Definition:The kinetic energy of the body is called the energy of its progressive movement.

If the external force acts on a resting body, the latter acquires some speed and can work itself. This stock of work is called kinetic energy Body. We write the equation of motion of the material point:

where  Resulting force. Multiply the equation of motion is scalar

, then

In the right part of the equation, we got an elementary work, in the left - an expression that can be converted to the type of complete differential:

As a result, have

. elementary workPerfect by force is equal to the increment of magnitude

defined with an accuracy of an arbitrary constant. It turns out that the force makes some work, and the kinetic energy of the body increases on the same number (normal designation T. or W. kin. ). With a negative work of force, the kinetic energy of the body decreases: energy is consumed to overcome the current force. It is usually believed that the boring body does not possess kinetic energy, so that an arbitrary constant is supposed to be zero:

.

§17.Potential energy.

Definition:The potential energy is part of the energy of the mechanical system, depending only on its configuration, i.e. From the mutual location of all particles of the system and, from location in an external potential field.

The loss of potential energy when moving the system from an arbitrary position "1" to another position "2" is measured by the work A 12, which all potential: internal and external, forces acting on the system are performed:

U.(1)  U.(2) \u003d a 12 or  U. =  BUT 12 ,

where  U. = U.(2)  U.(1)  changing the potential energy of the mechanical system,

U.(1), U.(2)  Remarks for the potential energy of the mechanical system in the provisions "1" and "2".

Accordingly, the work of potential forces with a small change in the system configuration   A \u003d. du.

These ratios are valid for the case of stationary (independent) external potential field. For the simplest case, finding a material point in an external potential field, the force with which this field acts on the point is calculated by the formula:

where

It is noticeable by a gradient of a scalar function (in this case, theential energy). Gradientto vector magnitude aimed towards the growth of function values U.. The resulting formula appears the "" sign, which indicates that the strength is aimed at losing the values \u200b\u200bof the function U..

The reverse relationship that allows for the known expression of the potential force, calculate the value of potential energy, obviously

.

The above formula makes it possible to determine explicit expressions of potential energy for special cases. When calculating this integral, one of the limits seek to choose such that potential energy In the point under consideration was zero.

Example number 1.

The strength of the gravitational interaction of the two bodies is equal

,

With the concept of work, another fundamental physical concept is connected - the concept of energy. Since mechanics are studied, firstly, the movement of bodies, and secondly, the interaction of bodies among themselves, it is customary to distinguish between two types of mechanical energy: kinetic energydue to the movement of the body and potential energycaused by the interaction of the body with other bodies.

Kinetic energy mechanical system call energy, savilovaya from the speeds of the movement of the points of this system.

An expression for kinetic energy can be found by determining the work of the automatic force attached to the material point. Based on (2.24), we write the formula for the elementary work of the resultant force:

Because

, then Da \u003d MυDυ. (2.25)

To find a job of a resulting force when changing the body velocity from υ 1 to υ 2 Integrating the expression (2.29):

(2.26)

Since the work is a measure of energy transmission from one body to another, then on

base (2.30) Write that the value There is kinetic energy

body:

where instead (1.44) get

(2.27)

Theorem, expressed by formula (2.30), is customary called theorem on kinetic energy . In accordance with it, the work of the forces acting on the body (or system of tel) is equal to the change in the kinetic energy of this body (or body system).

From the kinetic energy theorem follows physical meaning of kinetic energy : the kinetic energy of the body is equal to the work that it can perform in the process of reducing its speed to zero.The greater the "stock" of kinetic energy at the body, the greater work it is capable of committing.

The kinetic energy of the system is equal to the sum of the kinetic energies of material points, of which this system consists of:

(2.28)

If the work of all the forces acting on the body is positive, then the kinetic energy of the body increases, if the work is negative, then the kinetic energy decreases.

It is obvious that the elementary work of the resultant all the forces attached to the body will be equal to the elementary change in the kinetic energy of the body:

dA \u003d 1 (2.29)

In conclusion, we note that the kinetic energy, as well as the speed of movement, is relative. For example, the kinetic energy of the passenger sitting on the train will be different if we consider the movement relative to the canvas or relative to the car.

§2.7 Potential Energy

The second type of mechanical energy is potential energy - Energy due to interaction of tel.

Potential energy characterizes not any interaction of bodies, but only such that is described by the forces that do not depend on the speed. Most of the forces (the strength of gravity, the force of elasticity, gravitational forces, etc.) are the following; Exceptions are only friction forces. The work of the forces under consideration does not depend on the form of the trajectory, but is determined only by its initial and end position. The work of such forces on a closed trajectory is zero.

Forces, whose work does not depend on the form of the trajectory, and depends only on the initial and final position of the material point (body) called potential or conservative forces .

If the body interacts with its surroundings by means potential forces, To characterize this interaction, you can enter the concept of potential energy.

Potential call energy due to interaction bodies and depending on their mutual location.

We find the potential energy of the body raised above the ground. Let the body mass M evenly move in the gravitational field from position 1 to position 2 over the surface, the cross section of which the drawing plane is shown in Fig. 2.8. This section is the trajectory of the material point (body). If the friction is absent, then three forces act on the point:

1) the power of N on the surface of the surface of the surface normally, the operation of this force is zero;

2) MG gravity, the work of this force A 12;

3) Press force F from some driving body (internal combustion engine, electric motor, person, etc.); The work of this force is denoted by t.

