This is the branch of physics that studies motion based on Newton's laws. Classical mechanics is subdivided into:
The basic concepts of classical mechanics are the concept of force, mass and motion. Mass in classical mechanics is defined as a measure of inertia, or the ability of a body to maintain a state of rest or uniform rectilinear motion in the absence of forces acting on it. On the other hand, the forces acting on the body change the state of its motion, causing acceleration. The interaction of these two effects is the main theme of Newtonian mechanics.
Other important concepts of this section of physics are energy, momentum, angular momentum, which can be transferred between objects in the process of interaction. The energy of a mechanical system consists of its kinetic (energy of motion) and potential (depending on the position of the body relative to other bodies) energies. Fundamental conservation laws apply to these physical quantities.
The foundations of classical mechanics were laid by Galileo, as well as Copernicus and Kepler in the study of the laws of motion of celestial bodies, and for a long time mechanics and physics were considered in the context of astronomical events.
In his works, Copernicus noted that the calculation of the laws of motion of celestial bodies can be greatly simplified if we deviate from the principles laid down by Aristotle and consider the Sun, and not the Earth, as the starting point for such calculations, i.e. make the transition from geocentric to heliocentric systems.
The ideas of the heliocentric system were further formalized by Kepler in his three laws of motion of celestial bodies. In particular, it followed from the second law that all the planets of the solar system move in elliptical orbits, having the Sun as one of their focuses.
The next important contribution to the foundation of classical mechanics was made by Galileo, who, investigating the fundamental laws of the mechanical motion of bodies, in particular under the influence of the forces of gravity, formulated five universal laws of motion.
But still, the laurels of the main founder of classical mechanics belong to Isaac Newton, who in his work “Mathematical Principles of Natural Philosophy” synthesized those concepts in the physics of mechanical motion that were formulated by his predecessors. Newton formulated three fundamental laws of motion, which were named after him, as well as the law of universal gravitation, which drew a line under Galileo's research on the phenomenon of free falling bodies. Thus, a new, to replace the outdated Aristotelian, picture of the world of its basic laws was created.
Classical mechanics gives accurate results for the systems we encounter in everyday life. But they become incorrect for systems whose speed approaches the speed of light, where it is replaced by relativistic mechanics, or for very small systems where the laws of quantum mechanics apply. For systems that combine both of these properties, instead of classical mechanics with both characteristics, quantum field theory. For systems with a very large number of components, or degrees of freedom, classical mechanics can also be adequate, but the methods of statistical mechanics are used.
Classical mechanics is retained because, firstly, it is much easier to apply than other theories, and, secondly, it has great possibilities for approximation and application for a very wide class of physical objects, starting from the usual ones, such as a spinning top or a ball. , many astronomical objects (planets, galaxies) and very microscopic ones).
Although classical mechanics is broadly compatible with other "classical theories" such as classical electrodynamics and thermodynamics, there are some inconsistencies between these theories that were found in the late 19th century. They can be solved by methods of more modern physics. In particular, classical electrodynamics predicts that the speed of light is constant, which is incompatible with classical mechanics and led to the creation of special relativity. The principles of classical mechanics are considered together with the statements of classical thermodynamics, which leads to the Gibbs paradox, according to which it is impossible to accurately determine the amount of entropy and to the ultraviolet catastrophe, in which a black body must radiate an infinite amount of energy. To overcome these inconsistencies, quantum mechanics was created.
Objects that are studied by mechanics are called mechanical systems. The task of mechanics is to study the properties of mechanical systems, in particular their evolution in time.
The basic mathematical apparatus of classical mechanics is differential and integral calculus, developed specifically for this by Newton and Leibniz. In the classical formulation, mechanics is based on Newton's three laws.
The following is an exposition of the basic concepts of classical mechanics. For simplicity, we will consider only the material point of the object, the dimensions of which can be neglected. The movement of a material point is characterized by several parameters: its position, mass, and the forces applied to it.
In reality, the dimensions of every object that classical mechanics deals with are non-zero. Material points, such as an electron, obey the laws of quantum mechanics. Objects of non-zero size can experience more complex motions, since their internal state can change, for example, the ball can also rotate. However, for such bodies, the results are obtained for material points, considering them as aggregates of a large number of interacting material points. Such complex bodies behave like material points if they are small on the scale of the problem under consideration.
Radius vector and its derivatives
The position of the material point object is determined relative to a fixed point in space, which is called the origin. It can be given by the coordinates of this point (for example, in a rectangular coordinate system) or by the radius vector r, drawn from the origin to that point. In reality, a material point can move over time, so the radius vector is generally a function of time. In classical mechanics, in contrast to relativistic, it is believed that the passage of time is the same in all frames of reference.
