Energy does not disappear. Universal Energy Conservation Law

03.08.2019

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. DAΔw.

where - change in the energy of the electromagnetic field in the volume under consideration during Δt, A vector S̅. E̅ = H̅ × calledpointing 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 Energy flows per unit time in the direction of the vector N̅. E̅ × ΔAWhere - Value E̅ Vector Pointing {!LANG-8cba48da69e59ff7364fe08f123a1918!} 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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If the bodies constituting closed mechanical system, interact with each other only through the forces of grave and elasticity, the work of these forces is equal to change potential energy teltaken with the opposite sign:

By the kinetic energy theorem, this work is equal to a change in the kinetic energy of bodies (see 1.19):

Consequently:

The sum of the kinetic and potential energy of bodies constituting a closed system and interacting with each other through the forces and strength of elasticity remains unchanged.

This statement expresses the law of conservation of energy in mechanical processes . It is a consequence of Newton's laws. Amount E. = E.k. + E.p. Call complete mechanical energy . The law of conservation of mechanical energy is performed only when the body in a closed system interacts with each other conservative forces, that is, by the forces for which the concept of potential energy can be introduced.

An example of the application of the law of energy conservation - finding the minimum strength of an easy non-aggressive thread holding the body mass m. With its rotation in the vertical plane (the task of the Guigens). Fig. 1.20.1 explains the solution to this problem.

The law of energy conservation for the body in the upper and lower points of the trajectory is written in the form:

We draw attention to the fact that the force of tension of the thread is always perpendicular to the velocity of the body; Therefore, it does not make work.

With a minimum rotation speed, the thread tension at the upper point is zero and, therefore, the centripetal acceleration of the body at the top point is reported only by force of gravity:

From these ratios follow:

The centripetal acceleration at the bottom point is created by the forces and directed to the opposite directions:

It follows that with the minimum body velocity at the top point, the thread tension at the bottom will be in the module equal to the module

The strength of the thread should obviously exceed this value.

It is very important to note that the law of conservation of mechanical energy allowed the relationship between the coordinates and body speeds at two different points of the trajectory without analyzing the law of the body's movement at all intermediate points. The application of the law of conservation of mechanical energy can greatly simplify the solution of many tasks.

In real conditions, almost always on moving bodies, along with the forces, the forces of elasticity and other conservative forces are the forces of friction or the strength of resistance of the medium.

The friction force is not conservative. The work of friction force depends on the length of the path.

If there is a friction force between the bodies that make up a closed system, mechanical energy is not saved. Part of the mechanical energy turns into the internal energy of the body (heating).

With any physical interactions, the energy does not occur and does not disappear. It only turns out of one form to another.

This experimentally established fact expresses the fundamental law of nature - the law of conservation and turning energy .

One of the consequences of the law of conservation and the transformation of energy is the approval of the impossibility of creating a "perpetual motor" (Perpetuum Mobile) - a machine that could be carried out indefinitely, without consuming energy (Fig. 1.20.2).

History keeps a considerable number of Eternal Engine projects. In some of them, the errors of the inventor are obvious, in other these errors are disguised as a complex design of the device, and it is very difficult to understand why this machine will not work. Besle attempts to create a "eternal engine" continue and in our time. All these attempts are doomed to failure, since the law of preserving and turning the energy "prohibits" obtaining work without energy costs.

If the body of some mass M moved under the action of the attached forces, and its speed has changed from to the strength made a certain job a.

The work of all applied forces is equal to the work of the resultant force

Between the change in body velocity and the work performed on the body applied to the body, there is a connection. This connection is the easiest to establish this connection, considering the movement of the body along a straight line under the action of constant strength In this case, the velocity of the speed of movement of speed and acceleration is directed along one straight, and the body performs a straight equivalent movement. By sending the coordinate axis along direct movement, it is possible to consider F, S, υ and A as algebraic values \u200b\u200b(positive or negative depending on the direction of the corresponding vector). Then the work of the force can be written as a \u003d fs. With an equalized movement, the movement s is expressed by the formula

This expression shows that the work performed by the force (or the resultant all forces) is associated with changing the square of the speed (and not the speed).

The physical quantity equal to half the body mass on the square of its speed is called kinetic energy Body:

This statement is called Theorem on kinetic energy. The theorem on kinetic energy is valid and in the general case when the body moves under the action of a changing force, the direction of which does not coincide with the direction of movement.

Kinetic energy is the energy of movement. The kinetic energy of the body M mass moving at a speed is equal to the work that the force applied to the resting body should make this speed:

In physics, along with the kinetic energy or energy of the movement, the concept plays an important role. potential energy or energy interaction tel.

