Law of conservation of mechanical energy. Mechanical energy

If the bodies that make up closed mechanical system, interact with each other only through the forces of gravity and elasticity, then the work of these forces is equal to the difference in potential energy:

According to the kinetic energy theorem, this work is equal to the change in the kinetic energy of bodies:

Hence:

or . (5.16)

The sum of kinetic and potential energy of bodies that make up a closed system and interact with each other through gravitational and elastic forces remains unchanged.

The sum E = E k + E p is the total mechanical energy. Got law of conservation of complete mechanical energy :

The law of conservation of mechanical energy is satisfied only when bodies in a closed system interact with each other by conservative forces, that is, forces for which the concept of potential energy can be introduced.

IN real conditions Almost always, moving bodies, along with gravitational forces, elastic forces and other conservative forces, are acted upon by frictional forces or environmental resistance forces.

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

If friction forces act between the bodies that make up a closed system, then mechanical energy is not conserved. Part of the mechanical energy is converted into internal energy of bodies (heating).

For any physical interactions energy neither arises nor disappears. It just changes from one form to another.

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

The law of conservation of mechanical energy and the law of conservation of momentum make it possible to find solutions to mechanical problems in cases where active forces unknown. An example of this type of problem is the impact interaction of bodies.

An impact (or collision) is usually called a short-term interaction of bodies, as a result of which their speeds experience significant changes. During a collision of bodies, short-term impact forces act between them, the magnitude of which, as a rule, is unknown. Therefore, impact interaction cannot be considered directly using Newton's laws. The application of the laws of conservation of energy and momentum in many cases makes it possible to exclude the collision process itself from consideration and obtain a connection between the velocities of bodies before and after the collision, bypassing all intermediate values ​​of these quantities.

In mechanics, two models of impact interaction are often used - absolutely elastic and absolutely inelastic impacts.

An absolutely inelastic impact is an impact interaction in which bodies connect (stick together) with each other and move on as one body.

With absolutely no elastic impact mechanical energy is not conserved. It partially or completely turns into the internal energy of bodies (heating).

An absolutely elastic impact is a collision in which the mechanical energy of a system of bodies is conserved.

With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is satisfied.

In all phenomena occurring in nature, energy neither appears nor disappears. It only transforms from one type to another, while its meaning remains the same.

Law of energy conservation- a fundamental law of nature, which is that for an isolated physical system a scalar can be introduced physical quantity, which is a function of the system parameters and is called energy, which is conserved over time. Since the law of conservation of energy does not apply to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it can be called not a law, but the principle of conservation of energy.

Law of conservation of mechanical energy

In mechanics, the law of conservation of energy states that in a closed system of particles, total energy, which is the sum of kinetic and potential energy and does not depend on time, that is, it is an 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 forces of interaction between bodies for which the law of conservation of mechanical energy is satisfied are called conservative forces. The law of conservation of mechanical energy is not satisfied for friction forces, since in the presence of friction forces, mechanical energy is converted into thermal energy.

Mathematical formulation

Evolution mechanical system material points with masses \(m_i\) according to Newton’s second law satisfies the system of equations

\[ m_i\dot(\mathbf(v)_i) = \mathbf(F)_i \]

Where
\(\mathbf(v)_i \) are the velocities of material points, and \(\mathbf(F)_i \) are the forces acting on these points.

If we submit the forces as the sum of potential forces \(\mathbf(F)_i^p \) and non-potential forces \(\mathbf(F)_i^d \) , and write the potential forces in the form

\[ \mathbf(F)_i^p = - \nabla_i U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]

then, multiplying all equations by \(\mathbf(v)_i \) we can get

\[ \frac(d)(dt) \sum_i \frac(mv_i^2)(2) = - \sum_i \frac(d\mathbf(r)_i)(dt)\cdot \nabla_i U(\mathbf(r )_1, \mathbf(r)_2, \ldots \mathbf(r)_N) + \sum_i \frac(d\mathbf(r)_i)(dt) \cdot \mathbf(F)_i^d \]

The first sum on the right side of the equation is nothing more than the time derivative of a complex function, and therefore, if we introduce the notation

\[ E = \sum_i \frac(mv_i^2)(2) + U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]

and name this value mechanical energy, then by integrating the equations from time t=0 to time t, we can obtain

\[ E(t) - E(0) = \int_L \mathbf(F)_i^d \cdot d\mathbf(r)_i \]

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

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

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

Law of conservation of energy for the electromagnetic field

In electrodynamics, the law of conservation of energy is historically formulated in the form of Poynting's theorem.

The change in electromagnetic energy contained in a certain volume over a certain time interval is equal to the flow of electromagnetic energy through the surface limiting this volume and the amount of thermal energy released in given volume, taken with the opposite sign.