Consider the work of gravity when moving the body along the inclined plane length ℓ (Fig. 2.9). As can be seen from this picture, the work is equal

A "\u003d Mgℓ compassα \u003d mgℓ computers (90 ° + α) \u003d - Mgℓ sinα

From the triangle of ADD we have ℓ sinα \u003d h, so from the last formula it follows:

The trajectory of the body movement (see Fig. 2.8) can be schematically present in small sections of the inclined plane, so for, the work of gravity over the entire trajectory 1 -2 is true

A 12 \u003d Mg (H 1 -H 2) \u003d - (Mg H 2 - Mg H 1) (2.30)

So, the work of gravity does not depend on the trajectory of the body, and depends on the difference in the altitudes of the location of the initial and endpoint of the trajectory.

Magnitude

e n \u003d Mg H (2.31)

call potential energy the material point (body) weighing M raised above the ground H. Consequently, formula (2.30) can be rewritten so:

A 12 \u003d \u003d - (EN 2 - EN 1) or a 12 \u003d--Δen (2.32)

The work of gravity is equal to the reverse sign of a change in the potential energy of bodies, i.e. the difference of its final and initialvalues (potential energy theorem ).

Such reasoning can also be given for elastically deformed body.

(2.33)

Note that physical meaning It has the difference in potential energies as a value that determines the operation of the conservative forces. In connection with this, no matter what position, configuration, you should attribute zero potential energy.

From the potential energy theorem, one very important consequence can be obtained: conservative forces are always directed towards reducing potential energy.The prescribed pattern is manifested in that any system provided by itself always seeks to go to such a state in which its potential energy is of the smallest meaning. This is the principle of a minimum potential energy .

If the system in this state does not have the minimum potential energy, then this state is called energetically unprofitable.

If the ball is at the bottom of the concave bowl (Fig.2.10, a), where its potential energy is minimal (compared with its values \u200b\u200bin neighboring positions), then its condition is more profitable. Equilibrium ball in this case is sustainable: If you shift the ball aside and let go, it will return again in its original position.

Energetically unprofitable, for example, is the position of the ball on the top of the convex surface (Fig.2.10, b). The sum of the forces acting on the ball is zero, and therefore this ball will be in equilibrium. However, equilibrium is it unstable: The slightest impact so that he rolled down and thus moved to the state is energetically more profitable, i.e. Putting less

p substitial energy.

For indifferent equilibrium (Fig. 2.10, c) the potential energy of the body is equal to the potential energy of all its possible nearest states.

In Figure 2.11, you can specify some limited area of \u200b\u200bspace (for example, CD), in which the potential energy is less than outside it. This area was named potential pit .

Kinetic energy-corrive function, which is a measure of the material point of motion and depending only the ovasyimodulavosti-theaterial points forming the physical system under consideration, the energy mechanical system, depending on the discovery of its points in the selected reference. Often distinguished kinetic energetic factory.

More strictly, kinetic energy is the difference between the total energy of the system and its main energy; Thus, the kinetic energy is part of the capital energy, due to the flow.

Simple language, kinetic energy is the energy that bodyenimets only when driving. Everone moves, kinetic energy is zero.

Physical meaning

Consider a system consisting of one particle, and write Newton's second law:

There is a self-absorbing Assault, acting on the body. Multical intelligence of inactivity. Considering that, we get:

If the system is closed, i.e. there is no external for the system of force, or the equal to all forces is zero, , and the amount

it remains constant. This value is called kinetic energyparticles. If the system is isolated, then the kinetic energy is the method of movement.

For absolutely solid, teleparal kinetic energy can be written as the sum of the kinetic energy of the progressive and rotational movement:

Body mass

Masser speed center

Moment inertiaThekg · m²

Angle speed. Rad / S.

We find kinetic energy at various cases of movement:

1. Protective traffic

The speed of all points of the system is equal to the speed of the center of mass. Then

The kinetic energy of the system in progressive movement is equal to half of the mass of the system to the square of the velocity of the center of mass.

2. Rotary traffic (Fig. 77)

The speed of any point of the body :. Then

or using formula (15.3.1):

The kinetic energy of the body during rotation is equal to half the product of the inertia of the body relative to the axis of rotation per square of its angular velocity.

3. Flat-parallel motion

With this movement, the kinetic energy is made up of the energy of progressive and rotational movements

The general case of motion gives the formula to calculate the kinetic energy similar to the latter.

We have done the definition of work and power in paragraph 3 of chapters 14. Here we will consider examples of the calculation of the work and the power of the forces acting on the mechanical system.

Physical meaning of work

The operational of the forces acting on a particle during its movement goes to the increment of kinetic particle energy:

Properties of kinetic energy

Additivity.This property means that the kinetic energy of a mechanical system consisting of material points is equal to the sum of the kinetic energies of all material points that are included in the system.

Invariance with respect to the rotation of the reference system.Kinetic energy It does not depend on the position of the point, the direction of its speed and depends only on the speed module or, which is the same, from the square of its speed.

Preservation.Kinetic energy does not change in the interactions that change only the mechanical characteristics of the system. This property is invariant with respect to the transformations of the Galilee properties of the preservation of kinetic energy and the second law of Newton is enough to bring the mathematical formula of kinetic energy.

Relativism

At speeds close to the speed of light, the kinetic energy of any object is equal

Massage paper;

Speed \u200b\u200bsetting in the selected inertial reference system;

Speed \u200b\u200bof Light Vacuum (Sayer Energy).

This formula can be rewritten in the following form:

At low speeds (), the last ratio goes into a conventional formula.

The ratio of kinetic and internal energy

Kinetic energy depends on what positions the system is considered. If we consider the macroscopic object (for example, the solid body of visible sizes) as a whole, we can talk about such an energy form as internal energy. The kinetic energy in this case appears only when the body moves as an integer.

The same body considered from a microscopic point of view consists of atomolecules, and the internal energy is due to the movement of atoms and molecules and is considered as a consequence of particle motion motion, and the absolute body temperature is directly proportional to the average kinetic energy of such a movement of atoms and molecules. The proportionality coefficient -Poped boltzmann.