Potential energy is determined by the mutual position of the bodies (for example, the position of the body relative to the surface of the Earth). The concept of potential energy can only be introduced for forces, the work of which does not depend on the trajectory of the movement and is determined only by the initial and final provisions of the body.. Such forces are called conservative.

The work of conservative forces on a closed trajectory is zero. This statement explains the drawing below

The body of conservatism is the power of gravity and the power of elasticity. For these strength, you can enter the concept of potential energy.

If the body moves near the surface of the Earth, the strength permanent and direction is valid for it, the force of this force depends only on the vertical movement of the body. At any area of \u200b\u200bthe way, the work of gravity can be recorded in the projections of the movement vector on the Oy axis, directed vertically upwards:

This work is equal to a change in some physical magnitude of MGH, taken with the opposite sign. This physical value is called Potential energy bodies in the field of gravity

E p \u003d Mgh.

It is equal to the work that the power of gravity is performed when lowering the body to the zero level.

If we consider the movement of bodies in the field of land at considerable distances from it, then when determining the potential energy, it is necessary to take into account the dependence of the force from the distance to the center of the Earth (the law of the World Trial). For the strength of the world, the potential energy is conveniently counted from an infinitely remote point, that is, to assume the potential energy of the body in an infinitely remote point equal to zero. The formula expresses the potential body energy mass M at a distance of R from the center of the Earth, has the form:

where M - Earth weight, G - gravitational constant.

The concept of potential energy can be introduced for the force of elasticity. This force also has the property of conservatism. Stretching (or squeezing) spring, we can do it in various ways.

You can simply lengthen the spring by x, or first lengthen it by 2x, and then reduce the elongation to the value of X, etc. In all these cases, the force of elasticity performs the same work that depends only on the extension of the spring X in the final state If the original spring was undecked out. This work is equal to the work of the external force A taken with the opposite sign:

Potential energy of elastically deformed body It is equal to the work of the force of elasticity during the transition from this state to a state with zero deformation.

If the spring has already been deformed in the initial state, and its elongation was x 1, then, when switching to a new state with elongation X 2, the force of elasticity will work, equal to changing the potential energy taken with the opposite sign:

Potential energy with elastic deformation is the energy of the interaction of individual parts of the body among themselves by means of elastic forces.

The property of conservatism along with the force of gravity and the force of elasticity has some other types of forces, for example, the power of electrostatic interaction between charged bodies. The friction force does not have this property. The work of friction force depends on the path traveled. The concept of potential energy for the friction force is impossible to enter.

E k1 + E p1 \u003d E k2 + E P2.

The sum of the kinetic and potential energy of bodies constituting a closed system and interacting with each other through the forces and strength of elasticity remains unchanged.

This statement expresses The law of conservation of energy in mechanical processes. It is a consequence of Newton's laws. The amount e \u003d e k + e p is called Complete mechanical energy. The law of conservation of mechanical energy is performed only when the body in a closed system interacts with each other conservative forces, that is, by the forces for which the concept of potential energy can be introduced.

An example of the application of the law of energy conservation is to find the minimum strength of a light non-aggressive thread that holds the body weighing M during its rotation in the vertical plane (task H. Guigens). Fig. 1.20.1 explains the solution to this problem.

The law of energy conservation for the body in the upper and lower points of the trajectory is written in the form:

From these ratios follow:

The strength of the thread should obviously exceed this value.

The friction force is not conservative. The work of friction force depends on the length of the path.

If there is a friction force between the bodies that make up a closed system, mechanical energy is not saved. Part of the mechanical energy turns into the internal energy of the body (heating).

With any physical interactions, the energy does not occur and does not disappear. It only turns out of one form to another.

This experimentally established fact expresses the fundamental law of nature - The law of conservation and turning energy.

One of the consequences of the law of conservation and transformation of energy is the statement on the impossibility of creating a "perpetual engine" (Perpetuum Mobile) - a car that could have been done indefinitely, not spending energy

The law of conservation of energy, for any closed system, complete mechanical energy remains constant for any interactions of bodies inside the system. That is, energy does not arise from nowhere and does not disappear. It only passes from one form to another. This is true for closed systems in which the energy does not come from outside, and does not leave the system outside.

An approximate example of a closed system can serve as a drop in cargo relative to large mass, and small sizes on the ground from a small height. Suppose the load is fixed at some height. At the same time, it has potential energy. This energy depends on its mass and height on which the body is located.

Formula 1 - potential energy.

The kinetic energy of the cargo at the same time is zero, since the body is at rest. That is, the body speed is zero. At the same time, no third-party forces act on the system. In this case, only the force of gravity acting on the cargo is important for us.

Formula 2 - kinetic energy.

