The speed of a point moving along a straight line. Instant speed. Finding the coordinate from the known dependence of the speed on time.

The speed of movement - the movement of a point along a straight line or a given curved line, one has to speak both about the length of the path traveled by the point during any period of time, and about its movement during the same period; these values ​​\u200b\u200bmay not be the same if the movement took place in one direction or the other along the path

INSTANT SPEED()

is a vector physical quantity equal to the ratio of the displacement Δ made by the particle in a very small time interval Δt to this time interval.

A very small (or, as they say, physically infinitely small) time interval is understood here as such, during which the movement can be considered uniform and rectilinear with sufficient accuracy.

At each moment of time, the instantaneous velocity is directed tangentially to the trajectory along which the particle moves.

Its SI unit is the meter per second (m/s).

Vector and coordinate ways of moving a point. Speed ​​and acceleration.

The position of a point in space can be specified in two ways:

1) using coordinates,

2) using the radius vector.
In the first case, the position of the point is determined on the axes of the Cartesian coordinate system OX, OY, OZ, associated with the reference body (Fig. 3). To do this, from point A, it is necessary to lower the perpendiculars to the plane YZ (x coordinate), XZ (/y coordinate), XY (z coordinate), respectively. So, the position of the point can be determined by the records A (x, y, z), and for the case shown in Fig. C (x \u003d 6, y \u003d 10, z - 4.5), point A is indicated as follows: A (6, 10, 4.5).
On the contrary, if specific values ​​of the coordinates of a point in a given coordinate system are given, then to image the point, it is necessary to plot the coordinate values ​​on the corresponding axes and build a parallelepiped on three mutually perpendicular segments. Its vertex, opposite the origin O and placed on the diagonal of the parallelepiped, is point A.
If the point moves within the framework of any plane, then it is enough to draw two coordinate axes OX and OY through the reference * selected on the body at the point.

Velocity is a vector quantity equal to the ratio of the movement of the body to the time during which this movement occurred. With uneven movement, the speed of the body changes over time. With such a movement, the speed is determined by the instantaneous speed of the body. Instant speed - speed body at a given point in time or at a given point in the trajectory.

Acceleration. With uneven movement, the speed changes both in absolute value and in direction. Acceleration is the rate of change of speed. It is equal to the ratio of the change in the speed of the body to the time interval during which this movement occurred.

ballistic movement. Uniform motion of a material point along a circle. Curvilinear motion of a point in space.

Uniform circular motion.

The movement of a body along a circle is curvilinear, with it two coordinates and the direction of movement change. The instantaneous velocity of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at that point. Movement along any curvilinear trajectory can be represented as movement along the arcs of some circles. Uniform motion in a circle is motion with acceleration, although the absolute value of the speed does not change. Uniform circular motion is periodic motion.

The curvilinear ballistic motion of the body can be considered as the result of the addition of two rectilinear motions: uniform motion along the axis X and uniform movement along the axis at.

Kinetic energy of a system of material points, its connection with the work of forces. König's theorem.

The change in the kinetic energy of a body (material point) over a certain period of time is equal to the work done in the same time by the force acting on the body.

The kinetic energy of the system is the energy of motion of the center of mass plus the energy of motion relative to the center of mass:

,

where is the total kinetic energy, is the energy of the center of mass movement, is the relative kinetic energy.

In other words, the total kinetic energy of a body or a system of bodies in complex motion is equal to the sum of the system's energy in translational motion and the system's energy in rotational motion relative to the center of mass.

Potential energy in the field of central forces.

Such a force field is called central, in which the potential energy of the particle is a function only of the distance r to a certain point - the center of the field: U=U(r). The force acting on a particle in such a field also depends only on the distance r and is directed at each point in space along a radius drawn to this point from the center of the field.

The concept of moment of forces and moment of impulse, the relationship between them. Law of conservation of angular momentum. Moment of force (synonyms: torque; torque; torque) is a physical quantity that characterizes the rotational action of force on a rigid body.

In physics, a moment of force can be understood as a "rotating force". In the SI system, the units for moment of force are the newton meter, although the centinewton meter (cN m), foot-pound (ft lbf), inch-pound (lbf in), and inch-ounce (ozf in) are also often used to express moment of force. Symbol of moment of force τ (tau). The moment of force is sometimes called the moment of a pair of forces, this concept arose in the works of Archimedes on levers. The rotating counterparts of force, mass, and acceleration are moment of force, moment of inertia, and angular acceleration, respectively. The force applied to the lever, multiplied by the distance to the axis of the lever, is the moment of force. For example, a force of 3 newtons applied to a lever whose axis is 2 meters away is the same as 1 newton applied to a lever whose axis is 6 meters away. More precisely, the moment of force of a particle is defined as the cross product:

where is the force acting on the particle and r is the radius vector of the particle.