$ \frac(d)(dt)\int_(V)\omega_(em)dV=-\oint_(\partial V)\vec(S)d\vec(\sigma)-\int_V \vec(j)\ cdot \vec(E)dV $

An electromagnetic field has energy that is distributed in the space occupied by the field. When the field characteristics change, the energy distribution also changes. It flows from one area of ​​space to another, possibly transforming into other forms. Law of energy conservation for the electromagnetic field is a consequence of the field equations.

Inside some closed surface S, limiting the amount of space V occupied by the field contains energy W— electromagnetic field energy:

W=Σ(εε 0 E i 2 / 2 +μμ 0 H i 2 / 2)ΔV i .

If there are currents in this volume, then the electric field produces work on moving charges equal to

N=Σ ij̅ i ×E̅ i . ΔV i .

This is the amount of field energy that transforms into other forms. From Maxwell's equations it follows that

ΔW + NΔt = -Δt∮ SS̅ × n̅. dA,

Where ΔW— change in the energy of the electromagnetic field in the volume under consideration over time Δt, a vector S̅ = E̅ × H̅ called Poynting vector.

This law of conservation of energy in electrodynamics.

Through a small area the size ΔA with unit normal vector n̅ per unit time in the direction of the vector n̅ energy flows S̅ × n̅.ΔA, Where S̅- meaning Poynting vector within the site. The sum of these quantities over all elements of a closed surface (indicated by the integral sign), standing on the right side of the equality, represents the energy flowing out of the volume bounded by the surface per unit time (if this quantity is negative, then the energy flows into the volume). Poynting vector determines the flow of electromagnetic field energy through the site; it is non-zero wherever the vector product of the electric and magnetic field strength vectors is non-zero.

Three main directions can be distinguished practical application electricity: transmission and transformation of information (radio, television, computers), transmission of impulse and angular momentum (electric motors), transformation and transmission of energy (electric generators and power lines). Both momentum and energy are transferred by the field through empty space, the presence of the environment only leads to losses. Energy is not transmitted through wires! Current-carrying wires are needed to form electric and magnetic fields of such a configuration that the energy flow, determined by Poynting vectors at all points in space, is directed from the energy source to the consumer. Energy can be transmitted without wires; then it is carried by electromagnetic waves. (The internal energy of the Sun decreases and is carried away by electromagnetic waves, mainly light. Thanks to part of this energy, life on Earth is supported.)

Law of Conservation of Mechanical Energy

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

If the bodies that make up closed mechanical system, interact with each other only through the forces of gravity and elasticity, then the work of these forces is equal to the change in the potential energy of the bodies, taken with the opposite sign:

Hence

The sum of kinetic and potential energy of bodies that make up a closed system and interact with each other through gravitational and elastic forces remains unchanged.

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

During any physical interactions, energy does not appear or disappear, but only transforms from one form to another.

b. Taking into account friction

Looking closely at the movement of a ball bouncing on a slab (§ 102), you can find that after each blow the ball rises to a slightly lower height than before (Fig. 170), i.e. that is, the total energy does not remain exactly constant, but gradually decreases; this means that the law of conservation of energy in the form we formulated it is observed in this case only approximately. The reason is that in this experiment frictional forces arise: the resistance of the air in which the ball moves, and internal friction in the material of the ball and the plate itself. In general, in the presence of friction, the law of conservation of mechanical energy is always violated and the sum of the potential and kinetic energies of bodies decreases. Due to this loss of energy, work is done against the friction forces 1).

Reducing the height of the ball's rebound after many reflections from the slab.

For example, when a body falls from a great height, the speed of the body, due to the action of increasing resistance forces of the medium, soon becomes constant (§ 68); kinetic energy the body stops changing, but its potential energy of rising above the ground decreases. Work against the force of air resistance is done by gravity due to the potential energy of the body. Although some kinetic energy is imparted to the surrounding air, it is less than the decrease in the potential energy of the body, and, therefore, the total mechanical energy decreases.

Work against friction forces can also be performed due to kinetic energy. For example, when a boat moves, which is pushed away from the shore of a pond, the potential energy of the boat remains constant, but due to the resistance of the water, the speed of the boat, i.e., its kinetic energy, decreases, and the increase in the kinetic energy of the water observed in this case is less than the decrease in kinetic energy boat energy.

Friction forces between solid bodies act in a similar way. For example, the speed acquired by a load sliding down an inclined plane, and therefore its kinetic energy, is less than that which it would acquire in the absence of friction. You can choose the angle of inclination of the plane so that the load slides evenly. At the same time, its potential energy will decrease, but its kinetic energy will remain constant, and the work against the friction forces will be done due to the potential energy.