Next, the body is released, and it goes into a free drop. At the same time, its potential energy decreases. Since the height of the body is reduced above the ground. Also increases kinetic energy. Due to the fact that the body began to move and acquired some speed. The cargo moves to the ground with an acceleration of free fall, which means with the passage of a certain distance, its kinetic energy increases due to an increase in speed.

Figure 1 - Free Falling Body.

Since the load with small sizes, the air resistance is quite small and the energy for its overcoming of small and it can be neglected. The velocity of the body is not high and at a small distance does not reach the moment when it is balable by friction about the air and the acceleration stops.

At the time of the collision with the Earth, the kinetic energy is maximum. Since the body has the maximum speed for it. And the potential energy is zero, since the body has reached the surface of the Earth and the height is zero. That is, what happens, the maximum potential energy at the upper point, as it moves into the kinetic, which in turn reaches the maximum at the bottom point. But the sum of all energies in the system during the movement remains constant. As far as the potential energy has decreased, the kinetic one has increased.

Formula 3 - the total energy of the system.

Now, if the cargo attack the parachute. Thus, we will increase the force of friction about the air, and the system ceases to be closed. As before, the cargo moves to the ground, but its speed remains constant. Since gravity is balanced by the force of friction about the air surface of the parachute. Thus, the potential energy decreases with a decrease in height. And kinetic, throughout the fall remains constant. Since the mass of the body and its speed is unchanged.

Figure 2 - Slow Falling Body.

Surplus the potential energy arising from the decrease in body height is spent on overcoming air friction forces. Thereby reducing its final reduction rate. That is, the potential energy passes into the thermal, heating surface of the parachute and the ambient air.

The law of conservation of energy argues that the energy of the body never disappears and does not appear again, it can only turn from one species to another. This law is universal. In various sections of physics, it has its own wording. Classical mechanics considers the law of conservation of mechanical energy.

The complete mechanical energy of a closed system of physical bodies, between which conservative forces apply is the value of constant. So the law of conservation of energy in Newton's mechanics is formulated.

Closed, or isolated, it is customary to consider the physical system to which external forces do not apply. It does not exchange energy with the surrounding space, and its own energy, which it has, remains unchanged, that is, is preserved. In such a system, there are only internal forces, and the bodies interact with each other. It can only convert potential energy into kinetic and vice versa.

The simplest example of a closed system is a sniper rifle and bullet.

Types of mechanical forces

Forces that act inside the mechanical system are made to divide on conservative and non-mechanical.

Conservative The forces whose work does not depend on the trajectory of the body movement to which they are applied, but is determined only by the initial and end position of this body. Conservative forces are also called potential. The work of such forces on a closed contour is zero. Examples of conservative forces - gravity, strength of elasticity.

All other forces are called unconscious. These include friction force and resistance force. They are also called dissipative Forces. These forces, with any movements in a closed mechanical system, do negative work, and with their action, the complete mechanical energy of the system decreases (dissipates). It goes into other, not mechanical types of energy, for example, in warmth. Therefore, the law of conservation of energy in a closed mechanical system can be performed only if there are no non-conservative forces in it.

The total energy of the mechanical system consists of kinetic and potential energy and is their sum. These types of energies can turn into each other.

Potential energy

Potential energy Call the energy of the interaction of physical bodies or their parts between themselves. It is determined by their mutual location, that is, the distance between them, and is equal to the work that needs to be done to move the body from the point of reference to another point in the field of action of the conservative forces.

Potential energy has any fixed physical body raised to some height, since the strength of gravity acts on it, which is conservative. Such an energy has water on the edge of a waterfall, sledge on the top of the mountain.

Where did this energy come from? While the physical body was raised to height, they were working and spent energy. Here is this energy and stuck in the raised body. And now this energy is ready for work.

The magnitude of the potential energy of the body is determined by the height on which the body is relative to some initial level. For the point of reference, we can accept any point selected by us.

If we consider the position of the body relative to the ground, then the potential energy of the body on the surface of the earth is zero. And at the height h. It is calculated by the formula:

E p \u003d m ɡ h. ,

where m. - body mass

ɡ - acceleration of gravity

h. - Height of the center of mass body relative to the earth

ɡ \u003d 9.8 m / s 2

When falling the body with height h 1. to height h 2. The strength of gravity makes a job. This work is equal to the change in potential energy and has a negative value, since the magnitude of the potential energy when the body falls decreases.

A \u003d - ( E p2 - E p1) \u003d - δ E P. ,

where E p1 - Potential body energy at height h 1. ,

E p2 - potential body energy at height h 2. .

If the body is raised at some height, they make work against gravity. In this case, it has a positive value. And the magnitude of the potential energy of the body increases.