The angular momentum (kinetic momentum, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed about the axis of rotation, and how fast the rotation occurs.

It should be noted that rotation here is understood in a broad sense, not only as a regular rotation around an axis. For example, even with a rectilinear motion of a body past an arbitrary imaginary point, it also has an angular momentum. The angular momentum plays the greatest role in describing the actual rotational motion.

The angular momentum of a closed system is conserved.

The angular momentum of a particle with respect to some origin is determined by the vector product of its radius vector and momentum:

where is the radius vector of the particle relative to the selected reference point, is the momentum of the particle.

In the SI system, angular momentum is measured in units of joule-second; J s

From the definition of angular momentum follows its additivity. So, for a system of particles, the following expression is true:

.

Within the framework of the law of conservation of angular momentum, the conservative quantity is the angular momentum of rotation of the mass - it does not change in the absence of an applied moment of force or torque - the projection of the force vector onto the plane of rotation, perpendicular to the radius of rotation, multiplied by the lever (distance to the axis of rotation). The most common example of the law of conservation of angular momentum is a figure skater performing a rotation figure with acceleration. The athlete enters the rotation slowly enough, spreading her arms and legs wide, and then, as she gathers her body mass closer to the axis of rotation, pressing the limbs closer to the body, the rotation speed increases many times due to a decrease in the moment of inertia while maintaining the moment rotation. Here we see clearly that the smaller the moment of inertia, the higher the angular velocity and, as a result, the shorter the period of rotation, which is inversely proportional to it.

Law of conservation of angular momentum: The angular momentum of a system of bodies is conserved if the resulting moment of external forces acting on the system is zero:

.

If the resulting moment of external forces is not equal to zero, but the projection of this moment onto a certain axis is zero, then the projection of the angular momentum of the system onto this axis does not change.

Moment of inertia. Huygens-Steiner theorem. Moment of inertia and kinetic energy of rotation of a rigid body around a fixed axis.

^ Moment of inertia of a point- a value equal to the product of the mass m of a point and the square of its shortest distance r to the axis (center) of rotation: J z = m r 2 , J = m r 2 , kg. m 2.

Steiner's theorem: The moment of inertia of a rigid body about any axis is equal to the sum of the moment of inertia about the axis passing through the center of mass and the product of the mass of this body by the square of the distance between the axes. I=I 0 +md 2. The value of I, equal to the sum of the products of elementary masses by the squares of their distance from some axis, is called the moment of inertia of the body about the given axis. I=m i R i 2 Summation is performed over all elementary masses into which the body can be divided.

Jump to: navigation, search

Kinetic energy of rotational motion- the energy of the body associated with its rotation.

The main kinematic characteristics of the rotational motion of a body are its angular velocity () and angular acceleration. The main dynamic characteristics of rotational motion are the angular momentum about the rotation axis z:

and kinetic energy

where I z is the moment of inertia of the body about the axis of rotation.

A similar example can be found when considering a rotating molecule with principal axes of inertia I 1, I 2 and I 3. The rotational energy of such a molecule is given by the expression

where ω 1, ω 2, and ω 3 are the principal components of the angular velocity.

In the general case, the energy during rotation with angular velocity is found by the formula:

, where is the inertia tensor

Invariance of the laws of dynamics in ISO. The frame of reference moves forward and accelerates. The frame of reference rotates uniformly. (The material point is at rest in NISO, the material point moves in NISO.). Coriolis theorem.

Coriolis force- one of the forces of inertia that exists in a non-inertial frame of reference due to rotation and the laws of inertia, which manifests itself when moving in a direction at an angle to the axis of rotation. It is named after the French scientist Gustave Gaspard Coriolis, who first described it. The Coriolis acceleration was obtained by Coriolis in 1833, Gauss in 1803 and Euler in 1765.

The reason for the appearance of the Coriolis force is in the Coriolis (rotary) acceleration. In inertial reference frames, the law of inertia operates, that is, each body tends to move in a straight line and at a constant speed. If we consider the movement of a body, uniform along a certain rotating radius and directed from the center, it becomes clear that in order for it to be realized, it is required to give the body acceleration, since the farther from the center, the greater the tangential rotation speed should be. This means that from the point of view of the rotating frame of reference, some force will try to move the body from the radius.

