Mechanical, nuclear, electromagnetic, etc. However, so far we will consider only one form - mechanical. Moreover, from the point of view of the history of physics, it began with studying forces and work. At one of the stages of the formation of science, the law of conservation of energy was opened.

When considering mechanical phenomena, the concepts of kinetic and experimentally have been established that energy does not disappear without a trace, it turns into another from one species. We can assume that what has been said in general formulates the law of preservation

First, it should be noted that in the amount of potential and the bodies are called mechanical energy. Next, it is necessary to keep in mind that the law of conservation is fair in the absence of external influence and additional losses caused, for example, overcoming resistance forces. If some of these requirements are impaired, then it will lose its loss when the energy changes.

The easiest experiment confirming these boundary conditions, each can spend independently. Raise the ball to the height and release it. Having hitting the floor, he jumps and then falls again to the floor, and again jumbles. But with each time the height of its lift will be smaller and smaller while the ball will not be filled out on the floor.

What do we see in this experience? When the ball is stationary and is on top, it has only potential energy. When the fall begins, it appears speed, and it means that kinetic energy appears. But as the height falls, with which the movement began, it becomes less and, accordingly, it becomes less than its potential energy, i.e. It turns into kinetic. If we carry out calculations, it turns out that the energy values \u200b\u200bare equal, which means that the law of energy conservation under such conditions is performed.

However, in such an example there are violations of two previously established conditions. The ball moves surrounded by air and is experiencing resistance from his side, albeit a small one. And energy is spent on overcoming resistance. In addition, the ball faces the floor and bounces, i.e. It is experiencing an external impact, and this is the second violation of the boundary conditions that the law of energy conservation be fair.

In the end, the jump is stopped, and he will stop. All existing initial energy will be spent on overcoming air resistance and external influence. However, in addition to the transformation of energy, work will be performed on overcoming the forces of friction. This will lead to heating the body itself. Often, the amount of heating is not very significant, and it can only be determined when measuring with accurate devices, but such a change in temperature exists.

In addition to mechanical, there are other types of energy - light, electromagnetic, chemical. However, for all varieties of energy, it is fair that the transition to another is possible from one species, and that with such transformations the total energy of all types remains constant. This is a confirmation of the universal nature of energy conservation.

Here it is necessary to take into account that the transition of energy can mean and its useless loss. With mechanical phenomena, the evidence of this will be heating the environment or interacting surfaces.

Thus, the simplest mechanical phenomenon allowed us to determine the law of conservation of energy and the boundary conditions that ensure its execution. It was found that it is carried out from the available species in any other, and the universal nature of the law mentioned is revealed.

mechanical energy. Turning energy

Since the movement and interaction are interrelated (the interaction determines the movement of material objects, and the movement of objects, in turn, affects their interaction), there must be a single measure that characterizes the movement and interaction of matter.

Energy is a single scalar quantitative measure of various forms of movement and interaction of matter. Different forms of movement and interaction correspond to various types of energy: mechanical, internal, electromagnetic, nuclear, etc. The simplest type of energy corresponding to the simplest - mechanical - form of movement and interaction of matter is mechanical energy.

One of the most important laws of all natural science is universal Energy Conservation Law. He argues that energy does not arise nowhere and does not disappear without a trace, but only passes from one form to another.

The law of conservation of mechanical energy is the special case of the universal law of energy conservation.

The complete mechanical energy of the material point (particle) and particle systems folds from two parts. The first component of the particle energy is caused by its movement, is called kinetic energy and is calculated by the formula

where m. - mass of particles, - Her speed.

The kinetic energy of the particles changes, if when the particle moves to it, the strength (strength) performs work.

In the simplest case, when power constant in magnitude and in the direction, and the trajectory of movement is straightforward, then work A.accomplished by this force when moving
, determined by the formula

where s. - traveled path equal to rectilinear movement movement module
,
- scalar product of vectors and
equal to the product of the modules of these vectors on the cosine of the angle
between them.

Work can be positive if angle
acute
90 °), negative if angle
stupid (90 °
180 °), and can be zero if the angle
straight (
\u003d 90 °).

