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Dimensionless material point and different reference systems. What is a material point? How is a material point designated?

Dimensionless material point and different reference systems. What is a material point? How is a material point designated?

Material point

Material point(particle) - the simplest physical model in mechanics - an ideal body whose dimensions are equal to zero, one can also consider the dimensions of the body to be infinitely small compared to other dimensions 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.

With a rectilinear motion of a body, one coordinate axis is sufficient to determine its position.

Peculiarities

The mass, position and speed of a material point at any particular moment of 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 body motion described by a material point, which consists in changing its distance from some instantaneous center of rotation and two Euler angles that 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 a material point is evident 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. In 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 molecular collisions, followed by emission.

Notes

Wikimedia Foundation. 2010 .

mechanical movement
Absolutely rigid body

See what "Material point" is in other dictionaries:

MATERIAL POINT is a point with mass. In mechanics, the concept of a material point is used in cases where the dimensions and shape of a body do not play a role in studying its motion, but only the mass is important. Almost any body can be considered as a material point, if ... ... Big Encyclopedic Dictionary

MATERIAL POINT- a concept introduced in mechanics to designate an object, which is considered as a point having a mass. The position of M. t. in the right is defined as the position of the geom. points, which greatly simplifies the solution of problems in mechanics. In practice, the body can be considered ... ... Physical Encyclopedia

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

MATERIAL POINT Modern Encyclopedia

MATERIAL POINT- In mechanics: an infinitely small body. Dictionary of 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, the size and shape of which 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

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

Material point- geometric point with mass; material point is an abstract image of a material body that has mass and does not have dimensions ... Beginnings 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, fpranc. point mass, m; point matériel, m … Fizikos terminų žodynas

material point- A point with a mass ... Polytechnic terminological explanatory dictionary

Books

A set of tables. Physics. Grade 9 (20 tables), . Educational album of 20 sheets. Material point. moving body coordinates. Acceleration. Newton's laws. The law of universal gravitation. Rectilinear and curvilinear motion. Body movement along...

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.

With a rectilinear motion of a body, one coordinate axis is sufficient to determine its position.

Peculiarities

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

Consequences

Restrictions

The limitations of the application of the concept of a material point can be seen 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. In 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 molecular collisions, followed by emission.

Notes

Wikimedia Foundation. 2010 .

mechanical movement
Absolutely rigid body

See what "Material point" is in other dictionaries:

MATERIAL POINT Modern Encyclopedia

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

Material point- geometric point with mass; material point is an abstract image of a material body that has mass and does not have dimensions ... Beginnings of modern natural science

material point- A point with a mass ... Polytechnic terminological explanatory dictionary

Books

A set of tables. Physics. Grade 9 (20 tables), . Educational album of 20 sheets. Material point. moving body coordinates. Acceleration. Newton's laws. The law of universal gravitation. Rectilinear and curvilinear motion. Body movement along...

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.

With a rectilinear motion of a body, one coordinate axis is sufficient to determine its position.

Peculiarities

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

Consequences

Restrictions

Notes

Wikimedia Foundation. 2010 .

See what "Material point" is in other dictionaries:

A point that has mass. In mechanics, the concept of a material point is used in cases where the dimensions and shape of a body do not play a role in studying its motion, but only the 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, which is considered as a point having a mass. The position of M. t. in the right is defined as the position of the geom. points, which greatly simplifies the solution of problems in mechanics. In practice, the body can be considered ... ... Physical Encyclopedia

Modern Encyclopedia

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

A concept introduced in mechanics for an object of infinitely small 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 problems in mechanics. Almost any body can ... ... encyclopedic Dictionary

Material point- geometric point with mass; material point is an abstract image of a material body that has mass and does not have dimensions ... Beginnings of modern natural science

material point- A point with a mass ... Polytechnic terminological explanatory dictionary

Books

A set of tables. Physics. Grade 9 (20 tables), . Educational album of 20 sheets. Material point. moving body coordinates. Acceleration. Newton's laws. The law of universal gravitation. Rectilinear and curvilinear motion. Body movement along...

INTRODUCTION

The didactic material is intended for students of all specialties of the correspondence department of the GUTsMiZ, who study the course of mechanics according to 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 education of part-time students, examples of solutions typical tasks, questions and tasks similar to those offered to students in exams, reference material.

The purpose of such material is to help a part-time student independently master the kinematic description of translational and rotational movements in a short time, 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 a deeper and more conscious assimilation of the fundamentals of physics, which is necessary in the study of special disciplines. They help to understand the meaning of occurring natural phenomena, to understand the essence of physical laws and to clarify the scope of their application.

