Energy- the most universal value for describing physical phenomena.
Energy is the maximum amount of work that a body can do.
There are several types of energy. For example, in mechanics:

Potential energy of gravity,
determined by height h.

- Potential energy of elastic deformation,
determined by the amount of deformation X.

- Kinetic energy - the energy of the movement of bodies,
determined by the speed of the body v.

Energy can be transferred from one body to another, and can also be transformed from one type to another.

- Complete mechanical energy.

Law of energy conservation: in closed body system complete energy does not change at any interactions within this system of bodies. The law imposes restrictions on the course of processes in nature. Nature does not allow energy to appear from nowhere and disappear into nowhere. Perhaps it turns out only this way: how much one body loses energy, how much another acquires; how much one type of energy decreases, so much is added to another type.
In mechanics, to determine the types of energy, it is necessary to pay attention to three quantities: height lifting the body above the earth h, deformation x, speed body v.

Energy- a universal measure of various forms of movement and interaction.

A change in the mechanical motion of a body is caused by forces that act on it from other bodies. In order to quantitatively describe the process of energy exchange between interacting bodies, the concept is introduced in mechanics work force.

If a body is moving in a straight line and a constant force is acting on it F, making some angle α with the direction of movement, then the work of this force is equal to the projection of the force F s on the direction of movement (F s = Fcosα), multiplied by the corresponding displacement of the point of application of the force:

If we take a section of the trajectory from point 1 to point 2, then the work on it is equal to the algebraic sum of elementary works on separate infinitesimal sections of the path. Therefore, this sum can be reduced to the integral

Unit of work - joule(J): 1 J - work done by a force of 1 N on a path of 1 m (1 J = 1 N m).
To characterize the rate of doing work, the concept of power is introduced:
Over time dt force F does the job F d r, and the power developed by this force at a given time
i.e., it is equal to the scalar product of the force vector and the velocity vector with which the point of application of this force moves; N is a scalar value.
Power unit - watt(W): 1 W - power at which 1 J work is done in 1 s (1 W = 1 J / s)

Kinetic and potential energy.

Kinetic energy of a mechanical system is the energy of the mechanical movement of the system under consideration.
Strength F, acting on a body at rest and setting it in motion, does work, and the energy of the moving body increases by the amount of work expended. So the work done by the force F on the path that the body has traveled during the increase in speed from 0 to v, is spent on increasing the kinetic energy dT of the body, i.e.

Using Newton's second law and multiplying by the displacement d r we get
(1)
From formula (1) it can be seen that kinetic energy depends only on the mass and speed of the body (or point), i.e., the kinetic energy of the body depends only on the state of its motion.
Potential energy- mechanical energy body systems, which is determined by the nature of the forces of interaction between them and their mutual arrangement.
Let the interaction of bodies on each other be carried out by force fields (for example, fields of elastic forces, fields of gravitational forces), which are characterized by the fact that the work done by the forces acting in the system when moving the body from the first position to the second does not depend on the trajectory along which it movement has occurred, but depends only on initial and final positions of the system. Such fields are called potential, and the forces acting in them - conservative. If the work of a force depends on the trajectory of the body moving from one position to another, then such a force is called dissipative; an example of a dissipative force is the force of friction.
The specific form of the function P depends on the form of the force field. For example, the potential energy of a body of mass m, raised to a height h above the Earth's surface, is (7)

The total mechanical energy of a system is the energy of mechanical motion and interaction:
i.e., equal to the sum of the kinetic and potential energies.

Law of Conservation of Energy.

i.e., the total mechanical energy of the system remains constant. Expression (3) is law of conservation of mechanical energy: in a system of bodies between which only conservative forces act, the total mechanical energy is conserved, i.e. does not change over time.

Mechanical systems, on the bodies of which only conservative forces (both internal and external) act, are called conservative systems , and we formulate the law of conservation of mechanical energy as follows: in conservative systems, total mechanical energy is conserved.
9. Impact of absolutely elastic and inelastic bodies.

Hit is the collision of two or more bodies interacting for a very short time.