In nature, all movements (with the exception of movements in complete emptiness, for example the movements of celestial bodies) are accompanied by friction. Therefore, during such movements, the law of conservation of mechanical energy is violated, and this violation always occurs in one direction - towards a decrease in the total energy.

"In general, in the presence of friction 1. the law of conservation of mechanical energy is always violated and 2. the sum of the potential and kinetic energies of bodies decreases." The second is true. The first is a blatant lie! The law is not broken. Dura lex sed lex.

This video lesson is intended for self-acquaintance with the topic “The Law of Conservation of Mechanical Energy.” First, let's define total energy and a closed system. Then we will formulate the Law of Conservation of Mechanical Energy and consider in which areas of physics it can be applied. We will also define work and learn how to define it by looking at the formulas associated with it.

The topic of the lesson is one of the fundamental laws of nature - law of conservation of mechanical energy.

We previously talked about potential and kinetic energy, and also that a body can have both potential and kinetic energy together. Before talking about the law of conservation of mechanical energy, let us remember what total energy is. Total mechanical energy is the sum of the potential and kinetic energies of a body.

Also remember what is called a closed system. Closed system- this is a system in which there is a strictly defined number of bodies interacting with each other and no other bodies from the outside act on this system.

When we have defined the concept of total energy and a closed system, we can talk about the law of conservation of mechanical energy. So, the total mechanical energy in a closed system of bodies interacting with each other through gravitational forces or elastic forces (conservative forces) remains unchanged during any movement of these bodies.

We have already studied the law of conservation of momentum (LCM):

It often happens that the assigned problems can be solved only with the help of the laws of conservation of energy and momentum.

It is convenient to consider the conservation of energy using the example of a free fall of a body from a certain height. If a body is at rest at a certain height relative to the ground, then this body has potential energy. As soon as the body begins to move, the height of the body decreases, and the potential energy decreases. At the same time, speed begins to increase, and kinetic energy appears. When the body approaches the ground, the height of the body is 0, the potential energy is also 0, and the maximum will be the kinetic energy of the body. This is where the transformation of potential energy into kinetic energy is visible (Fig. 1). The same can be said about the movement of the body in reverse, from bottom to top, when the body is thrown vertically upward.

Rice. 1. Free fall of a body from a certain height

Additional task 1. “On the fall of a body from a certain height”

Problem 1

Condition

The body is at a height from the Earth's surface and begins to fall freely. Determine the speed of the body at the moment of contact with the ground.

Solution 1:

Initial speed of the body. Need to find .

Let's consider the law of conservation of energy.

Rice. 2. Body movement (task 1)

At the top point the body has only potential energy: . When the body approaches the ground, the height of the body above the ground will be equal to 0, which means that the potential energy of the body has disappeared, it has turned into kinetic energy:

According to the law of conservation of energy, we can write:

Body weight is reduced. Transforming the above equation, we obtain: .

The final answer will be: . If we substitute the entire value, we get: .

Answer: .

An example of how to solve a problem:

Rice. 3. Example of a solution to problem No. 1

This problem can be solved in another way, as vertical movement with free fall acceleration.

Solution 2 :

Let us write the equation of motion of the body in projection onto the axis:

When the body approaches the surface of the Earth, its coordinate will be equal to 0:

The gravitational acceleration is preceded by a “-” sign because it is directed against the chosen axis.

Substituting known values, we find that the body fell over time. Now let's write the equation for speed:

Assuming the free fall acceleration to be equal, we obtain:

The minus sign means that the body moves against the direction of the selected axis.

Answer: .

An example of solving problem No. 1 using the second method.

Rice. 4. Example of a solution to problem No. 1 (method 2)

Also, to solve this problem, you could use a formula that does not depend on time:

Of course, it should be noted that this example we considered taking into account the absence of friction forces, which in reality act in any system. Let's turn to the formulas and see how the law of conservation of mechanical energy is written:

Additional task 2

A body falls freely from a height. Determine at what height the kinetic energy is equal to a third of the potential energy ().

Rice. 5. Illustration for problem No. 2

Solution:

When a body is at a height, it has potential energy, and only potential energy. This energy is determined by the formula: . This will be the total energy of the body.

When a body begins to move downward, the potential energy decreases, but at the same time the kinetic energy increases. At the height that needs to be determined, the body will already have a certain speed V. For the point corresponding to the height h, the kinetic energy has the form:

The potential energy at this height will be denoted as follows: .

According to the law of conservation of energy, our total energy is conserved. This energy remains a constant value. For a point we can write the following relation: (according to Z.S.E.).

Remembering that the kinetic energy according to the conditions of the problem is , we can write the following: .