Trajectory
A trajectory is a set of all positions of a material point moving in the general case, it is a curved line, the form of which depends on the nature of the movement of the point and the selected frame of reference.
moving
Displacement is a vector connecting the initial and final position of a material point.
Speed
Speed, or the ratio of movement to time during which it occurs, is defined as the first derivative of movement to time:
In classical mechanics, speeds can be added and subtracted. For example, if one car is traveling west at a speed of 60 km / h, and catches up with another, which is moving in the same direction at a speed of 50 km / h, then relative to the second car, the first one is moving west at a speed of 60-50 = 10 km / h But in the future, fast cars, moving slowly at a speed of 10 km / h to the east.
To determine the relative speed in any case, the rules of vector algebra for compiling speed vectors are applied.
Acceleration
Acceleration, or rate of change of speed, is the derivative of velocity to time, or the second derivative of displacement to time:
The acceleration vector can change in magnitude as well as in direction. In particular, if the speed decreases, sometimes the acceleration is a deceleration, but in general any change in speed.
Forces. Newton's second law
Newton's second law states that the acceleration of a material point is directly proportional to the force acting on it, and the acceleration vector is directed along the line of action of this force. In other words, this law relates the force that acts on the body with its mass and acceleration. Then Newton's second law looks like this:
Value m v called momentum. Usually, mass m does not change with time, and Newton's law can be written in a simplified form
Where a acceleration as defined above. Body mass m Not always over time. For example, the mass of a rocket decreases as the fuel is used up. Under such circumstances, the latter expression does not apply, and the full form of Newton's second law should be used.
Newton's second law is not enough to describe the motion of a particle. It requires the definition of the force that acts on it. For example, a typical expression for the friction force when a body moves in a gas or liquid is defined as follows:
Where? some constant called the coefficient of friction.
After all the forces are determined, on the basis of Newton's second law, we obtain a differential equation called the equation of motion. In our example with only one force acting on the particle, we get:
After integrating, we get:
Where is the starting speed. This means that the speed of our object decreases exponentially to zero. This expression, in turn, can be integrated again to obtain an expression for the radius vector r of the body as a function of time.
If several forces act on a particle, then they are added according to the rules of vector addition.
Energy
If strength F acts on the particle, which as a result of this moves to? r, then the work done is equal to:
If the mass of the particle has become, then yearning for the work performed by all forces, from Newton's second law
Where T kinetic energy. For a material point is defined as
For complex objects from many particles, the kinetic energy of the body is equal to the sum of the kinetic energies of all particles.
A special class of conservative forces can be expressed by the gradient of a scalar function known as potential energy V:
If all forces acting on a particle are conservative, and V the total potential energy obtained by adding the potential energies of all forces, then
Those. total energy E=T+V is preserved in time. This is a manifestation of one of the fundamental physical laws of conservation. In classical mechanics, it can be useful in practice, because many varieties of forces in nature are conservative.
Newton's laws have several important implications for solids (see angular momentum)
There are also two important alternative formulations of classical mechanics: Lagrange mechanics and Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are sometimes useful for analyzing certain problems. They, like other modern formulations, do not use the concept of force, instead referring to other physical quantities such as energy.
Mechanics is a branch of physics that studies the simplest form of motion of matter - mechanical movement, which consists in changing the position of bodies or their parts over time. The fact that mechanical phenomena occur in space and time is reflected in any law of mechanics that explicitly or implicitly contains space-time relationships - distances and time intervals.
Mechanics sets itself two main tasks:
the study of various movements and the generalization of the results obtained in the form of laws with the help of which the nature of the movement in each specific case can be predicted. The solution of this problem led to the establishment by I. Newton and A. Einstein of the so-called dynamic laws;
search for common properties inherent in any mechanical system in the process of its movement. As a result of solving this problem, the laws of conservation of such fundamental quantities as energy, momentum, and angular momentum were discovered.
Dynamic laws and the laws of conservation of energy, momentum and angular momentum are the basic laws of mechanics and constitute the content of this chapter.
§one. Mechanical movement: basic concepts
Classical mechanics consists of three main sections - statics, kinematics and dynamics. In statics, the laws of the addition of forces and the conditions for the equilibrium of bodies are considered. In kinematics, a mathematical description is given of all kinds of mechanical motion, regardless of the reasons that cause it. In dynamics, the influence of the interaction between bodies on their mechanical motion is studied.