You can prove that the change in kinetic energy
particles when it moves from point 1 to point 2 is equal to the sum of work performed by all the forces acting on this particle, with this movement:

, (6.13)

where
- kinetic particle energy in the initial and endpoints, - work perfect by force (i.=1, 2, ... n.) With this movement.

Kinetic energy system
of N. Particles are called the sum of the kinetic energies of all particles of the system. Its change in any change in the system configuration, that is, an arbitrary movement of particles, equal to the total work
committed by all the forces acting on particles of the system when they are moved:

. (6.14)

The second component of mechanical energy is the energy of interaction, called potential energy. In the mechanics, the concept of potential energy can be introduced not for any interactions, but only for their specific class.

Let in each point where the particle may be, on it, as a result of interaction with other bodies, the force is valid, depending only on the coordinates x, Y, Z particles and perhaps on time t.:
. Then they say that the particle is in the force field of interaction with other bodies. Examples: The material point moving in the gravitational field of the Earth; Electron moving in an electrostatic field of a fixed charged body. In these examples, the force acting on a particle at each point of space on time does not depend:
. Such fields are called stationary.

If, for example, the electron will be in the electrical field of the condenser, the voltage between the plates of which changes, then at each point of space the force will depend on time:
. This field is called nonstationary.

The force acting on a particle is called a conservative, and the corresponding field is a field of conservative power, if the work performed by this force when the particle is moved along an arbitrary closed contour will be zero.

The conservative forces and the corresponding fields include the power of global gravity and, in particular, the strength of gravity (gravitational field), the culon force (electrostatic field), the strength of elasticity (the field of forces acting on the body attached to a certain point by elastic relationship).

Examples of non-conservative forces are the strength of friction, the strength of resistance of the body movement.

Only for interactions, which correspond to conservative forces, the concept of potential energy can be introduced.

Under potential energy
the mechanical system is understood as the magnitude, the decline of which (the difference of initial and final values) with an arbitrary change in the configuration of the system (change in the position of particles in space) is equal to work
committed by all internal conservative forces acting between particles of this system:

, (6.15)

where
- The potential energy of the system in the initial and final configuration.

Note that decrease
equal to the opposite sign (change)
potential energy and therefore the ratio (6.15) can be written as

. (6.16)

Such a determination of the potential energy of the particle system allows you to find it when changing the system configuration, but not the value of the potential energy energy at a given configuration. Therefore, in all specific cases, they are ascended, with which configuration of the system (zero configuration) its potential energy
is taken equal to zero (
). Then the potential energy of the system at any configuration
and from (6.15) it follows that

, (6.17)

that is, the potential energy of a system of particles of some configuration is equal to work
committed by internal conservative forces when changing the system configuration from this to zero.

The potential energy of the body located in a homogeneous field of gravity near the surface of the Earth is taken equal to zero when the body is on the ground surface. Then the potential energy of attraction to the Earth of the body located at height h., equal to the work of gravity
committed when moving the body from this height to the surface of the Earth, that is, h. vertically:

The potential energy of the body attached to the fixed point by an elastic bond (spring) is taken equal to zero with undeclared communication. Then the potential energy of elastically deformed (stretched or compressed by magnitude
) Springs with stiffness coefficient k. equal

. (6.19)

The potential energy of the gravitational interaction of material points and the electrostatic interaction of point charges is made equal to zero if these points (charges) are removed to an infinite distance from each other. Therefore, the energy of gravitational interaction of material points by the masses and
located at a distance r. from each other, equal to the work of the world of world
committed when changing the distance x. Between points OT x \u003d R. before
:

. (6.20)

From (6.20), it follows that the potential energy of the gravitational interaction of material points with the specified choice of zero configuration (infinite removal) turns out to be negative when placing points at a finite distance from each other. This is due to the fact that the world's strength is the strength of attraction, and its work when removing points from each other is negative. The negativeness of potential energy means that when the system is moving from an arbitrary configuration to zero (when removing points from the final distance to infinite), its potential energy increases.

Similarly, the potential energy of electrostatic interaction of point charges in vacuum is equal

(6.21)

and negative for attracting variepete charges (signs and different) and positive for repulsive charges of the same name (signs and same).