Please note: the mass and acceleration of gravity are reduced, after simple transformations we find that the height at which this relationship is satisfied is .

Answer:

Example of task 2.

Rice. 6. Formalization of the solution to problem No. 2

Imagine that a body in a certain frame of reference has kinetic and potential energy. If the system is closed, then with any change a redistribution has occurred, the transformation of one type of energy into another, but the total energy remains the same in value (Fig. 7).

Rice. 7. Law of conservation of energy

Imagine a situation where a car is moving along a horizontal road. The driver turns off the engine and continues driving with the engine turned off. What happens in this case (Fig. 8)?

Rice. 8. Car movement

IN in this case a car has kinetic energy. But you know very well that over time the car will stop. Where did the energy go in this case? After all, the potential energy of the body in this case also did not change; it was some kind of constant value relative to the Earth. How did the energy change occur? In this case, the energy was used to overcome friction forces. If friction occurs in a system, it also affects the energy of that system. Let's see how the change in energy is recorded in this case.

The energy changes, and this change in energy is determined by the work against the friction force. We can determine the work of the friction force using the formula, which is known from class 7 (force and displacement are directed in opposite directions):

So, when we talk about energy and work, we must understand that each time we must take into account the fact that part of the energy is spent on overcoming friction forces. Work is being done to overcome friction forces. Work is a quantity that characterizes the change in the energy of a body.

To conclude the lesson, I would like to say that work and energy are essentially related quantities through acting forces.

Additional task 3

Two bodies - a block of mass and a plasticine ball of mass - move towards each other with the same speeds (). After the collision, the plasticine ball sticks to the block, the two bodies continue to move together. Determine what part of the mechanical energy turned into the internal energy of these bodies, taking into account the fact that the mass of the block is 3 times greater than the mass of the plasticine ball ().

Solution:

Change internal energy can be designated . As you know, there are several types of energy. In addition to mechanical energy, there is also thermal, internal energy.

 Theory: Energy does not disappear anywhere, it transforms from one type to another, and it does not appear out of nowhere.
Energy can be converted into mechanical work or into.
The total energy of a closed system is a constant value: E=E k +E p

For example: we raise a body weighing 2 kg to a height of 1 meter, at this height the body’s potential E p =mgh=20 J, as the body falls, the height decreases, the potential energy also decreases. At the same time, the speed of the body begins to increase, as a result of which the kinetic energy increases. It turns out that energy goes from potential to kinetic. At the moment of touching the surface, the potential energy is zero, the kinetic energy is maximum and is equal to the same as at the beginning of 20 J. If the body is elastically reflected, then as it rises to a height, the kinetic energy will decrease and turn into potential.

Tasks:  A ball is thrown vertically upward from the surface of the Earth. Air resistance is negligible. When the initial speed of the ball increases by 2 times, the height of the ball rises
  1) will increase by √ 2 times
  2) will increase 2 times
  3) will increase 4 times
  4) will not change

Exercise: A bullet moving at a speed of 600 m/s pierced a board 1.5 cm thick and exited the board with a speed of 300 m/s. Determine the mass of the bullet if the average drag force acting on the bullet in the board is 81 kN.

A body of mass m, thrown vertically upward from the Earth with an initial speed υ 0, rose to a height h 0. Air resistance is negligible. The total mechanical energy of the body at some intermediate height h is equal to

Solution: Since air resistance is negligible, therefore the total energy of the system does not change. The total mechanical energy of the body at some intermediate height h is equal to the energy at the maximum height mgh 0.
Answer: 2
OGE assignment in physics (fipi): The ball moves down the inclined chute without friction. Which of the following statements about the ball's energy is true during this motion?
1) The kinetic energy of the ball increases, its total mechanical energy does not change.
2) The potential energy of the ball increases, its total mechanical energy does not change.
3) Both the kinetic energy and the total mechanical energy of the ball increase.
4) Both the potential energy and the total mechanical energy of the ball decrease.
Solution: As it moves down, the speed of the ball increases. Therefore the kinetic energy increases. Since there is no friction and the system can be considered closed, the total mechanical energy does not change.
Answer: 1
OGE assignment in physics (fipi): A freight car moving along a horizontal track at low speed collides with another car and stops. In this case, the buffer spring is compressed. Which of the following energy transformations occurs in this process?
1) the kinetic energy of the car is converted into potential energy springs
2) the kinetic energy of the car is converted into its potential energy
3) the potential energy of the spring is converted into its kinetic energy
4) the internal energy of the spring is converted into kinetic energy of the car
Solution: At first the car was moving, which means it had kinetic energy. During the collision, the spring compressed, i.e. the kinetic energy of the car is converted into potential energy of the spring