In practice, everything physical problems are solved approximately: real complex movement considered as a set of simple movements, a real object replaced by an idealized model this object, etc. For example, when considering the motion of the Earth around the Sun, one can neglect the size of the Earth. In this case, the description of the movement is greatly simplified - the position of the Earth in space can be determined by one point. Among the models of mechanics, the determining ones are material point and absolutely rigid body.
Material point (or particle) is a body, the shape and dimensions of which can be neglected under the conditions of this problem. Any body can be mentally divided into a very large number of parts, arbitrarily small compared to the size of the whole body. Each of these parts can be considered as a material point, and the body itself - as a system of material points.
If the deformations of the body during its interaction with other bodies are negligible, then it is described by the model absolutely rigid body.
Absolutely rigid body (or rigid body) is a body, the distance between any two points of which does not change in the process of motion. In other words, this is a body, the shape and dimensions of which do not change during its movement. An absolutely rigid body can be considered as a system of material points rigidly interconnected.
The position of a body in space can only be determined in relation to some other bodies. For example, it makes sense to talk about the position of a planet in relation to the Sun, an aircraft or a ship in relation to the Earth, but one cannot indicate their position in space without regard to any particular body. An absolutely rigid body, which serves to determine the position of an object of interest to us, is called a reference body. To describe the movement of an object, a reference body is associated with any coordinate system, for example, a rectangular Cartesian coordinate system. The coordinates of an object allow you to set its position in space. The smallest number of independent coordinates that must be set to fully determine the position of the body in space is called the number of degrees of freedom. For example, a material point freely moving in space has three degrees of freedom: a point can make three independent movements along the axes of a Cartesian rectangular coordinate system. An absolutely rigid body has six degrees of freedom: to determine its position in space, three degrees of freedom are needed to describe translational motion along the coordinate axes and three to describe rotation about the same axes. The coordinate system is equipped with a clock to keep time.
The set of the reference body, the coordinate system associated with it and the set of clocks synchronized with each other form the reference frame.
Mechanics- this is a part of physics that studies the laws of mechanical movement and the reasons that cause or change this movement.
Mechanics, in turn, is divided into kinematics, dynamics and statics.
mechanical movement- this is a change in the relative position of bodies or body parts over time.
Weight is a scalar physical quantity that quantitatively characterizes the inert and gravitational properties of matter.
inertia- this is the desire of the body to maintain a state of rest or uniform rectilinear motion.
inertial mass characterizes the ability of a body to resist a change in its state (rest or motion), for example, in Newton's second law
gravitational mass characterizes the body's ability to create a gravitational field, which is characterized by a vector quantity called tension. The intensity of the gravitational field of a point mass is equal to:
The gravitational mass characterizes the body's ability to interact with the gravitational field:
P equivalence principle gravitational and inertial masses: each mass is both inertial and gravitational at the same time.
The mass of the body depends on the density of the substance ρ and the size of the body (body volume V):
The concept of mass is not identical to the concepts of weight and gravity. It does not depend on the fields of gravity and accelerations.
Moment of inertia is a tensor physical quantity that quantitatively characterizes the inertia of a solid body, which manifests itself in rotational motion.
When describing the rotational motion, it is not enough to specify the mass. The inertia of a body in rotational motion depends not only on the mass, but also on its distribution relative to the axis of rotation.
1. Moment of inertia of a material point
where m is the mass of a material point; r is the distance from the point to the axis of rotation.
2. Moment of inertia of the system of material points
3. Moment of inertia of a perfectly rigid body
Strength- this is a vector physical quantity, which is a measure of the mechanical impact on the body from other bodies or fields, as a result of which the body acquires acceleration or deforms (changes its shape or size).
Mechanics uses various models to describe mechanical motion.
Material point(m.t.) is a body with a mass, the dimensions of which can be neglected in this problem.
Absolutely rigid body(a.t.t.) is a body that does not deform in the process of movement, that is, the distance between any two points in the process of movement remains unchanged.
§ 2. Laws of motion.
First Law n newton : any material point (body) retains a state of rest or uniform rectilinear motion until the impact from other bodies makes it change this state.
Newton's second law (the main law of the dynamics of translational motion): the rate of change in the momentum of a material point (body) is equal to the sum of the forces acting on it
Newton's third law : any action of material points (bodies) on each other has the character of interaction; the forces with which the material points act on each other are always equal in absolute value, oppositely directed and act along the straight line connecting these points
here is the force acting on the first material point from the second; - the force acting on the second material point from the side of the first. These forces are applied to different material points (bodies), always act in pairs and are forces of the same nature.