In order for the body to move with Coriolis acceleration, it is necessary to apply a force to the body equal to , where is the Coriolis acceleration. Accordingly, the body acts according to Newton's third law with a force of the opposite direction. The force that acts from the side of the body will be called the Coriolis force. The Coriolis force should not be confused with another force of inertia - the centrifugal force, which is directed along the radius of a rotating circle.

If the rotation is clockwise, then the body moving from the center of rotation will tend to leave the radius to the left. If the rotation is counterclockwise, then to the right.

HARMONIC OSCILLATOR

- a system that performs harmonic oscillations

Fluctuations are usually associated with the alternating transformation of energy of one form (kind) into energy of another form (different type). In a mechanical pendulum, energy is converted from kinetic to potential. In electrical LC circuits (that is, inductive-capacitive circuits), energy is converted from the electrical energy of the capacitance (energy electric field capacitor) into the magnetic energy of the inductor (the energy of the magnetic field of the solenoid)

Examples of harmonic oscillators (physical pendulum, mathematical pendulum, torsion pendulum)

physical pendulum- an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

Mathematical pendulum- an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces [

Torsional pendulum(also torsion pendulum, rotary pendulum) - a mechanical system, which is a body suspended in a gravitational field on a thin thread and having only one degree of freedom: rotation around an axis specified by a fixed thread

Areas of use

The capillary effect is used in non-destructive testing (capillary testing or testing by penetrating substances) to detect defects that have access to the surface of the controlled product. Allows you to detect cracks with an opening of 1 micron, which are not visible to the naked eye.

cohesion(from lat. cohaesus - connected, linked), adhesion of molecules (ions) of a physical body under the influence of attractive forces. These are the forces of intermolecular interaction, hydrogen bonding and (or) other chemical bonding. They determine the totality of the physical and physico-chemical properties of a substance: state of aggregation, volatility, solubility, mechanical properties, etc. The intensity of intermolecular and interatomic interactions (and, consequently, cohesive forces) decreases sharply with distance. The strongest cohesion is in solids and liquids, that is, in condensed phases, where the distance between molecules (ions) is small - on the order of several molecular sizes. In gases, the average distances between molecules are large compared to their sizes, and therefore cohesion in them is negligible. The measure of the intensity of intermolecular interaction is the energy density of cohesion. It is equivalent to the work of removing mutually attracted molecules to an infinite distance from each other, which practically corresponds to the evaporation or sublimation of a substance

Adhesion(from lat. adhaesio- sticking) in physics - adhesion of surfaces of dissimilar solid and / or liquid bodies. Adhesion is due to intermolecular interaction (van der Waals, polar, sometimes - the formation of chemical bonds or mutual diffusion) in the surface layer and is characterized by the specific work required to separate the surfaces. In some cases, adhesion may be stronger than cohesion, that is, adhesion within a homogeneous material, in such cases, when a tearing force is applied, a cohesive gap occurs, that is, a gap in the volume of the less strong of the contacting materials.

Concept of liquid (gas) flow and continuity equation. Derivation of the Bernoulli equation.

In hydraulics, a flow is considered such a mass movement when this mass is limited:

1) hard surfaces;

2) surfaces that separate different liquids;

3) free surfaces.

Depending on what kind of surfaces or their combinations a moving fluid is limited to, the following types of flows are distinguished:

1) non-pressure, when the flow is limited by a combination of solid and free surfaces, for example, a river, a canal, a pipe with an incomplete section;

2) pressure, for example, a pipe with a full section;

3) hydraulic jets, which are limited to a liquid (as we will see later, such jets are called flooded) or gaseous medium.

Free section and hydraulic radius of the flow. Continuity equation in hydraulic form

The Gromeka equation is suitable for describing the motion of a fluid if the components of the motion function contain some vortex quantity. For example, this vortex quantity is contained in the components ωx, ωy, ωz of the angular velocity w.

The condition that the movement is steady is the absence of acceleration, that is, the condition that the partial derivatives of all velocity components are equal to zero:

Now if we fold

then we get

If we project the displacement by an infinitesimal value dl onto the coordinate axes, we get:

dx=Uxdt; dy = Uy dt; dz = Uzdt. (3)

Now we multiply each equation (3) by dx, dy, dz, respectively, and add them:

Assuming that right part is zero, and this is possible if the second or third rows are zero, we get:

We have obtained the Bernoulli equation

Analysis of the Bernoulli equation

this equation is nothing but the equation of a streamline in steady motion.