Upon impact, the body is deformed. The concept of impact implies that the kinetic energy of the relative motion of the impacting bodies is converted for a short time into the energy of elastic deformation. During the impact, there is a redistribution of energy between the colliding bodies. Experiments show that the relative velocity of bodies after a collision does not reach its value before the collision. This is explained by the fact that there are no ideally elastic bodies and ideally smooth surfaces. The ratio of the normal component of the relative velocity of the bodies after the impact to the normal component of the relative velocity of the bodies before the impact is called recovery factorε: ε = ν n "/ν n where ν n" - after impact; ν n - before impact.

If for colliding bodies ε=0, then such bodies are called absolutely inelastic, if ε=1 - absolutely elastic. In practice, for all bodies 0<ε<1. Но в некоторых случаях тела можно с большой степенью точности рассматривать либо как абсолютно неупругие, либо как абсолютно упругие.

strike line called a straight line passing through the point of contact of the bodies and perpendicular to the surface of their contact. The beat is called central, if the colliding bodies before the impact move along a straight line passing through the centers of their masses. Here we consider only central absolutely elastic and absolutely inelastic impacts.
Absolutely elastic impact- a collision of two bodies, as a result of which no deformations remain in both bodies participating in the collision and the entire kinetic energy of the bodies before the impact after the impact is again converted into the original kinetic energy.
For an absolutely elastic impact, the law of conservation of kinetic energy and the law of conservation of momentum are satisfied.

Absolutely inelastic impact- the collision of two bodies, as a result of which the bodies are connected, moving further as a single whole. Absolutely inelastic impact can be demonstrated using plasticine (clay) balls that move towards each other.

If forces, friction and resistance forces do not act in a closed system, then the sum of the kinetic and potential energies of all bodies of the system remains constant.

Consider an example of the manifestation of this law. Let the body raised above the Earth have potential energy E 1 = mgh 1 and speed v 1 directed downwards. As a result of free fall, the body moved to a point with a height h 2 (E 2 = mgh 2), while its speed increased from v 1 to v 2. Therefore, its kinetic energy has increased from

Let's write the equation of kinematics:

Multiplying both sides of the equation by mg, we get:

After transformation we get:

Consider the restrictions that were formulated in the law of conservation of total mechanical energy.

What happens to mechanical energy if a friction force acts in the system?

In real processes, where friction forces act, there is a deviation from the law of conservation of mechanical energy. For example, when a body falls to the Earth, the kinetic energy of the body first increases as the speed increases. The resistance force also increases, which increases with increasing speed. Over time, it will compensate for gravity, and in the future, with a decrease in potential energy relative to the Earth, the kinetic energy does not increase.

This phenomenon is beyond the scope of mechanics, since the work of resistance forces leads to a change in body temperature. The heating of bodies under the action of friction is easy to detect by rubbing the palms together.

Thus, in mechanics, the law of conservation of energy has rather rigid boundaries.

The change in thermal (or internal) energy occurs as a result of the work of friction or resistance forces. It is equal to the change in mechanical energy. Thus, the sum of the total energy of bodies during interaction is a constant value (taking into account the transformation of mechanical energy into internal energy).

Energy is measured in the same units as work. As a result, we note that there is only one way to change mechanical energy - to do work.

Energy- a measure of the movement of matter in all its forms. The main property of all types of energy is interconvertibility. The amount of energy that a body possesses is determined by the maximum work that the body can do, having used up its energy completely. Energy is numerically equal to the maximum work that the body can do, and is measured in the same units as the work. During the transition of energy from one type to another, it is necessary to calculate the energy of the body or system before and after the transition and take their difference. This difference is called work:

Thus, the physical quantity characterizing the ability of a body to perform work is called energy.

The mechanical energy of a body can be due either to the movement of the body at a certain speed, or to the presence of the body in a potential field of forces.

Kinetic energy.

The energy possessed by a body due to its motion is called kinetic. The work done on the body is equal to the increment of its kinetic energy.

Let's find this work for the case when the resultant of all forces applied to the body is equal to .

The work done by the body due to kinetic energy is equal to the loss of this energy.

Potential energy.

If other bodies act on the body at each point in space, then the body is said to be in a field of forces or a force field.