,
here is the gravitational constant. .
Conservation laws in classical mechanics.
The laws of conservation are fulfilled in closed systems of interacting bodies.
A system is called closed if no external forces act on the system.
Pulse - vector physical quantity that quantitatively characterizes the stock of translational motion:
Law of conservation of momentum systems of material points(m.t.): in closed systems, m.t. total momentum is conserved
where is the speed of the i-th material point before the interaction; is its speed after interaction.
angular momentum is a physical vector quantity that quantitatively characterizes the reserve of rotational motion.
is the momentum of the material point, is the radius vector of the material point.
Law of conservation of angular momentum
:
in a closed system, the total angular momentum is conserved:
The physical quantity that characterizes the ability of a body or system of bodies to do work is called energy.
Energy is a scalar physical quantity, which is the most general characteristic of the state of the system.
The state of the system is determined by its movement and configuration, i.e., by the mutual arrangement of its parts. The motion of the system is characterized by the kinetic energy K, and the configuration (being in the potential field of forces) is characterized by the potential energy U.
total energy defined as the sum:
E = K + U + E int,
where E ext is the internal energy of the body.
The kinetic and potential energies add up to mechanical energy .
Einstein formula(relationship of energy and mass):
In the reference frame associated with the center of mass of the m.t. system, m \u003d m 0 is the rest mass, and E \u003d E 0 \u003d m 0. c 2 - rest energy.
Internal energy is determined in the frame of reference associated with the body itself, that is, the internal energy is at the same time the rest energy.
Kinetic energy is the energy of the mechanical movement of a body or system of bodies. The relativistic kinetic energy is determined by the formula
At low speeds v
.
Potential energy is a scalar physical quantity that characterizes the interaction of bodies with other bodies or with fields.
Examples:
potential energy of elastic interaction
potential energy of gravitational interaction of point masses
Law of energy conservation : the total energy of a closed system of material points is conserved
In the absence of dissipation (scattering) of energy, both total and mechanical energies are conserved. In dissipative systems, total energy is conserved, while mechanical energy is not conserved.
§ 2. Basic concepts of classical electrodynamics.
The source of the electromagnetic field is an electric charge.
Electric charge is the property of some elementary particles to enter into electromagnetic interaction.
Electric charge properties :
1. The electric charge can be positive and negative (it is generally accepted that the proton is positively charged, and the electron is negatively charged).
2. Electric charge is quantized. A quantum of electric charge is an elementary electric charge (е = 1.610 –19 C). In the free state, all charges are multiples of an integer number of elementary electric charges:
3. The law of conservation of charge: the total electric charge of a closed system is preserved in all processes involving charged particles:
q 1 + q 2 +...+ q N = q 1 * + q 2 * +...+ q N * .
4. relativistic invariance: the value of the total charge of the system does not depend on the motion of charge carriers (the charge of moving and resting particles is the same). In other words, in all ISOs, the charge of any particle or body is the same.
Description of the electromagnetic field.
The charges interact with each other (Fig. 1). The magnitude of the force with which charges of the same sign repel each other, and charges of opposite signs attract each other, is determined using the empirically established Coulomb's law:
Here, is the electric constant.
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|
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Fig.1 |
And what is the mechanism of interaction of charged bodies? One can put forward the following hypothesis: bodies with an electric charge generate an electromagnetic field. In turn, the electromagnetic field acts on other charged bodies that are in this field. A new material object emerged – an electromagnetic field.
Experience shows that in any electromagnetic field, a force acts on a stationary charge, the magnitude of which depends only on the magnitude of the charge (the magnitude of the force is proportional to the magnitude of the charge) and its position in the field. It is possible to assign to each point of the field a certain vector , which is the coefficient of proportionality between the force acting on a fixed charge in the field and the charge . Then the force with which the field acts on a fixed charge can be determined by the formula:
The force acting from the side of the electromagnetic field on a fixed charge is called electric force. The vector quantity characterizing the state of the field that causes the action is called the electric strength of the electromagnetic field.