Potential energy has a elastically deformed body (compressed or stretched spring). Its value depends on the rigidity of the spring and on which length it was squeezed or stretched, and is determined by the formula:

E p \u003d k · (Δx) 2/2 ,

where k. - the stiffness coefficient

Δx. - elongation or body compression.

Potential energy of the spring can work.

Kinetic energy

Translated from Greek "Kinema" means "movement". The energy that the physical body receives due to its movement is called kinetic. Its value depends on the speed of movement.

Rolling on the field a soccer ball, having risen from the mountain and continuing sledges, released from Luke Arrow - all of them possess kinetic energy.

If the body is at rest, its kinetic energy is zero. Once the strength or several strength works on the body, it will start moving. And since the body moves, the power acting on it makes work. Work force under the influence of which the body from the state of rest will move into motion and change its speed from zero to ν , called kinetic energy body mass m. .

If at the initial moment of time the body was already in motion, and its speed was meaning ν 1. and at the end moment she was equal ν 2. The work performed by the force or forces acting on the body will be equal to the increment of the kinetic energy of the body.

∆ E k \u003d. E k 2 - E K 1

If the direction of force coincides with the direction of movement, then positive work is performed, and the kinetic energy of the body increases. And if the force is directed towards, the opposite direction of movement, then negative work is performed, and the body gives kinetic energy.

Mechanical energy conservation law

E. K. 1 + E p1= E. K. 2 + E p2.

Any physical body located at some height has potential energy. But when you fall, it starts to lose this energy. Where does she go? It turns out that it does not disappear anywhere, but turns into the kinetic energy of the same body.

Suppose , at some height, the load is fixedly fixed. Its potential energy at this point is equal to the maximum value.If we let go, he will start falling at a certain speed. Consequently, it will begin to acquire kinetic energy. But at the same time, its potential energy will begin to decrease. At the point of fall, the kinetic energy of the body will reach the maximum, and the potential will decrease to zero.

The potential energy of the ball abandoned from height is reduced, and the kinetic energy increases. Sledge, which are at rest on top of the mountain, have potential energy. Their kinetic energy at this moment is zero. But when they start rolling down, the kinetic energy will increase, and the potential to decrease the same value. And the sum of their values \u200b\u200bwill remain unchanged. The potential energy of the apple hanging on the tree, when falling, turns into its kinetic energy.

These examples clearly confirm the law of conservation of energy, which says that the total energy of the mechanical system is the magnitude of constant . The value of the total energy of the system does not change, and the potential energy goes into kinetic and vice versa.

What magnitude the potential energy will decrease, the kinetic will increase on the same. Their amount will not change.

Equality is fair for a closed system of physical bodies
E k1 + E p1 \u003d E k2 + E p2,
- Value E k1, E p1 - kinetic and potential energy of the system to any interaction, E k2, E p2 - Appropriate energies after it.

The process of converting kinetic energy into a potential and on the contrary can be seen by watching the swinging pendulum.

Click on the picture

Being in an extremely right position, the pendulum seems to be free. At this point, its height above the reference point is maximal. Consequently, the maximum and potential energy. And the kinetic is zero, since it does not move. But the next moment the pendulum begins to move down. It increases its speed, and, it means, kinetic energy increases. But the height decreases, and potential energy decreases. At the bottom point, it will become equal to zero, and the kinetic energy will reach the maximum value. The pendulum will fly this point and starts to rise up to the left. It will increase its potential energy, and the kinetic will decrease. Etc.

To demonstrate the transformations of Energy Isaac, Newton came up with a mechanical system called cradle Newton or Newton's balls .

Click on the picture

If you reject to the side, then let go of the first ball, then its energy and impulse are transferred to the last through three intermediate balls that will remain fixed. And the last ball will deviate at the same speed and rises to the same height as the first. Then the last ball will transmit its energy and the pulse through the intermediate balls the first and so on.

The ball reserved to the side has the maximum potential energy. His kinetic energy at this moment is zero. Starting the movement, it loses potential energy and acquires the kinetic, which at the time of the collision with the second ball reaches the maximum, and the potential becomes equal to zero. Next, kinetic energy is transmitted by the second, then the third, fourth and fifth balls. The latter, having received kinetic energy, begins to move and rises to the same height, on which there was a first ball at the beginning of the movement. Its kinetic energy at this moment is zero, and the potential is equal to the maximum value. Further, it begins to fall and just also transmits the energy with the balls in the reverse order.

So continues quite a long time and could continue indefinitely if there were no non-conservative forces. But in reality, dissipative forces act in the system, under the action of which the balls lose their energy. Gradually reduces their speed and amplitude. And in the end, they stop. This confirms that the law of energy conservation is performed only in the absence of non-consistent forces.