From this follows the conclusions:

1) if the motion is steady, then the first and third rows in the Bernoulli equation are proportional.

2) rows 1 and 2 are proportional, i.e.

Equation (2) is the vortex line equation. The conclusions from (2) are similar to the conclusions from (1), only the streamlines replace the vortex lines. In a word, in this case condition (2) is satisfied for vortex lines;

3) the corresponding members of rows 2 and 3 are proportional, i.e.

where a is some constant value; if we substitute (3) into (2), then we obtain the streamline equation (1), since from (3) it follows:

ωx = aUx; ωy = aUy; ωz = aUz. (four)

Here follows an interesting conclusion that the vectors of linear velocity and angular velocity are co-directed, that is, parallel.

In a broader sense, one must imagine the following: since the motion under consideration is steady, it turns out that the particles of the liquid move in a spiral and their trajectories along the spiral form streamlines. Therefore, streamlines and particle trajectories are one and the same. This kind of movement is called screw.

4) the second row of the determinant (more precisely, the members of the second row) is equal to zero, i.e.

ω x = ω y = ω z = 0. (5)

But the absence of angular velocity is equivalent to the absence of vortex motion.

5) let line 3 be equal to zero, i.e.

Ux = Uy = Uz = 0.

But this, as we already know, is the condition for the equilibrium of the liquid.

The analysis of the Bernoulli equation is completed.

Galilean transformation. Mechanical principle of relativity. Postulates of special (private theory) relativity. Lorentz transformation and consequences from them.

The basic principle on which classical mechanics is based is the principle of relativity, formulated on the basis of empirical observations by G. Galileo. According to this principle, there are infinitely many frames of reference in which a free body is at rest or moves with a constant speed in absolute value and direction. These frames of reference are called inertial and move relative to each other uniformly and rectilinearly. In all inertial frames of reference, the properties of space and time are the same, and all processes in mechanical systems obey the same laws. This principle can also be formulated as the absence of absolute reference systems, that is, reference systems that are somehow distinguished relative to others.

The principle of relativity- a fundamental physical principle, according to which all physical processes in inertial reference frames proceed in the same way, regardless of whether the system is stationary or it is in a state of uniform and rectilinear motion.

Special theory of relativity (ONE HUNDRED; also private theory of relativity) is a theory that describes movement, the laws of mechanics and space-time relations at arbitrary speeds of movement, less than the speed of light in a vacuum, including those close to the speed of light. Within the framework of special relativity, Newton's classical mechanics is an approximation of low velocities. The generalization of SRT for gravitational fields is called the general theory of relativity.

The deviations in the course of physical processes from the predictions of classical mechanics described by the special theory of relativity are called relativistic effects, and the rates at which such effects become significant are relativistic speeds

Lorentz transformations- linear (or affine) transformations of a vector (respectively, affine) pseudo-Euclidean space that preserves lengths or, equivalently, the scalar product of vectors.

The Lorentz transformations of the pseudo-Euclidean signature space are widely used in physics, in particular, in the special theory of relativity (SRT), where the four-dimensional space-time continuum (Minkowski space) acts as an affine pseudo-Euclidean space

Transfer phenomenon.

In a gas that is in a non-equilibrium state, irreversible processes occur, called transport phenomena. In the course of these processes, there is a spatial transfer of matter (diffusion), energy (thermal conductivity), and momentum of directed motion (viscous friction). If the course of the process does not change with time, then such a process is called stationary. Otherwise, it is a non-stationary process. Stationary processes are possible only under stationary external conditions. In a thermodynamically isolated system, only non-stationary transport phenomena can occur, aimed at establishing an equilibrium state

Subject and method of thermodynamics. Basic concepts. First law of thermodynamics.

The principle of construction of thermodynamics is quite simple. It is based on three experimental laws and the equation of state: the first law (the first law of thermodynamics) - the law of conservation and transformation of energy; the second law (the second law of thermodynamics) indicates the direction in which natural phenomena proceed in nature; the third law (the third law of thermodynamics) states that the absolute zero of temperature is unattainable. Thermodynamics, unlike statistical physics, does not consider specific molecular patterns. On the basis of experimental data, the basic laws (principles or beginnings) are formulated. These laws and their consequences are applied to specific physical phenomena associated with the transformation of energy in a macroscopic way (without taking into account the atomic and molecular structure), they study the properties of bodies of specific sizes. The thermodynamic method is used in physics, chemistry, and a number of technical sciences.

