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

a) the car entering the garage cannot be considered such an object, since the size difference between the car and the garage is relatively small;
b) Auto on the highway Moscow - Rostov can be viewed as such a point, since the size of the vehicle is about less than the path.

Second task

a) a boy walking home from school (path 1 km);
b) Boy doing exercise?
a) Since the path from school to the house is a kilometer, the boy can be considered as a point, since it is very small in its size relative to the distance.
b) When the same child performs the morning exercise, it cannot be taken for the material point.

Material point

Practically under the material point understands the body weight, size and form of which can be neglected when solving this task.

With the straight-line movement of the body, one coordinate axis is enough to determine its position.

Features

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

Corollary

Restrictions

Notes

Wikimedia Foundation. 2010.

Watch what is a "material point" in other dictionaries:

Massive point. In the mechanics, the material point is used in cases where the sizes and shape of the body do not play roles when studying its movement, but only mass is important. Almost any body can be viewed as a material point if ... ... Big Encyclopedic Dictionary

The concept administered in the mechanics to designate the object is considered as a point having a mass. The position of M. t. In Pré is defined as the position of the geom. Points that significantly simplifies the solution of the problems of mechanics. Practically body can be considered ... ... Physical encyclopedia

Modern encyclopedia

In the mechanics: an infinitely small body. A dictionary of foreign words included in the Russian language. Chudinov A.N., 1910 ... Dictionary of foreign words of the Russian language

The concept administered in the mechanics for the object of infinitely small sizes having a mass. The position of the material point in space is defined as the position of the geometric point, which simplifies the solution of the mechanics problems. Virtually any body can ... ... encyclopedic Dictionary

Material point - geometric point with mass; Material dot abstract image of a material body, having a mass and having no size ... The start of modern natural science

material point - point having a lot of ... Polytechnic Terminology Dictionary

Books

Set of tables. Physics. Grade 9 (20 tables) ,. An academic album of 20 sheets. Material point. Coordinates of a moving body. Acceleration. Newton's laws. The law of global gravity. Straight and curvilinear movement. Body Movement

To describe the movement of the body, you need to know how different points are moving. However, in the case of a progressive movement, all the points of the body move equally. Therefore, to describe the progressive movement of the body, it is enough to describe the movement of one point of its point.

Also in many tasks of mechanics there is no need to indicate the positions of individual parts of the body. If the size of the body is small compared to distances to other bodies, then this body can be described as a point.

Definition

Material point It is called the body, the sizes of which in these conditions can be neglected.

The word "material" emphasizes here the difference between this point from the geometric. The geometric point does not have any physical properties. The material point may have a mass, electric charge and other physical characteristics.

The same body in some conditions can be considered a material point, and in others - no. So, for example, considering the movement of the ship from one seaport to another, the ship can be considered a material point. However, when studying the movement of the ball, which rolls along the deck of the ship, the ship is impossible to count the material point. The movement of a hare running through the woods from the wolf can be described by taking a hare for a material point. But it is impossible to consider the hare material point, describing his attempts to hide in Nora. When studying the movement of the planets around the Sun, they can be described by material points, and during the daily rotation there are such a model in its axis.

It is important to understand that in the nature of material points does not exist. The material point is an abstraction, a model for the description of the movement.

Examples of solving problems on the topic "Material Point"

Example 1.

Example 2.

The task

In which of the cases below, the body studied can be taken for the material point: a) calculate the pressure of the tractor to the ground; b) calculate the height on which the rocket rose; c) calculate the work when picked up in the horizontal position of the slab of the known mass on a given height; d) determine the volume of the steel ball using the measuring cylinder (minsurics).

Answer

a) when calculating the pressure of the tractor to the soil, the tractor cannot be taken for the material point, since in this case it is important to know the surface area of \u200b\u200bthe caterpillars;

b) When calculating the height of the rocket lift, the rocket can be considered a material point, since the rocket moves along and the distance traveled by the rocket. Much more of its size;

b) In this case, the slab overlap can be considered a material point. Since it makes a translational movement and to solve the problem, it is enough to know the movement of its mass center;

d) when determining the volume of the ball. The ball is considered a material point, because the size of the ball is essential in this problem.

Example 3.

The task	Is it possible to take the land for the material point when calculating: a) the distance from the ground to the Sun; b) the path covered by the Earth in orbit around the Sun; c) the length of the Earth's equator; d) speed of movement of the point of the equator at the daily rotation of the Earth around the axis; e) the speed of the earth in orbit around the sun?
Answer	a) In these conditions, the land can be taken for the material point, since its size is much less than the distance from it to the Sun; e) In this case, the land can be taken for the material point, since the size of the orbits is much superior to the size of the Earth.