If the lines of action of all these forces pass through one point - the force center of the field - and the magnitude of the force depends only on the distance to this center, then such forces are called central, and the field of such forces is called central (gravitational, electric field of a point charge).

The field of forces constant in time is called stationary.

A field in which the lines of action of forces are parallel straight lines located at the same distance from each other is homogeneous.

All forces in mechanics are divided into conservative and non-conservative (or dissipative).

Forces whose work does not depend on the shape of the trajectory, but is determined only by the initial and final position of the body in space, are called conservative.

The work of conservative forces along a closed path is zero. All central forces are conservative. The forces of elastic deformation are also conservative forces. If only conservative forces act in the field, the field is called potential (gravitational fields).

Forces whose work depends on the shape of the path are called non-conservative (friction forces).

Potential energy is the energy possessed by bodies or body parts due to their relative position.

The concept of potential energy is introduced as follows. If the body is in a potential field of forces (for example, in the gravitational field of the Earth), each point of the field can be associated with some function (called potential energy) so that the work A 12, performed over the body by the forces of the field when it moves from an arbitrary position 1 to another arbitrary position 2, was equal to the decrease of this function on the path 1®2:

,

where and are the values ​​of the potential energy of the system in positions 1 and 2.

In each specific problem, it is agreed to consider the potential energy of a certain position of the body equal to zero, and take the energy of other positions relative to the zero level. The specific form of the function depends on the nature of the force field and the choice of the zero level. Since the zero level is chosen arbitrarily, it can have negative values. For example, if we take as zero the potential energy of a body located on the surface of the Earth, then in the field of gravity forces near the earth's surface, the potential energy of a body of mass m, raised to a height h above the surface, is (Fig. 5).

where is the displacement of the body under the action of gravity;

The potential energy of the same body lying at the bottom of a well with depth H is equal to

In the considered example, it was about the potential energy of the Earth-body system.

Potential energy of gravity - the energy of a system of bodies (particles) due to their mutual gravitational attraction.

For two gravitating point bodies with masses m 1 and m 2, the potential energy of gravity is:

,

where \u003d 6.67 10 -11 - gravitational constant,

r is the distance between the centers of mass of the bodies.

The expression for the potential energy of gravity is obtained from Newton's law of gravity, provided that for infinitely distant bodies the gravitational energy is 0. The expression for the gravitational force is:

On the other hand, according to the definition of potential energy:

Then .

Potential energy can be possessed not only by a system of interacting bodies, but by a single body. In this case, the potential energy depends on the relative position of the body parts.

Let us express the potential energy of an elastically deformed body.

The potential energy of elastic deformation, if we assume that the potential energy of an undeformed body is zero;

where k- coefficient of elasticity, x- deformation of the body.

In the general case, a body can simultaneously possess both kinetic and potential energies. The sum of these energies is called full mechanical energy body: .

The total mechanical energy of a system is equal to the sum of its kinetic and potential energies. The total energy of the system is equal to the sum of all types of energy that the system possesses.

The law of conservation of energy is the result of a generalization of many experimental data. The idea of ​​this law belongs to Lomonosov, who stated the law of conservation of matter and motion, and the quantitative formulation was given by the German physician Mayer and the naturalist Helmholtz.

Law of conservation of mechanical energy: in the field of only conservative forces, the total mechanical energy remains constant in an isolated system of bodies. The presence of dissipative forces (friction forces) leads to dissipation (scattering) of energy, i.e. converting it into other types of energy and violating the law of conservation of mechanical energy.

The law of conservation and transformation of total energy: the total energy of an isolated system is a constant value.

Energy never disappears and does not appear again, but only changes from one form to another in equivalent quantities. This is the physical essence of the law of conservation and transformation of energy: the indestructibility of matter and its motion.

An example of the law of conservation of energy:

In the process of falling, potential energy is converted into kinetic energy, and the total energy, equal to mgH, remains constant.

If only conservative forces act on the system, then we can introduce for it the concept potential energy. Any arbitrary position of the system, characterized by setting the coordinates of its material points, we will conditionally take as zero. The work done by conservative forces during the transition of the system from the considered position to zero is called potential energy of the system in first position

The work of conservative forces does not depend on the transition path, and therefore the potential energy of the system at a fixed zero position depends only on the coordinates of the material points of the system in the considered position. In other words, potential energy of the systemUis a function of only its coordinates.