Thermodynamics - the doctrine of the connection and mutual transformations of various types of energy, heat and work.

The concept of thermodynamics comes from the Greek words "thermos" - warmth, heat; "dynamos" - strength, power.

In thermodynamics, a body is understood as a certain part of space filled with matter. The shape of a body, its color and other properties are not essential for thermodynamics, therefore, the thermodynamic concept of a body differs from the geometric one.

The internal energy U plays an important role in thermodynamics.

U is the sum of all types of energy contained in an isolated system (the energy of thermal motion of all microparticles of the system, the energy of interaction of particles, the energy of electric shells of atoms and ions, intranuclear energy, etc.).

The internal energy is a single-valued function of the state of the system: its change DU during the transition of the system from state 1 to state 2 does not depend on the type of process and is equal to ∆U = U 1 – U 2 . If the system performs a circular process, then:

The total change in its internal energy is 0.

The internal energy U of the system is determined by its state, i.e. U of the system is a function of the state parameters:

U = f(p,V,T) (1)

At not too high temperatures, the internal energy of an ideal gas can be considered equal to the sum of the molecular kinetic energies of the thermal motion of its molecules. The internal energy of a homogeneous, and in the first approximation, heterogeneous systems is an additive quantity - equal to the sum of the internal energies of all its macroscopic parts (or phases of the system).

adiabatic process. Poisson's equation, adiabat. Polytropic process, polytropic equation.

An adiabatic process is one in which there is no heat transfer.

Adiabatic, or adiabatic process(from other Greek ἀδιάβατος - "impassable") - a thermodynamic process in a macroscopic system, in which the system does not exchange thermal energy with the surrounding space. Serious study of adiabatic processes began in the 18th century.

An adiabatic process is a special case of a polytropic process, since in it the heat capacity of the gas is zero and, therefore, constant. Adiabatic processes are reversible only when the system remains in equilibrium at each moment of time (for example, the change in state occurs slowly enough) and there is no change in entropy. Some authors (in particular, L. D. Landau) called only quasi-static adiabatic processes adiabatic.

The adiabatic process for an ideal gas is described by the Poisson equation. The line depicting an adiabatic process on a thermodynamic diagram is called adiabatic. Processes in a number of natural phenomena can be considered adiabatic. Poisson equation is an elliptic partial differential equation that, among other things, describes

• electrostatic field,
• stationary temperature field,
• pressure field,
• velocity potential field in hydrodynamics.

It is named after the famous French physicist and mathematician Simeon Denis Poisson.

This equation looks like:

where is the Laplace operator or Laplacian, and is a real or complex function on some manifold.

In a three-dimensional Cartesian coordinate system, the equation takes the form:

In the Cartesian coordinate system, the Laplace operator is written in the form and the Poisson equation takes the form:

If a f tends to zero, then the Poisson equation turns into the Laplace equation (the Laplace equation is a special case of the Poisson equation):

Poisson's equation can be solved using the Green's function; see, for example, the article screened Poisson equation. There are various methods for obtaining numerical solutions. For example, an iterative algorithm is used - the "relaxation method".

Also, such processes have received a number of applications in technology.

Polytropic process, polytropic process- a thermodynamic process during which the specific heat capacity of a gas remains unchanged.

In accordance with the essence of the concept of heat capacity, the limiting particular phenomena of a polytropic process are an isothermal process () and an adiabatic process ().

In the case of an ideal gas, the isobaric process and the isochoric process are also polytropic ?

Polytropic equation. The isochoric, isobaric, isothermal and adiabatic processes discussed above have one common property - they have a constant heat capacity.

Ideal heat engine and Carnot cycle. K.P.D. ideal heat engine. The content of the second law of K.P.D. real heat engine.

The Carnot cycle is an ideal thermodynamic cycle. Carnot heat engine, operating according to this cycle, has the maximum efficiency of all machines for which the maximum and minimum temperatures of the ongoing cycle coincide, respectively, with the maximum and minimum temperatures of the Carnot cycle.