Complete mechanical energy system (mechanical energy system)
called the sum of its kinetic and potential energies

. (6.22)

From (6.22) it follows that the change in complete mechanical energy is made up of changes in its kinetic and potential energy

Substitute in formula (6.33) of formula (6.14) and (6.16). In Formula (6.14) overall work
all the forces acting on the point of the system will imagine as the amount of the work of forces external with respect to the system under consideration,
and the work of the inner forces, which, in turn, develops from the work of internal conservative and non-consistent forces,

:

After the substitution, we get that

For a closed system
0. If the system is also conservative, that is, there are only internal conservative forces in it, then
\u003d 0. In this case, equation (6.24) takes
, which means that

Equation (6.2) is the mathematical record of the law of conservation of mechanical energy, which says: complete mechanical energy of a closed conservative system is constant, that is, does not change over time.

Condition
0 is performed if non-conservative forces act in the system, but their operation is zero, such as, for example, with the presence of peace friction. In this case, for a closed system, the law of conservation of mechanical energy is also applicable.

Note that
separate terms of mechanical energy: kinetic and potential energy - are not required to remain constant. They may vary, which is accompanied by the performance of work by conservative internal forces, but changes in potential and kinetic energy.
and
equal to the module and opposite by the sign. For example, due to the internal conservative forces of work on particles of the system, its kinetic energy will increase, but at the same time its potential energy will decrease on an equal value.

If non-conservative forces do in the system, this is necessarily accompanied by mutual transformations of mechanical and other types of energy. Thus, the operation of the work by the non-conservative forces of friction of sliding or resistance of the medium is necessarily accompanied by heat release, that is, the transition of the mechanical energy into the inner (thermal) energy. The non-conservative forces whose work leads to the transition of mechanical energy into thermal, is called dissipative, and the process of transition of mechanical energy in thermal - dissipation of mechanical energy.

There are many non-consistent forces, whose work, on the contrary, leads to an increase in the mechanical energy of the system at the expense of other types of energy. For example, as a result of chemical reactions, a projectile explosion occurs; At the same time, fragments receive an increase of mechanical (kinetic) energy due to the operation of the non-conservative pressure of the expanding gases - the explosion products. In this case, by performing the work of non-conservative forces, a chemical energy transition occurred into mechanical. The scheme of mutual transformations of energy when performing work conservative and non-consistent forces is presented in Figure 6.3.

Thus, the work is a quantitative measure of the transformation of one types of energy to others. The operation of the conservative forces is equal to the amount of potential energy that has passed into a kinetic or vice versa (the overall mechanical energy does not change), the work of non-conservative forces is equal to the number of mechanical energy that has passed into other types of energy or vice versa.

Figure 6.3 - Energy transformation scheme.

The universal law of conservation of energy is in fact there is a law of non-profitability of movement in nature, and the law of conservation of mechanical energy is the law of non-profitability of mechanical movement under certain conditions. The change in the same mechanical energy at the failure of these conditions does not mean the destruction of the movement or its appearance of nowhere, but indicates the transformation of one forms of movement and the interaction of matter to others.

Pay attention to the difference of the designations of infinitely small values. For example, dX Indicates the infinitely small increment of the coordinate,
- speeds dE. - Energy, and infinitely small work is denoted
. This difference has a deep meaning. The coordinates and speed of the particle, its energy and many other physical quantities are the function of the state of the particle (system of particles), that is, are determined by the current state of the particle (particle system) and do not depend on what preceding states, and on what kind of particle ( The system) came to the current state. The change in this value can be represented as the difference between the values \u200b\u200bof this value in the final and initial states. An infinitely small change in such a value (state functions) is called a complete differential and for the value X. denotes dX..

The same values \u200b\u200bas work or the amount of heat, characterize the state of the system, but the method to which the transition from one state of the system to another was implemented. For example, it is pointless to talk about the availability of work at the particle system in some given state, but we can talk about the work performed by the forces acting on the system, when it moves from one state to another. Thus, it does not make sense to talk about the difference of values \u200b\u200bof such a value in the final and initial states. Infinitely small amount of magnitude Y.not a function of the state is denoted
.