Further experiments with charges show that the vector does not completely characterize the electromagnetic field. If the charge begins to move, then some additional force appears, the magnitude and direction of which are in no way related to the magnitude and direction of the vector. The additional force that occurs when a charge moves in an electromagnetic field is called magnetic force. Experience shows that the magnetic force depends on the charge and on the magnitude and direction of the velocity vector. If we move a test charge through any fixed point of the field with the same velocity, but in different directions, then the magnetic force will be different each time. However, always. Further analysis of the experimental facts made it possible to establish that for each point of the electromagnetic field there is a single direction MN (Fig. 2), which has the following properties:
Fig.2
If a certain vector is directed along the MN direction, which has the meaning of the coefficient of proportionality between the magnetic force and the product, then setting , and uniquely characterizes the state of the field that causes the appearance of . The vector was called the vector of electromagnetic induction. Since and , then
In an electromagnetic field, an electromagnetic Lorentz force acts on a charge moving at a speed q (Fig. 3):
.
The vectors and , that is, the six numbers , are equal components of a single electromagnetic field (components of the electromagnetic field tensor). In a particular case, it may turn out that all or all ; then the electromagnetic field is reduced to either electric or magnetic fields.
The experiment confirmed the correctness of the constructed two-vector model of the electromagnetic field. In this model, each point of the electromagnetic field is given a pair of vectors and . The model we have constructed is a model of a continuous field, since the functions and describing the field are continuous functions of the coordinates.
The theory of electromagnetic phenomena using the continuous field model is called classical.
In reality, the field, like matter, is discrete. But this begins to affect only at distances comparable to the sizes of elementary particles. The discreteness of the electromagnetic field is taken into account in quantum theory.
The principle of superposition.
Fields are usually depicted using lines of force.
force line is a line, the tangent to which at each point coincides with the field strength vector.
D
For point immobile charges, the pattern of force lines of the electrostatic field is shown in fig. 6.
The intensity vector of the electrostatic field created by a point charge is determined by the formula (Fig. 7 a and b) the magnetic field line is constructed so that at each point of the line of force the vector is directed tangentially to this line. The lines of force of the magnetic field are closed (Fig. 8). This suggests that the magnetic field is a vortex field.
Rice. eight
And if the field creates not one, but several point charges? Do the charges influence each other, or does each of the system's charges contribute to the resulting field independently of the others? Will the electromagnetic field created by the i-th charge in the absence of other charges be the same as the field created by the i-th charge in the presence of other charges?
Superposition principle : the electromagnetic field of an arbitrary system of charges is the result of the addition of fields that would be created by each of the elementary charges of this system in the absence of the others:
and .
Laws of the electromagnetic field
The laws of the electromagnetic field are formulated as a system of Maxwell's equations.
First
It follows from Maxwell's first equation that electrostatic field - potential (converging or diverging) and its source are motionless electric charges.
Second Maxwell's equation for a magnetostatic field:
It follows from Maxwell's second equation that the magnetostatic field is vortex non-potential and has no point sources.
Third Maxwell's equation for an electrostatic field:
It follows from Maxwell's third equation that the electrostatic field is not vortex.
In electrodynamics (for a variable electromagnetic field), Maxwell's third equation is:
i.e. the electric field is not potential (not Coulomb), but vortex and is created by a variable flux of the magnetic field induction vector.
Fourth Maxwell's equation for a magnetostatic field
It follows from the fourth Maxwell equation in magnetostatics that the magnetic field is vortex and is created by direct electric currents or moving charges. The direction of twisting of the magnetic field lines is determined by the right screw rule (Fig. 9).
R
Fig.9
In electrodynamics, Maxwell's fourth equation is:
The first term in this equation is the conduction current I associated with the movement of charges and creating a magnetic field.
The second term in this equation is the "displacement current in vacuum", i.e., the variable flux of the electric field strength vector.
The main provisions and conclusions of Maxwell's theory are as follows.
A change in time of the electric field leads to the appearance of a magnetic field and vice versa. Therefore, there are electromagnetic waves.
The transfer of electromagnetic energy occurs at a finite speed . The speed of transmission of electromagnetic waves is equal to the speed of light. From this followed the fundamental identity of electromagnetic and optical phenomena.
B E D E N I E
Physics is the science of nature that studies the most general properties of the material world, the most general forms of the motion of matter, which underlie all natural phenomena. Physics establishes the laws that govern these phenomena.
Physics also studies the properties and structure of material bodies and indicates the ways of practical application of physical laws in technology.
In accordance with the variety of forms of matter and its movement, physics is divided into a number of sections: mechanics, thermodynamics, electrodynamics, physics of oscillations and waves, optics, physics of the atom, nucleus and elementary particles.
At the intersection of physics and other natural sciences, new sciences arose: astrophysics, biophysics, geophysics, physical chemistry, etc.