Didactic material may be useful for full-time students.

KINEMATICS

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

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

1. Material point. Reference system. Trajectory.

Path. Displacement vector

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

In order to mathematically describe the motion of a body, it is necessary to determine the frame of reference. Reference system (CO) consists of reference body and related coordinate systems and hours. If there are no special instructions in the condition of the problem, it is considered that the coordinate system is associated with the Earth's surface. The most commonly used coordinate system is Cartesian system.

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

Trajectory of movement A curve in space along which a body moves is called. According to the type of trajectory, rectilinear, curvilinear motion and circular motion are distinguished.

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

Displacement vector material point Δ r is the difference of vectors r 2 and r 1 , i.e.

Δ r = r 2 - r 1.

The modulus of this vector, called displacement, is the shortest distance between positions BUT and AT(initial and final) moving point. Obviously, Δs ≥ Δ r, and the equality holds for rectilinear motion.

When a material point moves, the value of the path traveled, the radius vector and its coordinates change with time. Kinematic equations of motion (further motion equations) 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 it is possible to find the speed of its movement, acceleration, etc., which we will see below.

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

2. Kinematics of translational motion

Translational called such a movement in which any straight line, rigidly connected with a moving body, remains parallel to itself .

Speed characterizes the speed of movement and the direction of movement.

medium speed motion in the time interval Δ t is called the quantity

(1)

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

instantaneous speed movements (speed at a given time) is called the value, the modulus of which 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 measurement is m/s.

The value of speed depends on the choice of reference system. If a person is sitting in a train car, he, along with the train, moves relative to the CO associated with the ground, but is at rest relative to the CO associated with the car. If a person walks along the car at a speed , then his speed relative to CO "ground"  s depends on the direction of movement. Along the movement of the train  z \u003d  trains +  , against   z \u003d  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 velocity projections on the coordinate axes are known, the velocity modulus can be determined using the Pythagorean theorem:

(3)

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

Acceleration - a physical quantity that characterizes the rate of change in velocity in magnitude and direction Average acceleration defined as

(4)

where Δυ is the change in speed over 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 constituents

a = a τ + a n. (6)

tangential (or tangential) acceleration a τ characterizes the speed of change in magnitude, its modulus

.(7)

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

Normal (centripetal) acceleration a n characterizes the change in speed in the direction, its modulus

(8)

where R- radius of curvature of the trajectory.

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

The total acceleration module is determined by the Pythagorean theorem

. (9)

Direction of the full acceleration vector a is determined by the vector sum of the vectors of normal and tangential accelerations (Fig. 3)

equivariable called movement from permanent acceleration . If the acceleration is positive, then it is uniformly accelerated motion if it is negative, equally slow .

In a straight line aם =0 and a = aτ . If a aם =0 and aτ = 0, the body moves straight and even; at aם =0 and aτ = const movement rectilinear equally variable.

At uniform motion the distance traveled is calculated by the formula:

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

where s 0 - initial path for t = 0. The last formula must be remembered.

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

For uniform motion  = ∫ a d t = a∫d t, hence

= at +  0 , (11)

where  0 - initial speed at t=0.

Distance traveled s= ∫d t = ∫(at +  0)d t. Solving this integral, we get

s = at 2/2 +  0 t + s 0 , (12)

where s 0 - initial path (for t= 0). Formulas (11), (12) are recommended to be remembered.

Graphic dependencies a(t), υ (t) and s(t) are shown in Fig.5.

To uniformly variable motion with free fall acceleration g= 9.81 m/s 2 applies free movement bodies in a vertical plane: bodies fall down from g›0, when moving up, the acceleration g‹ 0. The speed of movement and the distance traveled in this case change according to (11):

 =  0 + gt; (13)

h = gt 2/2 +  0 t +h 0 . (14)

Consider the motion 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 vertical along the axis OU(Fig. 6). Along the horizontal axis, in the absence of environmental resistance, the movement is uniform; along the vertical axis - equally variable: uniformly slowed down to the maximum point of ascent 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 BUT(origin). Its components along the selected axes:

 0x =  x =  0 cos α = const; (15)

 0у =  0 sinα. (16)

According to formula (13), for our example, at any point of the trajectory to the point FROM

 y =  0y - g t=  0 sinα. - g t ;

 x =  0x =  0 cos α = const.