A distinctive feature of the function of the state is that their changes in the processes in which the system, coming out of the initial state, is returned to it, equal to zero. The mechanical state of the particle system is set by their coordinates and speeds. Therefore, if as a result of some process, the mechanical system is returned to its original state, the coordinates and velocities of all particles of the system take initial values. Mechanical energy, as a value, depending only on the coordinates and velocities of particles, also takes the initial value, that is, will not change. At the same time, the work performed by the forces acting on the particles will be different from zero, and its value can be different depending on the type of trajectories described by the particles of the system.

Full mechanical energy of a closed system of tel remains unchanged.

In all phenomena occurring in nature, the energy does not occur and does not disappear. It only turns into one species in another, while its value is preserved.

Law of energy conservation - The fundamental law of nature, which consists in the fact that for an isolated physical system, a scalar physical value can be introduced, which is the function of system parameters and called energy, which is preserved over time. Since the law of conservation of energy applies not to specific values \u200b\u200band phenomena, but reflects the general, applicable everywhere, and always, the pattern, it can be called not a law, but the principle of energy conservation.

Mechanical energy conservation law

In the mechanics, the law of energy conservation claims that in a closed particle system, full energy, which is the sum of kinetic and potential energy and does not depend on the time, that is, the integral of motion. The law of conservation of energy is valid only for closed systems, that is, in the absence of external fields or interactions.

The interaction forces between the bodies, for which the law of conservation of mechanical energy is carried out are called conservative forces. The law of conservation of mechanical energy is not performed for the forces of friction, since if there is a friction force, there is a transformation of mechanical energy into thermal.

Mathematical formulation

The evolution of the mechanical system of material points with the masses \\ (m_i \\) according to the second law of Newton satisfies the system of equations

\\ [m_i \\ dot (\\ mathbf (v) _i) \u003d \\ mathbf (f) _i \\]

where
\\ (\\ mathbf (v) _i \\) - the velocities of material points, and \\ (\\ mathbf (f) _i \\) - the forces acting on these points.

If you submit for forces as the sum of the potential forces \\ (\\ mathbf (f) _i ^ p \\) and the unprofitable forces \\ (\\ mathbf (f) _i ^ d \\), and the potential forces are recorded as

\\ [\\ MathBF (f) _i ^ p \u003d - \\ nabla_i u (\\ mathbf (r) _1, \\ mathbf (r) _2, \\ ldots \\ mathbf (r) _n) \\]

then domineering all equations on \\ (\\ mathbf (v) _i \\) can be obtained

\\ [\\ FRAC (D) (DT) \\ Sum_i \\ Frac (MV_i ^ 2) (2) \u003d - \\ Sum_i \\ FRAC (D \\ MathBF (R) _i) (DT) \\ CDOT \\ NABLA_I U (\\ MathBF (R ) _1, \\ mathbf (r) _2, \\ ldots \\ mathbf (r) _n) + \\ Sum_i \\ FRAC (D \\ MathBF (R) _i) (DT) \\ Cdot \\ MathBF (f) _i ^ d \\]

The first sum in the right part of the equation is nothing but the time derivative from a complex function, and therefore, if you enter the designations

\\ [E \u003d \\ Sum_i \\ FRAC (MV_i ^ 2) (2) + U (\\ MathBF (R) _1, \\ MathBF (R) _2, \\ ldots \\ mathbf (r) _n) \\]

and call this magnitude mechanical energy, then integrating the equations from time to T \u003d 0 until T, you can get

\\ [E (T) - E (0) \u003d \\ int_l \\ mathbf (f) _i ^ d \\ cdot d \\ mathbf (r) _i \\]

where integration is carried out along the trajectories of motion of material points.

Thus, the change in the mechanical energy of the system of material points over time is equal to the work of non-optical forces.

The law of energy conservation in the mechanics is performed only for systems in which all the forces are potential.

Energy conservation law for electromagnetic field

In electrodynamics, the law of conservation of energy is historically formulated in the form of the Pinging theorem.

The change in the electromagnetic energy concluded in a certain amount, for a certain time interval is equal to the stream of electromagnetic energy through the surface that limits this volume, and the amount of thermal energy released in this amount taken with the opposite sign.