Physics is the theoretical basis of technology. The development of physics served as the foundation for the creation of such new branches of technology as space technology, nuclear technology, quantum electronics, etc. In turn, the development of technical sciences contributes to the creation of completely new methods of physical research that determine the progress of physics and related sciences.
PHYSICAL FOUNDATIONS OF CLASSICAL MECHANICS
I. Mechanics. General concepts
Mechanics is a branch of physics that considers the simplest form of motion of matter - mechanical motion.
Mechanical movement is understood as a change in the position of the body under study in space over time relative to a certain goal or system of bodies that are conditionally considered motionless. Such a system of bodies, together with a clock, for which any periodic process can be chosen, is called reference system(S.O.). S.O. often chosen for reasons of convenience.
For a mathematical description of motion with S.O. they associate a coordinate system, often rectangular.
The simplest body in mechanics is a material point. This is a body whose dimensions can be neglected under the conditions of a given task.
Any body whose dimensions cannot be neglected is considered as a system of material points.
Mechanics is subdivided into kinematics, which deals with the geometric description of motion without studying its causes, dynamics, which studies the laws of motion of bodies under the action of forces, and statics, which studies the conditions for the equilibrium of bodies.
2. Point kinematics
Kinematics studies the space-time movement of bodies. It operates with such concepts as displacement, path, time t, speed, acceleration.
The line that a material point describes during its movement is called a trajectory. According to the shape of the trajectory of movement, they are divided into rectilinear and curvilinear. Vector , connecting the initial I and final 2 points is called displacement (Fig. I.I).
Each moment of time t has its own radius vector:
Thus, the movement of a point can be described by a vector function.
which we define vector way to specify the movement, or three scalar functions
x= x(t); y= y(t); z= z(t) , (1.2)
which are called kinematic equations. They determine the task of movement coordinate way.
The movement of the point will also be determined if for each moment of time the position of the point on the trajectory is set, i.e. addiction
It determines the task of movement natural way.
Each of these formulas is law point movement.
3. Speed
If the moment of time t 1 corresponds to the radius vector , and , then for the interval the body will receive displacement . In this case average speed for t they call the value
which, in relation to the trajectory, is a secant passing through points I and 2. speed at time t is called a vector
From this definition it follows that the speed at each point of the trajectory is directed tangentially to it. From (1.5) it follows that the projections and the modulus of the velocity vector are determined by the expressions:
If the law of motion (1.3) is given, then the modulus of the velocity vector is determined as follows:
Thus, knowing the law of motion (I.I), (1.2), (1.3), one can calculate the vector and module of the doctor of speed, and, conversely, knowing the speed from formulas (1.6), (1.7), one can calculate the coordinates and path.
4. Acceleration
With arbitrary movement, the velocity vector changes continuously. The value characterizing the rate of change of the velocity vector is called acceleration.
If in. moment of time t 1 is the speed of the point, and at t 2 - , then the speed increment will be (Fig. 1.2). The average acceleration at the same time
but instant
For the projection and acceleration module we have: , (1.10)
If the natural way of motion is given, then the acceleration can be determined in this way. The speed varies in magnitude and direction, the increment in speed is decomposed into two values; - directed along (increment of speed in magnitude) and - directed perpendicularly (increment. speed in direction), i.e. = + (Fig.I.3). From (1.9) we get:
Tangential (tangential) acceleration characterizes the rate of change in magnitude (1.13)
normal (centripetal acceleration) characterizes the speed of change in direction. To calculate a n consider
OMN and MPQ under the condition of a small movement of the point along the trajectory. From the similarity of these triangles we find PQ:MP=MN:OM:
The total acceleration in this case is determined as follows:
5. Examples
I. Equal-variable rectilinear motion. This is a movement with constant acceleration() . From (1.8) we find
or where v 0 - speed at time t 0 . Assuming t 0 =0, we find , and the distance traveled S from formula (I.7):
where S 0 is a constant determined from the initial conditions.
2. Uniform movement in a circle. In this case, the speed changes only in direction, that is, centripetal acceleration.
I. Basic concepts
The movement of bodies in space is the result of their mechanical interaction with each other, as a result of which there is a change in the movement of bodies or their deformation. As a mara of mechanical interaction in dynamics, a quantity is introduced - force . For a given body, force is an external factor, and the nature of the movement also depends on the property of the body itself - compliance with the external influence exerted on it or the degree of inertia of the body. The measure of the inertia of a body is its mass. t depending on the amount of matter in the body.