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

 y =  0y - g t=  0 sinα. - g t = 0 → t =  0 sinα/ g. (17)

Knowing this time, it is possible to determine the maximum height of the body lifting by (14):

h max =  0у t- gt 2 /2= 0 sinα  0 sinα/ g– g( 0 sinα /g) 2 /2 = ( 0 sinα) 2 /(2 g) (18)

Since the trajectory of movement is symmetrical, the total time of movement to the end point AT equals

t 1 =2 t= 2 0 sinα / g. (19)

Range of flight AB taking into account (15) and (19) is determined as follows:

AB=  x t 1 =  0 cosα 2 0 sinα/ g= 2 0 2 cosα sinα/ g. (20)

The total acceleration of a moving body at any point in the trajectory is equal to the free fall acceleration g; it can be decomposed into normal and tangential, as shown in Fig.3.

The concept of a material point. Trajectory. Path and movement. Reference system. Velocity and acceleration in curvilinear motion. Normal and tangential accelerations. Classification of mechanical movements.

The subject of mechanics . Mechanics is a branch of physics devoted to the study of the laws of the simplest form of motion of matter - mechanical motion.

Mechanics consists of three subsections: kinematics, dynamics and statics.

Kinematics studies the motion of bodies without taking into account the causes that cause it. It operates with such quantities as displacement, distance traveled, time, speed and acceleration.

Dynamics explores the laws and causes that cause the movement of bodies, i.e. studies the motion of material bodies under the action of forces applied to them. To the kinematic quantities are added quantities - force and mass.

ATstatic investigate the equilibrium conditions for a system of bodies.

Mechanical movement body is called the change in its position in space relative to other bodies over time.

Material point - a body, the size and shape of which can be neglected under the given conditions of motion, considering the mass of the body concentrated at a given point. The material point model is the simplest model of body motion in physics. A body can be considered a material point when its dimensions are much smaller than the characteristic distances in the problem.

To describe the mechanical movement, it is necessary to indicate the body relative to which the movement is considered. An arbitrarily chosen motionless body, in relation to which the motion of this body is considered, is called reference body .

Reference system - the reference body together with the coordinate system and clock associated with it.

Consider the motion of a material point M in a rectangular coordinate system, placing the origin at point O.

The position of the point M relative to the reference system can be set not only with the help of three Cartesian coordinates, but also with the help of one vector quantity - the radius vector of the point M drawn to this point from the origin of the coordinate system (Fig. 1.1). If are unit vectors (orts) of the axes of a rectangular Cartesian coordinate system, then

or the time dependence of the radius vector of this point

Three scalar equations (1.2) or one vector equation (1.3) equivalent to them are called kinematic equations of motion of a material point .

trajectory a material point is a line described in space by this point during its movement (the locus of the ends of the radius vector of the particle). Depending on the shape of the trajectory, rectilinear and curvilinear motions of a point are distinguished. If all parts of the trajectory of the point lie in the same plane, then the movement of the point is called flat.

Equations (1.2) and (1.3) define the trajectory of a point in the so-called parametric form. The role of the parameter is played by time t. Solving these equations jointly and excluding the time t from them, we find the trajectory equation.

long way material point is the sum of the lengths of all sections of the trajectory traversed by the point during the considered period of time.

Displacement vector material point is a vector connecting the initial and final position of the material point, i.e. increment of the radius-vector of a point for the considered time interval

With rectilinear motion, the displacement vector coincides with the corresponding section of the trajectory. From the fact that displacement is a vector, the law of independence of motions, confirmed by experience, follows: if a material point participates in several motions, then the resulting displacement of the point is equal to the vector sum of its displacements performed by it for the same time in each of the movements separately

To characterize the movement of a material point, a vector physical quantity is introduced - speed , a quantity that determines both the speed of movement and the direction of movement at a given time.

Let a material point move along a curvilinear trajectory MN so that at time t it is at point M, and at time at point N. The radius vectors of points M and N, respectively, are equal, and the length of the arc MN is (Fig. 1.3 ).

Average speed vector points in the time interval from t before t+Δ t called the ratio of the increment of the radius-vector of a point over this period of time to its value:

The average velocity vector is directed in the same way as the displacement vector i.e. along the chord MN.

Instantaneous speed or speed at a given time . If in expression (1.5) we pass to the limit, tending to zero, then we will obtain an expression for the velocity vector of the m.t. at the time t of its passage through the t.M trajectory.

In the process of decreasing the value, the point N approaches t.M, and the chord MN, turning around t.M, in the limit coincides in direction with the tangent to the trajectory at the point M. Therefore, the vectorand speedvmoving point directed along a tangent trajectory in the direction of motion. The velocity vector v of a material point can be decomposed into three components directed along the axes of a rectangular Cartesian coordinate system.