$ \\ FRAC (D) (DT) \\ int_ (V) \\ Omega_ (EM) DV \u003d - \\ OINT _ (\\ Partial V) \\ VEC (S) D \\ VEC (\\ Sigma) - \\ int_v \\ VEC (j) \\ The electromagnetic field has the energy that is distributed in the space occupied by the field. When changing the characteristics of the field, the distribution of energy changes. It flows from one area of \u200b\u200bspace to another, moving, possibly in other forms.

For the electromagnetic field is a consequence of field equations. Law of energy conservation Inside some closed surface

S, Limiting spacev. occupied by the field contains energyW. - Energy of the electromagnetic field:W \u003d.

E i 2/2 +Σ(εε 0 H i 2/2)μμ 0 ΔV i.If there are currents in this volume, the electric field produces work on moving charges, per unit of time

N \u003d

I.Σ J̅ I × E̅ i. ΔV i.this is the magnitude of the energy of the field that goes into other forms. From the Maxwell equations it follows that

ΔW + nΔt \u003d -Δt∮ S.S̅ × n̅. DA

where Δw. - change in the energy of the electromagnetic field in the volume under consideration during Δt, A vector S̅. = E̅. × H̅called pointing vector.

it energy conservation law in electrodynamics.

Via a small area of \u200b\u200bthe magnitude ΔA. with a single normal vector n̅. per unit time in the direction of the vector n̅. Energy flows S̅. × n̅.ΔA Where S̅. - Value vector Pointing within the site. The sum of these values \u200b\u200bin all elements of the closed surface (designated by the integral sign), standing in the right part of equality, is an energy flowing from the volume bounded by the surface, per unit of time (if this value is negative, then the energy flows into the volume). Vector Pointing Determines the flow of energy of the electromagnetic field through the pad, it is different from zero everywhere, where the vector product of the vectors of electric and magnetic fields is different from zero.

Three main directions of the practical application of electricity can be distinguished: transmission and transformation of information (radio, television, computers), impulse transmission and momentum (electric motors), transformation and power transmission (electric generators and power lines). Both the pulse and energy are transferred to the field through an empty space, the presence of the medium only leads to losses. Energy is not transmitted by wires! The wires with the current are needed to form electrical and magnetic fields of such a configuration so that the energy flow, defined by the Pointing vectors in all points of space, was directed from the energy source to the consumer. Energy can be transmitted without wires, then electromagnetic waves are then transferred. (The inner energy of the Sun decreases, is carried out by electromagnetic waves, mainly light. Thanks to the part of this energy, life is maintained on earth.)

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To make calculations, you must resolve the elements of ActiveX! Energy - The most universal value for describing physical phenomena.
Energy is the maximum amount of work that the body can make.
There are several types of energy. For example, in mechanics:

Potential energy of gravity
Determined by height h..

- potential energy of elastic deformation,
Deformation is determined h..

- kinetic energy - energy movement energy,
Determined by the speed of the body v..

Energy can be transmitted from alone bodies to others, as well as turn from one species to another.

- Full mechanical energy.

Law of energy conservation: in closed Complete body system energy does not change With any interactions within this system tel. The law imposes restrictions on processes in nature. Nature does not allow the emergence of energy nowhere and disappearing to nowhere. Perhaps it turns out only like this: how much one body loses energy, so much another acquires; How many of the same type of energy decreases, as much as the other type is added.
In the mechanics, it is necessary to pay attention to three quantities to determine the types of energy: height lifting body over ground h, deformation x., speed Body v..

Potential energy is rather an abstract value, because any item that has some height above the surface of the Earth will already have a certain amount of potential energy. It is calculated by multiplying the speed of free fall to height above the ground, as well as for mass. If the body moves, you can talk about the presence of kinetic energy.

Formula and description of the law

The result of the addition of kinetic and potential energy in a system closed from an external influence of the system, the parts of which interact due to the forces of elasticity and grave, does not change - this is the law of energy conservation in classical mechanics. The formula of this law looks like this: EK1 + EP1 \u003d EC2 + EP2. Here, EC1 is the kinetic energy of a certain physical body at a particular point in time, and EP1 is potential. The same is true for EC2 and EP2, but in the next time interval. But this law is faithful only if the system in which it acts is closed (or conservative). This suggests that the value of complete mechanical energy does not change when only conservative forces act on the system. When non-conservative forces come into effect, part of the energy changes, taking other forms. Such systems were called dissipative. The energy of energy conservation works when the forces outside do not act on the body.