Maximum efficiency is achieved with a reversible cycle. In order for the cycle to be reversible, heat transfer in the presence of a temperature difference must be excluded from it. To prove this fact, suppose that heat transfer occurs at a temperature difference. This transfer occurs from a hotter body to a colder one. If we assume the process is reversible, then this would mean the possibility of transferring heat back from a colder body to a hotter one, which is impossible, therefore the process is irreversible. Accordingly, the conversion of heat into work can only occur isothermally [Comm 4] . In this case, the reverse transition of the engine to the starting point only by an isothermal process is impossible, since in this case all the work received will be spent on restoring the initial position. Since it was shown above that the adiabatic process can be reversible, this kind of adiabatic process is suitable for use in the Carnot cycle.

In total, two adiabatic processes occur during the Carnot cycle:

1. Adiabatic (isentropic) expansion(in the figure - process 2→3). The working fluid is detached from the heater and continues to expand without heat exchange with the environment. At the same time, its temperature decreases to the temperature of the refrigerator.

2. Adiabatic (isentropic) compression(in the figure - process 4→1). The working fluid is detached from the refrigerator and compressed without heat exchange with the environment. At the same time, its temperature increases to the temperature of the heater.

Boundary conditions En and Еt.

In a conducting body in an electrostatic field, all points of the body have the same potential, the surface of the conducting body is an equipotential surface, and the field strength lines in the dielectric are normal to it. Denoting through E n and E t the normal and tangent to the conductor surface, the components of the field strength vector in the dielectric near the conductor surface, these conditions can be written as:

E t = 0; E = E n = -¶U/¶n; D = -e*¶U/¶n = s,

where s is the surface density of the electric charge on the surface of the conductor.

Thus, at the interface between the conducting body and the dielectric, there is no tangent to the surface (tangential) component of the electric field strength, and the electric displacement vector at any point directly adjacent to the surface of the conducting body is numerically equal to the electric charge density s on the surface of the conductor

Clausius theorem, Clausius inequality. entropy, its physical meaning. Entropy change in irreversible processes. Basic Equation of Thermodynamics.

the sum of the reduced heats during the transition from one state to another does not depend on the form (path) of the transition in the case of reversible processes. The last statement is called Clausius theorems.

Considering the processes of converting heat into work, R. Clausius formulated the thermodynamic inequality that bears his name.

"The reduced amount of heat received by the system during an arbitrary circular process cannot be greater than zero"

where dQ is the amount of heat received by the system at temperature T, dQ 1 is the amount of heat received by the system from the sections environment with temperature T 1, dQ ¢ 2 - the amount of heat given off by the system to areas of the environment at temperature T 2. The Clausius inequality allows you to set an upper limit on thermal efficiency. at variable temperatures of the heater and refrigerator.

It follows from the expression for the reversible Carnot cycle that or , i.e. for a reversible cycle, the Clausius inequality turns into an equality. This means that the reduced amount of heat received by the system in the course of a reversible process does not depend on the type of process, but is determined only by the initial and final states of the system. Therefore, the reduced amount of heat received by the system in the course of a reversible process serves as a measure of the change in the state function of the system, called entropy.

The entropy of a system is a function of its state, defined up to an arbitrary constant. The increase in entropy is equal to the reduced amount of heat that must be reported to the system in order to transfer it from the initial state to the final state in any reversible process.

, .

An important feature of entropy is its increase in isolated

Methods for specifying the movement of a point.

Set Point Movement - this means to indicate a rule by which at any moment of time it is possible to determine its position in given system reference.

The mathematical expression for this rule is called the law of motion , or motion equation points.

There are three ways to specify the movement of a point:

vector;

coordinate;

natural.

To set the movement in a vector way, need:

à select a fixed center;

à determine the position of the point using the radius vector , starting at the fixed center and ending at the moving point M;

à define this radius vector as a function of time t: .

Expression

called vector law of motion dots, or vector equation of motion.

!! Radius vector - this is the distance (vector modulus) + direction from the center O to the point M, which can be determined in different ways, for example, by angles with given directions.

To set movement coordinate way , need:

à select and fix a coordinate system (any: Cartesian, polar, spherical, cylindrical, etc.);

à determine the position of the point using the appropriate coordinates;

à set these coordinates as functions of time t.

In the Cartesian coordinate system, therefore, it is necessary to specify the functions

In the polar coordinate system, the polar radius and polar angle should be defined as functions of time:

In general, with the coordinate method of setting, one should set as a function of time those coordinates with which the current position of the point is determined.