From a comparison of expressions (1.7) and (1.8), it follows that the projections of the velocity of a material point on the axes of a rectangular Cartesian coordinate system are equal to the first time derivatives of the corresponding coordinates of the point:

A movement in which the direction of the velocity of a material point does not change is called rectilinear. If the numerical value of the instantaneous velocity of a point remains unchanged during the movement, then such movement is called uniform.

If, in arbitrary equal time intervals, a point passes paths of different lengths, then the numerical value of its instantaneous velocity changes over time. Such movement is called uneven.

In this case, a scalar value is often used, called the average ground speed of uneven movement in a given section of the trajectory. It is equal to the numerical value of the speed of such a uniform movement, at which the same time is spent on the passage of the path, as with a given uneven movement:

Because only in the case of rectilinear motion with a constant speed in the direction, then in the general case:

The value of the path traveled by a point can be represented graphically by the area of the figure of a bounded curve v = f (t), direct t = t 1 and t = t 1 and the time axis on the speed graph.

The law of addition of speeds . If a material point simultaneously participates in several movements, then the resulting displacement, in accordance with the law of independence of motion, is equal to the vector (geometric) sum of elementary displacements due to each of these movements separately:

According to definition (1.6):

Thus, the speed of the resulting movement is equal to the geometric sum of the velocities of all movements in which the material point participates (this provision is called the law of addition of velocities).

When a point moves, the instantaneous speed can change both in magnitude and in direction. Acceleration characterizes the rate of change in the module and direction of the velocity vector, i.e. change in the magnitude of the velocity vector per unit of time.

Mean acceleration vector . The ratio of the speed increment to the time interval during which this increment occurred expresses the average acceleration:

The vector of the average acceleration coincides in direction with the vector .

Acceleration, or instantaneous acceleration is equal to the limit of the average acceleration when the time interval tends to zero:

In projections onto the corresponding coordinates of the axis:

In rectilinear motion, the velocity and acceleration vectors coincide with the direction of the trajectory. Consider the motion of a material point along a curvilinear plane trajectory. The velocity vector at any point of the trajectory is directed tangentially to it. Let's assume that in t.M of the trajectory the speed was , and in t.M 1 it became . At the same time, we assume that the time interval during the transition of a point on the way from M to M 1 is so small that the change in acceleration in magnitude and direction can be neglected. In order to find the velocity change vector , it is necessary to determine the vector difference:

To do this, we move it parallel to itself, aligning its beginning with the point M. The difference of two vectors is equal to the vector connecting their ends is equal to the side of the AC MAC, built on the velocity vectors, as on the sides. We decompose the vector into two components AB and AD, and both, respectively, through and . Thus, the velocity change vector is equal to the vector sum of two vectors:

Thus, the acceleration of a material point can be represented as the vector sum of the normal and tangential accelerations of this point

By definition:

where - ground speed along the trajectory, coinciding with the absolute value of the instantaneous speed at a given moment. The vector of tangential acceleration is directed tangentially to the trajectory of the body.

If we use the notation for the unit tangent vector, then we can write the tangential acceleration in vector form:

Normal acceleration characterizes the rate of change of speed in direction. Let's calculate the vector:

To do this, we draw a perpendicular through the points M and M1 to the tangents to the trajectory (Fig. 1.4) We denote the intersection point by O. For a sufficiently small section of the curvilinear trajectory, we can consider it part of a circle of radius R. Triangles MOM1 and MBC are similar, because they are isosceles triangles with the same angles at the vertices. That's why:

But then:

Passing to the limit at and taking into account that at the same time , we find:

Since at angle , the direction of this acceleration coincides with the direction of the normal to the velocity , i.e. the acceleration vector is perpendicular to . Therefore, this acceleration is often called centripetal.

Normal acceleration(centripetal) is directed along the normal to the trajectory to the center of its curvature O and characterizes the rate of change in the direction of the point's velocity vector.

The total acceleration is determined by the vector sum of the tangential normal accelerations (1.15). Since the vectors of these accelerations are mutually perpendicular, the total acceleration module is equal to:

The direction of full acceleration is determined by the angle between the vectors and :

Classification of movements.

For classifications of movements, we use the formula for determining the total acceleration

Let's pretend that

Consequently,
This is a case of uniform rectilinear motion.

But

2)
Consequently

This is a case of uniform motion. In this case

At v 0 = 0 v t= at – speed of uniformly accelerated movement without initial speed.

Curvilinear motion at constant speed.