EntranceMaterial point. Reference system
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.
At rectilinear motion body, one coordinate axis is enough 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 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 good approximation as a material point, one can sometimes consider a monatomic molecule (inert gases, metal vapors, etc.), 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] Themes 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 abstract image a material body that has mass and no 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. Law gravity. Rectilinear and curvilinear motion. Body movement along...
What is mechanical movement?
Mechanical motion is a change in the relative position of bodies or their parts in space over time.
What is a reference system?
Reference system - a set of coordinate systems and clocks associated with the reference body.
What is a trajectory? Path?
The line that describes a material point is called a trajectory during its movement. The path is the length of the path.
What is a radius vector?
The radius vector is the vector connecting the origin O with the point M.
What is called the speed of movement of a material point? How is the velocity vector directed?
Speed is a vector quantity that determines both the speed of movement and its direction in this moment time. The vector is directed tangentially at the given point of the trajectory.
What is the acceleration of a material point? What is the direction of the acceleration vector?
Acceleration is a vector quantity that characterizes the rate of change of speed in absolute value and direction. Directed along the direction of velocity or perpendicular.
What is called angular velocity? What is the direction of the angular velocity vector?
Angular velocity directed along the axis of rotation, i.e. according to the rule of the right screw
What is called angular acceleration? What is the direction of the angular acceleration vector?
The vector is directed along the axis of rotation in the same direction as during accelerated rotation and in opposite side when decelerating
What characterizes normal acceleration?
Normal acceleration- characterizes the rate of change of speed in the direction, directed along the normal to the trajectory.
What characterizes tangential acceleration?
Tangential acceleration characterizes the rate of change of speed modulo, directed tangentially to the trajectory
What is called the force of gravity and body weight? What is the difference between gravity and body weight?
Gravity is the force with which the earth pulls objects towards itself. F=mg. Body weight - the force with which the body presses on the support or stretches the suspension as a result of gravity. P=mg . The force of gravity always acts, and the weight of the body manifests itself only when other forces act on the body besides the force of gravity.
What is Young's modulus?
Young's modulus - numerically equal to the stress at a relative elongation equal to 1. Depends on the material of the body.
What are inertial forces?
Forces of inertia - forces due to the accelerated motion of a non-inertial frame of reference (NFR) relative to an inertial frame of reference (ISR).
What is the moment of force about fixed point? What is the direction of the momentum vector?
The moment of force relative to a point is called a vector quantity equal to: M=.
What is the shoulder of strength?
The arm of the force is the shortest distance between the force and point O.
What is the moment of force about a fixed axis?
The moment of force about the axis is a scalar value equal to the product of the modulus of force F and the distance d from the straight line on which the vector F lies to the axis of rotation.
What is a pair of forces? What is the moment pair of forces?
A pair of forces is a lever. The sum of the moments of force is zero
What is the moment of inertia of a body? What does it depend on?
The moment of inertia of a body is a measure of the inertia of a body in rotational motion, it depends on the mass of the body, its distribution in the volume of the body and the choice of the axis of rotation.
What is the work done in rotation?
Angle of rotation
What is mechanical work?
What is called mechanical energy?
Energy is a universal measure of all forms of matter motion and interaction
What is equal to kinetic energy body?
What is the angular momentum of a particle with respect to a fixed point? What is the direction of the angular momentum vector?
The angular momentum of a material point relative to a fixed point O is called physical quantity, determined by the vector product: L==. Directed along the axis to the side determined by the rule of the right screw
What is called pressure?
Pressure is a scalar quantity equal to the force acting per unit area and directed perpendicularly. P=F/S
What is called resonance?
The phenomenon of a sharp increase in the amplitude of forced oscillations is called when the frequency of the driving force approaches a frequency equal to or close to the natural frequency of the oscillatory system.
What is called sublimation?
The process of molecules leaving the surface of a solid is called sublimation.
What is called potential?
Potential is a quantity equal to the potential energy of a single positive charge. Φ=W/q 0 .
What is called current strength?
The current strength is the charge passing through a unit cross-sectional area per unit of time.
What is called tension?
Voltage is a potential difference. U=φ 1 -φ 2 , U=A/q
What is inductance?
Current inductance is the proportionality factor between the magnetic flux and the amount of current that creates this magnetic flux. Ф=LI
What is called resonance?
Resonance is the phenomenon of a sharp increase in the amplitude of forced oscillations when the frequency of the driving force approaches a frequency equal to or close to the natural frequency of the oscillatory system.
heat engine efficiency
Short circuit
Occurs with a sharp increase in current and a decrease in resistance.
Force.
Force is a vector quantity, a measure of the action on a given body from other bodies or fields that appear during acceleration and deformation
Friction force.
The friction force is the force that occurs when moving or trying to cause the movement of one body on the surface of another and is directed along the contact of the surface against the movement A standing wave in some region of space is described by the equation 
. Write down the condition for the points of the medium at which the oscillation amplitude is minimal Average kinetic energy of ideal gas molecules.
Third party forces
Extraneous forces are forces of non-electric origin that can act on an electric charge.
The law of universal gravitation.
Hooke's law.
Law of Archimedes.
Archimedes' law: a body immersed in a liquid or gas is subjected to a buoyant force equal to the weight of the liquid or gas of the displaced body. F a \u003d F strand V t g
Avogadro's law.
Avogadro's law: for the same p and T, 1 mole of any gas occupies the same volume
Dalton's Law.
Dalton's Law: The pressure of a mixture of gases is equal to the sum of the partial pressures produced by each gas separately.