Introduction

Didactic material is designed to students of all specialties of the correspondence of the Faculty of Hutsmiz, studying the course of mechanics on the program for engineering and technical specialties.

The didactic material contains a summary of the theory on the topic under study, adapted to the level of training of vocational students, examples of solving typical tasks, questions and tasks similar to the proposed students on exams, reference material.

The purpose of such a material is to help the audiola student independently to assimilate the kinematic description of the progressive and rotational movements using the method of an analogy; Learn to solve numerical and qualitative tasks, deal with issues related to the dimension of physical quantities.

Special attention is paid to solving qualitative tasks, as one of the techniques of a deeper and conscious assimilation of the foundations of the physics necessary in the study of special disciplines. They help to understand the meaning of the occurrences of nature phenomena, to understand the essence of physical laws and clarify the scope of their application.

Didactic material may be useful to students of the day form of training.

KINEMATICS

Part of the physics studying the mechanical movement is called mechanics . Under mechanical movement understand the change over the time of the mutual arrangement of bodies or their parts.

Kinematics - The first section of mechanics, she studies the laws of the movement of bodies, not interested in the reasons that cause this movement.

1. The material point. Reference system. Trajectory.

Way. Vector of movement

The simplest kinematics model - material point . This is the body, the sizes of which can be neglected in this task. Any body can be represented as a totality of material points.

To mathematically describe the movement of the body, you need to decide on the reference system. Reference system (CO) consists of body countdown and associated with it coordinate systems and watch. If there are no special indications in the condition of the problem, it is believed that the coordinate system is associated with the surface of the Earth. As a coordinate system most often used descartovsystem.

Let it be necessary to describe the motion of the material point in the Cartesian coordinate system Hu.Z. (Fig.1). At some point in time t. 1 point is in position BUT. The position of the point in space can be characterized by a radius - vector r. 1, conducted from the beginning of the coordinates BUT, and coordinates x. 1 , y. 1 , z. one . Here and then, vector quantities are denoted by bold italics. By the time t. 2 = t. 1 + Δ. t. The material point will move to the position INwith a radius vector r. 2 and coordinates x. 2 , y. 2 , z. 2 .

Movement trajectory the curve in the space is called the body. According to the type of trajectory distinguish between straight, curvilinear movement and movement around the circle.

Path Length (or way ) - Length of the site AU, measured by the trajectory of motion, is denoted by ΔS (or s). The path in the international system of units (s) is measured in meters (m).

Vector of movement material point Δ r. represents the difference of vectors r. 2 and r. 1, i.e.

Δ r. = r. 2 - r. 1.

The module of this vector, called moving, is the shortest distance between the positions. BUT and IN (initial and finite) moving point. Obviously, ΔS ≥ Δ r.Moreover, equality is performed with a straightforward movement.

When the material point is moved, the value of the path passed, the radius-vector and its coordinate changes with time. Kinematic motion equations (further motion equations) Call them depending on time, i.e. View equations

s.\u003d S ( t.), r \u003d R. (t.), x.=h.(t.), y.=w.(t.), z.=z (T.).

If such an equation is known for a moving body, then at any time you can find the speed of its movement, acceleration, etc., whereby make sure.

Any body movement can be represented as a set progressand rotational movements.

2. Kinematics of translational movement

Additional call such a movement in which any straight, rigidly associated with a moving body remains parallel to itself .

Speed characterizes the speed of movement and direction of movement.

Average speed Movement in the time interval Δ t. called the magnitude

(1)

where - s Cut the path passed by the body during the time  t..

Instant speed movement (the speed at the moment time) is called the value, the module of which is determined by the first derivative of the time

(2)

Speed \u200b\u200b- vector magnitude. Instant speed vector is always directed by tangent To the trajectory of movement (Fig. 2). Speed \u200b\u200bmeasurement unit - m / s.

The speed value depends on the selection of the reference system. If a person is sitting in the train carriage, he moves along with the train with respect to the Earth-related, but resting relative to the CA associated with the car. If a person walks on a car at a speed , then its speed relative to the "earth"  з depends on the direction of movement. Along the train movement  z \u003d  trains + , against   з \u003d  trains - .

Projections of the velocity vector on the axis of the coordinate υ h. , υ u ,υ z. Defined as the first derivatives from the respective coordinates over time (Fig. 2):

If the speed projections on the axis of the coordinates are known, the speed module can be determined by the Pythagorean theorem:

(3)

Uniform call movement with a constant speed (υ \u003d const). If the direction of the speed vector does not change v., the movement will be uniform straightforward.