Thus, the basic concepts of mechanics are: moving matter, space and time as forms of the existence of moving matter, mass as a measure of inertia of bodies, force as a measure of mechanical interaction between bodies. Relationships between these concepts are determined by laws! movements that were formulated by Newton as a generalization and refinement of experimental facts.
2. Laws of mechanics
1st law. Any body maintains a state of rest or uniform rectilinear motion, while external influences do not change this state. The first law contains the law of inertia, as well as the definition of force as a cause that violates the inertial state of the body. To express it mathematically, Newton introduced the concept of momentum or momentum of a body:
then if
2nd law. The change in momentum is proportional to the applied force and occurs in the direction of this force. Selecting units of measurement m and so that the coefficient of proportionality is equal to unity, we obtain
If while driving m= const , then
In this case, the 2nd law is formulated as follows: the force is equal to the product of the mass of the body and its acceleration. This law is the basic law of dynamics and allows us to find the law of motion of bodies from given forces and initial conditions. 3rd law. The forces with which two bodies act on each other are equal and directed in opposite directions, i.e., (2.4)
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Classical (Newtonian) mechanics studies the motion of material objects at speeds that are much less than the speed of light in vacuum.
The beginning of the formation of classical mechanics is associated with the name of the Italian. scientist Galileo Galilei (1564-1642). For the first time, he moved from a natural-philosophical consideration of natural phenomena to a scientific-theoretical one.
The foundations of classical physics were laid by the works of Galileo, Kepler, Descartes, and the edifice of this science was built by the works of Newton.
Galileo
1. established the fundamental principle of classical mechanics - the principle of inertia
Movement is the proper and basic, natural state of bodies, while friction and the action of other external forces can change and even stop the movement of the body.
2. formulated another fundamental principle of classical mechanics - the principle of relativity - Equality of all IFRs.
According to this principle, inside a uniformly moving system, all mechanical processes occur as if the system were at rest.
3. The principle of relativity of motion sets the rules for the transition from one IFR to another.
These rules are called Galilean transformations and they consist in projecting one IFR onto another.
Galilean transformations impose a certain requirement on the formulation of the laws of mechanical motion: these laws must be formulated in such a way that they remain invariant in any IFR.
Let some body A be assigned to the Cartesian system, the coordinates of which are denoted by x, y, z, and we need to determine the parameters of the body in a parallel coordinate system with strokes (xl, yl, zl). For simplicity, we will determine the parameters of one point of the body, and we will combine the x1 coordinate axis with the x axis. We also assume that the coordinate system with strokes is at rest, and without strokes, it moves uniformly and rectilinearly. Then the rules of Galilean transformations have the form
4. formulation of the law of free fall (the path of a free falling body is proportional to the acceleration equal to 9.81 m/s2.
Developing and deepening the research of Galileo, Newton formulated three laws of mechanics.
1. Every body is in a state of rest or uniform and rectilinear motion. Until the impact from other bodies makes him change this state.
The meaning of the first law is that if external forces do not act on the body, then there is a frame of reference in which it is at rest. But if the body is at rest in one frame, then there are many other frames of reference in which the body moves at a constant speed. These systems are called inertial (ISO).
Any frame of reference moving uniformly and rectilinearly relative to the IFR is also an IFR.
2. The second law considers the results of the action on the body of other bodies. For this, a physical quantity called force is introduced.
Force is a vector quantitative measure of the mechanical action of one body on another.
Mass is a measure of inertia (inertia is the ability of a body to resist a change in its state).
The greater the mass, the less acceleration the body will receive, other things being equal.
There is also a more general formulation of Newton's second law for another physical quantity - the momentum of the body. Momentum is the product of a body's mass and its speed:
In the absence of external forces, the momentum of the body remains unchanged, in other words, it is conserved. This situation is achieved if other bodies do not act on the body, or their action is compensated.
3. The actions of two material bodies on each other are numerically equal in strength and directed in opposite directions.
The forces act independently. The force with which several bodies act on any other body is the vector sum of the forces with which they would act separately.
This statement is superposition principle.
The dynamics of material points is based on Newton's laws, in particular, the law of conservation of momentum of the system.
The sum of the momenta of the particles that form a mechanical system is called the momentum of the system. Internal forces, i.e. the interactions of the bodies of the system with each other do not affect changes in the total momentum of the system. It follows from this law of conservation of momentum: in the absence of external forces, the momentum of the system of material points remains constant.
Another conserved quantity is energy- a general quantitative measure of the movement and interaction of all types of matter. Energy does not arise from nothing and does not disappear, it can only pass from one form to another.