Coulomb's law.
The interaction force F between two stationary charges in a vacuum is proportional to the charges and inversely proportional to the square of the distance between them
Wiedemann-Franz law
λ/γ=3(k/e) 2 , where λ is thermal conductivity, γ is specific conductivity
Ohm's law for current in gases
The principle of superposition of fields.
Lenz's rule.
The inductive current is always directed in such a way as to obstruct the cause that causes it to appear.
Newton's second law.
The force acting on the body is equal to the product of the mass m of the body and the acceleration imparted by this force: F=ma
wave equation.
Second law of thermodynamics
The process of spontaneous transfer of heat from a cold body to a hot one is impossible Electrical displacement vector.
When moving from one environment to another tension electric field changes abruptly, to characterize the continuous electrostatic field, the electric displacement vector (D) is introduced
Steiner's theorem.
Bernoulli equation.
Weight.
Mass is a measure of the inertia of a body, as well as the source and object of gravity
Ideal gas model.
Molecules - material points, do not interact with each other, the collision is elastic
Basic provisions of the ICB
All bodies are made up of atoms and molecules; molecules are constantly moving and interacting with each other
The basic equation of the MKT
P=1/3nm 0 V kv 2 =2/3nE k
EMF - the work of external forces to move a single positive charge along the electric circuit ε=C st /q
Maxwell distribution.
Maxwell's law on the distribution of molecules of an ideal gas by velocities: in a gas that is in equilibrium at a given temperature, a certain stationary distribution of molecules by velocities is established that does not change with time.
hydrostatic pressure.
The hydrostatic pressure is:
barometric formula
Hall phenomenon.
The Hall phenomenon is the occurrence of an electric field in a conductor or semiconductor with current when it moves in a magnetic field
Carnot cycle and its efficiency.
The Carnot cycle consists of two isotherms and two adiabats.
Tension vector circulation electrostatic field.
The circulation of the electrostatic field strength vector is numerically equal to the work that the electrostatic forces do when moving a single positive electric charge along a closed path. 
What is a material point?
A material point is a body whose dimensions can be neglected compared to the distance to another body, considered in this problem.
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.
INstatic 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
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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.
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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 instantaneous speed point remains unchanged during the movement, then such a 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:
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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 
A-priory:
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:
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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
Hence, 
This is a case of uniform rectilinear motion.
But 
2)
Hence 
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.
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 of the point in space where our material point is located.
On the other hand, a material point is the main reference 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 as 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 the problem of body motion, depends on the questions that the solution of the formulated problem must answer.
Possible different approaches to the question of using the material point model.
One of them is empirical. It is believed that the material point model is applicable when the dimensions of the moving bodies are negligible compared to the magnitude of the relative displacements of these bodies. As an illustration, one can solar system. If we assume that the Sun is a fixed material point and consider that it acts on another material point-planet according to the law of universal gravitation, then the problem of the motion of a point-planet has known solution. Among the possible trajectories of the movement of the point, there are those on which the laws of Kepler, empirically established for the planets of the solar system, are fulfilled.
Thus, when describing the orbital motions of the planets, the material point model is quite satisfactory. (However, the construction of a mathematical model of such phenomena as solar and lunar eclipses requires taking into account the real 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 body dimensions in other problems and, consequently, for the acceptability of the material point model? Practice shows that it is not.
For example, a small bullet l= 1 ÷ 2 cm flying distance L= 1 ÷ 2 km, i.e. ratio, however, the flight path (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 a projectile is often considered a material point, then this is accompanied by reservations of the series additional conditions, as a rule, empirically taking into account the real characteristics of the body.
If we turn to astronautics, then when a 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 SC have any noticeable effect on the trajectory. Only sometimes, when correcting the trajectory, it becomes necessary to ensure the exact 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 it depends on which "side" it enters the dense layers of the atmosphere whether it will deliver to desired point Cosmonauts' lands and returned materials.
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 issue of using the material point model is rational. According to the law of change in the momentum of the system, applied to a separate body, the center of mass C of the body has the same acceleration as some (let's call it equivalent) material point, which is affected by the same forces as the body, i.e.
Generally speaking, the resulting force can be represented as a sum , where depends only on and (the radius vector and the velocity of the point C), and - and on the angular velocity of the 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, the motion of the center of mass of the body is said to be independent of the rotational motion of the body. Thus, the possibility of using the material point model receives a mathematical rigorous (and not just empirical) justification.
Naturally, in practice the condition F 2 = 0 rarely and usually F 2 No. 0, but it may turn out that F 2 is somewhat 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 the replacement of the body with an equivalent material point is acceptable, otherwise such a replacement will lead to significant errors.
This can also take place when the body moves forward and from the point of view of kinematics it can be "replaced" by some equivalent point.
Naturally, the material point model is not suitable for answering questions such as “why does the Moon face the Earth with only one of its sides?” Similar phenomena are associated with the rotational movement of the body.
Vitaly Samsonov
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 solving typical problems, questions and tasks similar to those offered to students in exams, and 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 the occurring natural phenomena, to understand the essence physical laws and clarify their scope.
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 A. The position of a point in space can be characterized by a radius - a 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 denoted in bold italics. By the time t 2 = t 1 + ∆ t the material point will move to the position IN 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 A And IN(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 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 uniform motion with acceleration free fall 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 A(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 WITH
y = 0y - g t= 0 sinα. - g t ;
x = 0x = 0 cos α = const.
At the highest point of the trajectory, the point WITH, 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 IN 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.