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

(4)

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

Vector instant acceleration determined as a derivative of speed vector v. By time:

(5)

Since with curved motion, the speed can be changed both in magnitude and in the direction, it is customary to decompose the vector of acceleration into two mutually perpendicular Compound

but = but τ + but n. (6)

Tangential (or tangent) acceleration but τ characterizes the speed of change of speed by magnitude, its module

.(7)

Tangential acceleration is aimed at tangential to the trajectory of speed at speeds at an accelerated movement and against speed during slow motion (Fig. 3) ..

Normal (centripetal) acceleration but n characterizes the change in speed in the direction, its module

(8)

where R. - The radius of the curvature of the trajectory.

The normal acceleration vector is directed towards the center of the circle, which can be carried out regarding this path of the trajectory; It is always perpendicular to the tangential acceleration vector (Fig. 3).

The complete acceleration module is determined by the Pythagora theorem

. (9)

Full acceleration vector direction but Determined by the vector sum of vectors of normal and tangential accelerations (Fig.3)

Equipment Call the movement S. constantacceleration . If acceleration is positively, then it Equal asked movement if it is negative - equalized .

With straight movement but ם \u003d 0 and but = but τ. If a but ם \u003d 0 and but τ \u003d 0, the body moves straight and evenly; for but ם \u003d 0 and but τ \u003d const motion straight equestrone.

For uniform motion The traveled path is calculated by the formula:

d. s. \u003d d. t.→ s. \u003d ∫d. t. \u003d ∫d. t.=  t.+ s. 0 , (10)

where s. 0 - the initial path for t. = 0. The last formula needs to be remembered.

Graphic dependencies υ (t.) I. s.(t.) shown in Fig.4.

For equipment movement  = ∫ but D. t. = but∫ D. t.From here

= butt. +  0, (11)

where  0 - the initial speed when t.=0.

Distance traveled s.\u003d ∫d. t. = ∫(butt. +  0) D t.. Solving this integral, we get

s. = butt. 2/2 +  0 t. + s. 0 , (12)

where s. 0 - initial way (for t. \u003d 0). Formulas (11), (12) Recommend to remember.

Graphic dependencies but(t.), υ (t.) I. s.(t.) are shown in Fig. 5.

To equally-ground movement with speeding acceleration g. \u003d 9.81 m / s 2 refers free traffic bodies in the vertical plane: down body fall with g.\u003e 0, when moving up acceleration g.\u003c0. The speed of movement and the passage passed is changed according to (11):

 =  0 + g.t.; (13)

h. = g.t. 2/2 +  0 t. + H. 0 . (14)

Consider the movement of the body thrown at an angle to the horizon (ball, stone, cannon shell, ...). This complex movement consists of two simple: horizontally along the axis OH and vertically along the axis OU (Fig. 6). Along the horizontal axis in the absence of resistance of the medium, the movement is uniform; The vertical axis is equal: equible to the maximum lifting point and equivalent after it. The trajectory of movement has a parabola. Let  0 be the initial body velocity thrown at an angle α to the horizon from the point BUT (origin). Its components on the selected axes:

 0x \u003d  x \u003d  0 cos α \u003d const.; (15)

 0U \u003d  0 sinα. (sixteen)

According to the formula (13) we have for our example anywhere in the trajectory to the point FROM

 y \u003d  0u - g. t. \u003d  0 sinα. - G. t. ;

 x \u003d  0x \u003d  0 cos α \u003d const.

In the highest point of the trajectory, point FROM, the vertical component of the speed  y \u003d 0. From here you can find the time of movement to the point C:

 y \u003d  0u - g. t. \u003d  0 sinα. - G. t. = 0 → t. \u003d  0 sinα / G.. (17)

Knowing this time, you can determine the maximum height of the lifting of the body of software (14):

h. Max \u003d  0u t.- g.t. 2/2 \u003d  0 sinα  0 sinα / g.– g.( 0 sinα /g.) 2/2 \u003d ( 0 sinα) 2 / (2 g.) (18)

Since the trajectory of motion is symmetrical, then the full time of movement to the end point IN equally

t. 1 =2 t. \u003d 2 0 SINα / g.. (19)

Range of flight AU Given (15) and (19) will be determined:

AU\u003d  x. t. 1 \u003d  0 cosα 2 0 SINα / G. \u003d 2 0 2 cosα sinα / G.. (20)

Complete acceleration of the moving body at any point of the trajectory is equal to the acceleration of free fall G.; It can be decomposed on the normal and tangential, as shown in Fig.3.

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