The measure of energy change is work. In classical mechanics, work is defined as a measure of the action of a force, which depends on the magnitude and direction of the force, as well as on the displacement of the point of its application.
Law of conservation of energy: total mechanical energy remains unchanged (or is conserved) if the work of external forces in the system is zero.
In classical mechanics, it is believed that all mechanical processes are subject to the principle of strict determinism (determinism is the doctrine of universal causation and regularity of phenomena), which consists in recognizing the possibility of accurately determining the future state of a mechanical system by its previous state.
Newton introduced two abstract concepts - "absolute space" and "absolute time".
According to Newton, space is an absolute motionless homogeneous isotropic infinite receptacle of all bodies (that is, emptiness). And time is a pure homogeneous uniform and discontinuous duration of processes.
In classical physics, it was believed that the world can be decomposed into many independent elements by experimental methods. This method is, in principle, unlimited, since the whole world is a collection of a huge number of indivisible particles. The basis of the world is atoms, i.e. smallest, indivisible, structureless particles. Atoms move in absolute space and time. Time is considered as an independent substance, the properties of which are determined by itself. Space is also an independent substance.
Recall that a substance is an essence, something underlying. In the history of philosophy, substance has been interpreted in different ways: as a substratum, i.e. the basis of something; something that is capable of independent existence; as the basis and center of change of the subject; as a logical entity. When they say that time is a substance, they mean that it is able to exist independently.
Space in classical physics is absolute, which means that it is independent of matter and time. You can remove all material objects from space, and absolute space remains. The space is homogeneous, i.e. all its points are equivalent. Space is isotropic, i.e. all directions are equivalent. Time is also homogeneous, i.e. all its moments are equivalent.
The space is described by Euclid's geometry, according to which the shortest distance between two points is a straight line.
Space and time are infinite. The understanding of their infinity was borrowed from mathematical analysis.
The infinity of space means that no matter how large a system we take, we can always point to one that is even larger. The infinity of time means that no matter how long a given process lasts, one can always point to one in the world that will last longer.
The rules of Galilean transformations follow from the fragmentation and absoluteness of space and time.
From the isolation of moving bodies from space and time, the rule of addition of velocities in classical mechanics follows: it consists in a simple addition or subtraction of the velocities of two bodies moving relative to each other.
ux \u003d u "x + υ, uy \u003d u" y, uz \u003d u "z.
The laws of classical mechanics made it possible to formulate the first scientific picture of the world - mechanistic.
First of all, classical mechanics developed the scientific concept of the motion of matter. Now motion is interpreted as the eternal and natural state of bodies, as their basic state, which is directly opposite to pre-Galilean mechanics, in which motion was considered as introduced from outside. But at the same time, mechanical motion is absolutized in classical physics.
In fact, classical physics developed a peculiar understanding of matter, reducing it to a real, or weight, mass. At the same time, the mass of bodies remains unchanged under any conditions of motion and at any speed. Later, in mechanics, the rule of replacing bodies with an idealized image of material points was established.
The development of mechanics has led to a change in ideas about the physical properties of objects.
Classical physics considered the properties found during measurement to be inherent in the object and only to it (the principle of absoluteness of properties). Recall that the physical properties of an object are characterized qualitatively and quantitatively. The qualitative characteristic of a property is its essence (for example, speed, mass, energy, etc.). Classical physics proceeded from the fact that the means of cognition do not affect the objects under study. For various types of mechanical problems, the means of cognition is the frame of reference. Without its introduction, it is impossible to correctly formulate or solve a mechanical problem. If the properties of an object, neither qualitatively nor quantitatively, depend on the frame of reference, then they are called absolute. So, no matter what frame of reference we take for solving a specific mechanical problem, each of them will manifest qualitatively and quantitatively the mass of the object, the force acting on the object, acceleration, speed.
If the properties of an object depend on the reference system, then they are considered to be relative. Classical physics knew only one such quantity - the speed of an object according to a quantitative characteristic. This meant that it was meaningless to say that an object is moving at such and such a speed without specifying the frame of reference: in different frames of reference, the quantitative value of the mechanical speed of the object will be different. All other properties of the object were absolute both in terms of qualitative and quantitative characteristics.
Already the theory of relativity revealed the quantitative relativity of such properties as length, lifetime, mass. The quantitative value of these properties depends not only on the object itself, but also on the frame of reference. From this it followed that the quantitative definiteness of the properties of an object should be related not to the object itself, but to the system: object + frame of reference. But the object itself still remained the bearer of the qualitative definiteness of properties.