An example of the manifestation of the law

One of the typical examples illustrating the described law is to conduct experience with the steel ball, which falls on the slab from the same substance or on the glass, bouncing from it about the same height, where it was before the fall. This effect is achieved due to the fact that when the subject moves, the energy is converted several times. Initially, the value of potential energy begins to strive for zero, while kinetic increases, but after a collision, it becomes the potential energy of the elastic deformation of the ball.

It continues until the end of the subject in which he starts his movement up due to the strength of the elastic deformation of both the slab and the fallen object. But at the same time the potential energy of gravity takes into business. Since the ball is understood at about the same height with which he fell, the kinetic energy in it is the same. In addition, the sum of all energies acting on a moving subject remains the same during the entire described process, confirming the law of preserving full mechanical energy.

Elastic deformation - what is it?

In order to fully understand the above example, it is necessary to deal with more detail with what the potential energy of an elastic body is a concept means possessing elasticity that allows you to return to the state of peace when deforming all parts of this system, making some work on the bodies with which physical comes an object. The form of the elastic strength does not affect the form of the trajectory of the movement, since the work performed by them depends only on the position of the body at the beginning and at the end of the movement.

When external forces act

But the law of conservation does not apply to real processes in which friction force participates. In an example, you can bring the subject falling to the Earth. During the collision, the kinetic energy and the resistance force increase. This process does not fit into the framework of mechanics, since due to increasing resistance the body temperature increases. From the foregoing it follows that the law of energy conservation in the mechanics has serious limitations.

Thermodynamics

The first law of thermodynamics is reading: the difference between the amount of heat accumulated due to the work performed on the external objects is equal to the change in the internal energy of the non-conservative thermodynamic system.

But this statement is most often formulated in another form: the amount of heat obtained by the thermodynamic system is spent on the work performed on the objects outside the system, as well as to change the amount of energy inside the system. According to this law, it cannot disappear, turning from one form to another. From this it follows that the creation of a machine that does not consume energy (the so-called perpetual engine) is impossible, since the system will need energy from the outside. But many still persistently tried to create it, without taking into account the law of conservation of energy.

An example of the manifestation of the law of conservation in thermodynamics

Experiments show that thermodynamic processes cannot be reversed. An example of this may be the contact of bodies having a different temperature, at which the heated heat will be heated, and the second is to take it. The reverse process is not possible in principle. Another example is the gas transition from one part of the vessel to another after the opening of partitions between them, provided that the second part is empty. The substance in this case will never start moving in the opposite direction spontaneously. From the above, it follows that any thermodynamic system tends to the state of rest, in which its individual parts are in equilibrium and have the same temperature and pressure.

Hydrodynamics

The application of the law of conservation in hydrodynamic processes is expressed in principle described by Bernoulli. It sounds like this: the amount of pressure of both kinesthetic and potential energy per unit volume is one and the same in any particular point of flow of fluid or gas. This means that to measure the flow rate, it is enough to measure the pressure at two points. It is done, as a rule, a pressure gauge. But Bernoulli law is valid only if the liquid under consideration has a viscosity that is zero. In order to describe the current of real liquids, Bernoulli integral is used, which assumes the addition of the components that take into account the resistance.

Electrodynamics

During the electrification of the two bodies, the number of electrons in them remains unchanged, due to which the positive charge of one body is equal to the module to the negative charge of the other. Thus, the law of conservation of an electric charge says that in an electrically isolated system, the amount of charges does not change its bodies. This statement is true and then when the charged particles are experiencing transformations. Thus, when 2 neutrally charged particles are faced, the sum of their charges still remains zero, since together with a negatively charged particle appears and positively charged.

Conclusion

The law of conservation of mechanical energy, impulse and moment is fundamental physical laws related to the homogeneity of time and its isotropy. They are not limited to the framework of mechanics and apply both to the processes occurring in outer space and to quantum phenomena. The laws of preservation allow us to obtain data on various mechanical processes without their study using the equations of motion. If a process in the theory ignores these principles, then it is pointless to carry out experiments in this case, as they will be non-responding.