To be able to set the movement of the point natural way, you need to know it trajectory . Let us write down the definition of the trajectory of a point.

trajectory point is called set of its positions for any period of time(usually from 0 to +¥).

In the example with the wheel rolling on the road, the trajectory of point 1 is cycloid, and points 2 – roulette; in the reference frame associated with the center of the wheel, the trajectories of both points are circles.

To set the movement of a point in a natural way, you need to:

à know the trajectory of the point;

à on the trajectory, select the origin and the positive direction;

à determine the current position of the point by the length of the trajectory arc from the origin to this current position;

à specify this length as a function of time.

An expression that defines the above function,

called the law of motion of a point along a trajectory, or natural equation of motion points.

Depending on the type of function (4), a point along the trajectory can move in different ways.

3. Point trajectory and its definition.

The definition of the concept of "point trajectory" was given earlier in question 2. Consider the question of determining the trajectory of a point for different ways motion tasks.

natural way: the trajectory must be given, so it is not necessary to find it.

Vector way: you need to switch to the coordinate method according to the equalities

Coordinate method: it is necessary to exclude the time t from the equations of motion (2), or (3).

The coordinate equations of motion define the trajectory parametrically, through the parameter t (time). To obtain an explicit equation for the curve, the parameter must be excluded from the equations.

After excluding time from equations (2), two equations of cylindrical surfaces are obtained, for example, in the form

The intersection of these surfaces will be the trajectory of the point.

When a point moves along a plane, the problem is simplified: after eliminating time from the two equations

the trajectory equation will be in one of the following forms:

When will be, so the trajectory of the point will be the right branch of the parabola:

It follows from the equations of motion that

therefore, the trajectory of the point will be the part of the parabola located in the right half-plane:

Then we get

Since then the entire ellipse will be the trajectory of the point.

At the center of the ellipse will be at the origin O; when we get a circle; the parameter k does not affect the shape of the ellipse, it determines the speed of the point moving along the ellipse. If cos and sin are interchanged in the equations, then the trajectory will not change (the same ellipse), but the initial position of the point and the direction of movement will change.

The speed of a point characterizes the “speed” of changing its position. Formally: speed - movement of a point per unit of time.

Precise definition.

Then Attitude

The speed of a point is a vector that determines at each given moment the speed and direction of movement of the point.

The speed of uniform movement is determined by the ratio of the path traveled by a point in a certain period of time to the value of this period of time.

Speed; S- way; t- time.

The speed is measured in units of length divided by a unit of time: m/s; cm/s; km/h, etc.

In the case of rectilinear motion, the velocity vector is directed along the trajectory in the direction of its motion.

If a point travels unequal paths in equal intervals of time, then this movement is called uneven. Velocity is a variable and is a function of time.

The average speed of a point over a given period of time is the speed of such a uniform rectilinear motion at which the point would receive the same movement during this period of time as in its considered movement.

Consider a point M that moves along a curvilinear trajectory given by the law

During the time interval? t, the point M will move to the position M 1 along the arc MM 1. If the time interval? t is small, then the arc MM 1 can be replaced by a chord and, in the first approximation, find the average speed of the point

This speed is directed along the chord from point M to point M 1 . We find the true speed by going to the limit when? t> 0

When?t> 0, the direction of the chord in the limit coincides with the direction of the tangent to the trajectory at the point M.

Thus, the value of the speed of a point is defined as the limit of the ratio of the path increment to the corresponding time interval as the latter tends to zero. The direction of the velocity coincides with the tangent to the trajectory at the given point.

point acceleration

Note that in the general case, when moving along a curvilinear trajectory, the speed of a point changes both in direction and in magnitude. The change in speed per unit time is determined by acceleration. In other words, the acceleration of a point is a quantity that characterizes the rate of change of speed over time. If for a time interval? t the speed changes by a value, then the average acceleration

The true acceleration of a point at a given time t is the value to which the average acceleration tends when? t\u003e 0, that is

With a time interval tending to zero, the acceleration vector will change both in magnitude and direction, tending to its limit.

Dimension of acceleration

Acceleration can be expressed in m/s 2 ; cm/s 2 etc.

In the general case, when the motion of a point is given in a natural way, the acceleration vector is usually decomposed into two components directed along the tangent and along the normal to the point's trajectory.

Then the acceleration of a point at time t can be represented as

Let us denote the constituent limits by and.

The direction of the vector does not depend on the size of the time interval?t.