The potential energy of the system is not uniquely defined, but up to an arbitrary constant. This arbitrariness cannot affect physical conclusions, since the course of physical phenomena may depend not on the absolute values ​​of the potential energy itself, but only on its difference in various states. The same differences do not depend on the choice of an arbitrary constant.

conservative, then BUT 12 = BUT 1O2 = BUT 1O + BUT O2 = BUT 1O - BUT 2O. By definition of potential energy U 1 = A 1O , U 2 = A 2O. In this way,

A 12 = U 1 – U 2 , (3.10)

those. the work of conservative forces is equal to the decrease in the potential energy of the system.

Same job BUT 12 , as shown earlier in (3.7), can be expressed in terms of the kinetic energy increment by the formula

BUT 12 = To 2 – To 1 .

Equating their right-hand sides, we get To 2 – To 1 = U 1 – U 2 , whence

To 1 + U 1 = To 2 + U 2 .

The sum of the kinetic and potential energies of a system is called its total energy E. In this way, E 1 = E 2 , or

E K+U= const. (3.11)

In a system with only conservative forces, the total energy remains unchanged. Only transformations of potential energy into kinetic energy and vice versa can occur, but the total energy supply of the system cannot change. This position is called the law of conservation of energy in mechanics.

Let us calculate the potential energy in some simplest cases.

a) Potential energy of a body in a uniform gravitational field. If a material point located at a height h, will fall to the zero level (i.e. the level for which h= 0), then gravity will do work A=mgh. Therefore, on top h material point has potential energy U=mgh+C, where FROM is an additive constant. An arbitrary level can be taken as zero, for example, the level of the floor (if the experiment is carried out in a laboratory), sea level, etc. Constant FROM is equal to potential energy at zero level. Setting it equal to zero, we get

U=mgh. (3.12)

b) Potential energy of a stretched spring. The elastic forces that occur when a spring is stretched or compressed are central forces. Therefore, they are conservative, and it makes sense to talk about the potential energy of a deformed spring. They call her elastic energy. Denote by x spring extension, those. difference x = l – l 0 lengths of the spring in the deformed and undeformed states. Elastic force F depends on stretch. If stretching x not very large, then it is proportional to it: F = – kx(Hooke's law). When the spring returns from the deformed to the undeformed state, the force F does the job

.

If the elastic energy of the spring in the undeformed state is assumed to be equal to zero, then

. (3.13)

c) Potential energy of gravitational attraction of two material points. According to Newton's law of universal gravitation, the gravitational force of attraction of two point bodies is proportional to the product of their masses mm and is inversely proportional to the square of the distance between them:

,(3.14)

where Gis the gravitational constant.

The force of gravitational attraction, as a central force, is conservative. It makes sense for her to talk about potential energy. When calculating this energy, one of the masses, for example M, can be considered as stationary, and the other as moving in its gravitational field. When moving mass m from infinity, gravitational forces do work

,

where r- distance between masses M and m in final state.

This work is equal to the loss of potential energy:

.

Usually potential energy at infinity U is taken equal to zero. With such an agreement

. (3.15)

The quantity (3.15) is negative. This has a simple explanation. Attractive masses have maximum energy at an infinite distance between them. In this position, the potential energy is considered to be zero. In any other position, it is smaller, i.e. negative.

Let us now assume that, along with conservative forces, dissipative forces also act in the system. The work of all forces BUT 12 during the transition of the system from position 1 to position 2 is still equal to the increment of its kinetic energy To 2 – To one . But in the case under consideration, this work can be represented as the sum of the work of conservative forces
and work of dissipative forces
. The first work can be expressed in terms of the decrease in the potential energy of the system:
. That's why

.

Equating this expression to the increment of kinetic energy, we obtain

, (3.16)

where E=K+U is the total energy of the system. Thus, in the case under consideration, the mechanical energy E system does not remain constant, but decreases, since the work of dissipative forces
negative.