This acceleration always coincides with the direction of speed, that is, it is directed tangentially to the trajectory of the point and is therefore called tangential or tangential acceleration.

The second component of the acceleration of the point is directed perpendicular to the tangent to the trajectory at the given point towards the concavity of the curve and affects the change in the direction of the velocity vector. This component of acceleration is called normal acceleration.

Since the numerical value of the vector is equal to the increment of the point velocity over the considered time interval?t, then the numerical value of the tangential acceleration

The numerical value of the tangential acceleration of a point is equal to the time derivative of the numerical value of the speed. The numerical value of the normal acceleration of a point is equal to the square of the point's speed divided by the radius of curvature of the trajectory at the corresponding point on the curve

The total acceleration in case of non-uniform curvilinear motion of a point is geometrically composed of the tangential and normal accelerations.

If a material point is in motion, then its coordinates are subject to change. This process can be fast or slow.

Definition 1

The value that characterizes the rate of change in the position of the coordinate is called speed.

Definition 2

average speed is a vector quantity, numerically equal to the displacement per unit time, and co-directional with the displacement vector υ = ∆ r ∆ t ; υ ∆ r .

Picture 1 . The average speed is co-directed to the movement

The modulus of the average speed along the path is equal to υ = S ∆ t .

Instantaneous speed characterizes the movement at a certain point in time. The expression "velocity of a body at a given time" is considered incorrect, but applicable in mathematical calculations.

Definition 3

The instantaneous speed is the limit to which the average speed υ tends when the time interval ∆t tends to 0:

υ = l i m ∆ t ∆ r ∆ t = d r d t = r ˙ .

The direction of the vector υ is tangent to the curvilinear trajectory, because the infinitesimal displacement d r coincides with the infinitesimal element of the trajectory d s .

Figure 2. Instantaneous velocity vector υ

The existing expression υ = l i m ∆ t ∆ r ∆ t = d r d t = r ˙ in Cartesian coordinates is identical to the equations proposed below:

υ x = d x d t = x ˙ υ y = d y d t = y ˙ υ z = d z d t = z ˙ .

The record of the modulus of the vector υ will take the form:

υ \u003d υ \u003d υ x 2 + υ y 2 + υ z 2 \u003d x 2 + y 2 + z 2.

To go from Cartesian rectangular coordinates to curvilinear, apply the rules of differentiation of complex functions. If the radius vector r is a function of curvilinear coordinates r = r q 1 , q 2 , q 3 , then the velocity value is written as:

υ = d r d t = ∑ i = 1 3 ∂ r ∂ q i ∂ q i ∂ r = ∑ i = 1 3 ∂ r ∂ q i q ˙ i .

Figure 3. Displacement and instantaneous velocity in curvilinear coordinate systems

For spherical coordinates, suppose that q 1 = r ; q 2 \u003d φ; q 3 \u003d θ, then we get υ presented in this form:

υ = υ r e r + υ φ e φ + υ θ φ θ , where υ r = r ˙ ; υ φ = r φ ˙ sin θ ; υ θ = r θ ˙ ; r ˙ = d r d t ; φ ˙ = d φ d t ; θ ˙ = d θ d t ; υ \u003d r 1 + φ 2 sin 2 θ + θ 2.

Definition 4

instantaneous speed call the value of the derivative of the function of movement in time at a given moment, associated with the elementary movement by the relation d r = υ (t) d t

Example 1

Given the law of rectilinear motion of a point x (t) = 0 , 15 t 2 - 2 t + 8 . Determine its instantaneous speed 10 seconds after the start of movement.

Solution

The instantaneous velocity is usually called the first derivative of the radius vector with respect to time. Then its entry will look like:

υ (t) = x ˙ (t) = 0 . 3 t - 2 ; υ (10) = 0 . 3 × 10 - 2 = 1 m/s.

Answer: 1 m/s.

Example 2

The movement of a material point is given by the equation x = 4 t - 0 , 05 t 2 . Calculate the moment of time t about with t when the point stops moving, and its average ground speed υ.

Solution

Calculate the equation of instantaneous speed, substitute numerical expressions:

υ (t) = x ˙ (t) = 4 - 0 , 1 t .

4 - 0 , 1 t = 0 ; t about with t \u003d 40 s; υ 0 = υ (0) = 4; υ = ∆ υ ∆ t = 0 - 4 40 - 0 = 0 , 1 m / s.

Answer: given point stop after 40 seconds; the value of the average speed is 0.1 m/s.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter