Formulas on the topic of the law of conservation of energy. Law of conservation of mechanical energy. The essence of the law of conservation of energy

The principle of conservation of energy is absolutely accurate; no cases of its violation have been recorded. It is a fundamental law of nature from which others follow. Therefore, it is important to understand it correctly and be able to apply it in practice.

Fundamental Principle

There is no general definition for the concept of energy. There are different types of it: kinetic, thermal, potential, chemical. But this doesn’t clarify the point. Energy is a certain quantitative characteristic that, no matter what happens, remains constant for the entire system. You can watch the sliding puck stop and declare: the energy has changed! In fact, no: mechanical energy turned into thermal energy, part of which was dissipated in the air, and part of it went to melting the snow.

Rice. 1. Conversion of work spent on overcoming friction into thermal energy.

Mathematician Emmy Noether was able to prove that the constancy of energy is a manifestation of the uniformity of time. This quantity is invariant with respect to transport along the time coordinate, since the laws of nature do not change over time.

We will consider total mechanical energy (E) and its types - kinetic (T) and potential (V). If we add them up, we get an expression for the total mechanical energy:

$E = T + V_((q))$

By writing potential energy as $V_((q))$, we indicate that it depends solely on the configuration of the system. By q we mean generalized coordinates. These can be x, y, z in a rectangular Cartesian coordinate system, or they can be any others. Most often they deal with the Cartesian system.

Rice. 2. Potential energy in the gravitational field.

The mathematical formulation of the law of conservation of energy in mechanics looks like this:

$\frac (d)(dt)(T+V_((q))) = 0$ – the time derivative of the total mechanical energy is zero.

In its usual, integral form, the formula for the law of conservation of energy is written as follows:

In mechanics, restrictions are imposed on the law: the forces acting on the system must be conservative (their work depends only on the configuration of the system). In the presence of non-conservative forces, for example, friction, mechanical energy is converted into other types of energy (thermal, electrical).

Thermodynamics

Attempts to create a perpetual motion machine were especially characteristic of the 18th and 19th centuries - the era when the first steam engines were made. Failures, however, led to positive result: the first law of thermodynamics was formulated:

$Q = \Delta U + A$ – the heat expended is spent on doing work and changing internal energy. This is nothing more than the law of conservation of energy, but for heat engines.

Rice. 3. Scheme of a steam engine.

Tasks

A load weighing 1 kg, suspended on a thread L = 2 m, was deflected so that the lifting height turned out to be equal to 0.45 m, and was released without an initial speed. What will be the tension in the thread at the lowest point?

Solution:

Let's write Newton's second law in projection onto the y-axis at the moment when the body passes the bottom point:

$ma = T – mg$, but since $a = \frac (v^2)(L)$, it can be rewritten in a new form:

$m \cdot \frac (v^2)(L) = T – mg$

Now let’s write down the law of conservation of energy, taking into account that at the initial position the kinetic energy is equal to zero, and at the lowest point - potential energy equal to zero:

$m \cdot g \cdot h = \frac (m \cdot v^2)(2)$

Then the tension force of the thread is:

$T = \frac (m \cdot 2 \cdot g \cdot h)(L) + mg = 10 \cdot (0.45 + 1) = 14.5 \: H$

What have we learned?

During the lesson, we looked at a fundamental property of nature (uniformity of time), from which the law of conservation of energy follows, and looked at examples of this law in different branches of physics. To secure the material, we solved the problem with a pendulum.

Test on the topic

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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 the potential and kinetic energy, and also that a body can have both potential and kinetic energy. Before talking about the law of conservation of mechanical energy, let us remember what it is total energy. 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 we considered this example 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 happen? 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:

The change in internal energy can be denoted by . As you know, there are several types of energy. In addition to mechanical energy, there is also thermal, internal energy.

The law of conservation of energy states that the energy of a body never disappears or appears again, it can only be transformed from one type to another. This law is universal. It has its own formulation in various branches of physics. Classical mechanics considers the law of conservation of mechanical energy.

The total mechanical energy of a closed system of physical bodies between which conservative forces act is a constant value. This is how Newton's law of conservation of energy is formulated.

A closed, or isolated, physical system is considered to be one that is not affected by external forces. There is no exchange of energy with the surrounding space, and the own energy that it possesses remains unchanged, that is, it is conserved. In such a system, only internal forces act, and the bodies interact with each other. Only the transformation of potential energy into kinetic energy and vice versa can occur in it.

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

Types of mechanical forces

The forces that act inside a mechanical system are usually divided into conservative and non-conservative.

Conservative forces are considered whose work does not depend on the trajectory of the body to which they are applied, but is determined only by the initial and final position of this body. Conservative forces are also called potential. The work done by such forces along a closed loop is zero. Examples of conservative forces – gravity, elastic force.

All other forces are called non-conservative. These include friction force and resistance force. They are also called dissipative forces. These forces, during any movements in a closed mechanical system, perform negative work, and under their action, the total mechanical energy of the system decreases (dissipates). It turns into other, non-mechanical forms of energy, for example, into heat. Therefore, the law of conservation of energy in a closed mechanical system can be fulfilled only if there are no non-conservative forces in it.

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

Potential energy

Potential energy is called the energy of interaction of physical bodies or their parts with each other. It is determined by their relative position, that is, the distance between them, and is equal to the work that needs to be done to move the body from the reference point to another point in the field of action of conservative forces.

Any motionless physical body raised to some height has potential energy, since it is acted upon by gravity, which is a conservative force. Such energy is possessed by water at the edge of a waterfall, and a sled on a mountain top.

Where did this energy come from? While the physical body was raised to a height, work was done and energy was expended. It is this energy that is stored in the raised body. And now this energy is ready to do work.

The amount of potential energy of a body is determined by the height at which the body is located relative to some initial level. We can take any point we choose as a reference point.

If we consider the position of the body relative to the Earth, then the potential energy of the body on the Earth’s surface is zero. And on top h it is calculated by the formula:

E p = m ɡ h ,

Where m - body mass

ɡ - acceleration of gravity

h – height of the body’s center of mass relative to the Earth

ɡ = 9.8 m/s 2

When a body falls from a height h 1 up to height h 2 gravity does work. This work is equal to the change in potential energy and has negative meaning, since the amount of potential energy decreases when a body falls.

A = - ( E p2 – E p1) = - ∆ E p ,

Where E p1 – potential energy of the body at height h 1 ,

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

If the body is raised to a certain height, then work is done against the forces of gravity. In this case it has a positive value. And the amount of potential energy of the body increases.

An elastically deformed body (compressed or stretched spring) also has potential energy. Its value depends on the stiffness of the spring and on the length to which it was compressed or stretched, and is determined by the formula:

E p = k·(∆x) 2 /2 ,

Where k – stiffness coefficient,

∆x – lengthening or compression of the body.

The potential energy of a spring can do work.

Kinetic energy

Translated from Greek, “kinema” means “movement.” The energy that a physical body receives as a result of its movement is called kinetic. Its value depends on the speed of movement.

Rolling across the field soccer ball, a sled rolling down a mountain and continuing to move, an arrow shot from a bow - they all have kinetic energy.

If a body is at rest, its kinetic energy is zero. As soon as a force or several forces act on a body, it will begin to move. And since the body moves, the force acting on it does work. The work of force, under the influence of which a body from a state of rest goes into motion and changes 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 mattered ν 1 , and at the final moment it was equal to ν 2 , then the work done by the force or forces acting on the body will be equal to the increase in the kinetic energy of the body.

∆ E k = E k 2 - Ek 1

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

Law of conservation of mechanical energy

Ek 1 + E p1= E k 2 + E p2

Any physical body located at some height has potential energy. But when it falls, it begins 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 , the load is fixedly fixed at a certain height. Its potential energy at this point is equal to its maximum value. If we let go of it, it will begin to fall 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 impact, the kinetic energy of the body will reach a maximum, and the potential energy will decrease to zero.

The potential energy of a ball thrown from a height decreases, but its kinetic energy increases. A sled at rest on a mountain top has potential energy. Their kinetic energy at this moment is zero. But when they begin to roll down, the kinetic energy will increase, and the potential energy will decrease by the same amount. And the sum of their values ​​will remain unchanged. The potential energy of an apple hanging on a tree when it falls is converted into its kinetic energy.

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

By what amount the potential energy decreases, the kinetic energy increases by the same amount. Their amount will not change.

For a closed system of physical bodies the following equality is true:
E k1 + E p1 = E k2 + E p2,
Where E k1, E p1 - kinetic and potential energies of the system before any interaction, E k2 , E p2 - the corresponding energies after it.

The process of converting kinetic energy into potential energy and vice versa can be seen by watching a swinging pendulum.

Click on the picture

Being in the extreme right position, the pendulum seems to freeze. At this moment its height above the reference point is maximum. Therefore, the potential energy is also maximum. And the kinetic value is zero, since it is not moving. But the next moment the pendulum begins to move downwards. Its speed increases, and, therefore, its kinetic energy increases. But as the height decreases, so does the potential energy. At the lowest point it will become equal to zero, and the kinetic energy will reach its maximum value. The pendulum will fly past this point and begin to rise up to the left. Its potential energy will begin to increase, and its kinetic energy will decrease. Etc.

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

Click on the picture

If you deflect to the side and then release the first ball, its energy and momentum will be transferred to the last through three intermediate balls, which will remain motionless. And the last ball will deflect at the same speed and rise to the same height as the first. Then the last ball will transfer its energy and momentum through the intermediate balls to the first, etc.

The ball moved to the side has maximum potential energy. Its kinetic energy at this moment is zero. Having started to move, it loses potential energy and gains kinetic energy, which at the moment of collision with the second ball reaches a maximum, and potential energy becomes equal to zero. Next, the kinetic energy is transferred to the second, then the third, fourth and fifth balls. The latter, having received kinetic energy, begins to move and rises to the same height at which the first ball was at the beginning of its movement. Its kinetic energy at this moment is zero, and its potential energy is equal to its maximum value. Then it begins to fall and transfers energy to the balls in the same way in the reverse order.

This continues for quite a long time and could continue indefinitely if non-conservative forces did not exist. But in reality, dissipative forces act in the system, under the influence of which the balls lose their energy. Their speed and amplitude gradually decrease. And eventually they stop. This confirms that the law of conservation of energy is satisfied only in the absence of non-conservative forces.

Energy is a scalar quantity. The SI unit of energy is the Joule.

Kinetic and potential energy

There are two types of energy - kinetic and potential.

DEFINITION

Kinetic energy- this is the energy that a body possesses due to its movement:

DEFINITION

Potential energy is energy that is determined by the relative position of bodies, as well as the nature of the interaction forces between these bodies.

Potential energy in the Earth's gravitational field is the energy due to the gravitational interaction of a body with the Earth. It is determined by the position of the body relative to the Earth and is equal to the work of moving the body from this provision to zero level:

Potential energy is the energy caused by the interaction of body parts with each other. It is equal to the work of external forces in tension (compression) of an undeformed spring by the amount:

A body can simultaneously possess both kinetic and potential energy.

The total mechanical energy of a body or system of bodies is equal to the sum of the kinetic and potential energies of the body (system of bodies):

Law of energy conservation

For a closed system of bodies, the law of conservation of energy is valid:

In the case when a body (or a system of bodies) is acted upon by external forces, for example, the law of conservation of mechanical energy is not satisfied. In this case, the change in the total mechanical energy of the body (system of bodies) is equal to the external forces:

The law of conservation of energy allows us to establish a quantitative relationship between various forms movement of matter. Just like , it is valid not only for, but also for all natural phenomena. The law of conservation of energy says that energy in nature cannot be destroyed just as it cannot be created from nothing.

In the most general view The law of conservation of energy can be formulated as follows:

• Energy in nature does not disappear and is not created again, but only transforms from one type to another.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

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 with time:

Mechanical systems whose bodies are acted upon only by conservative forces (internal and external) are called conservative systems.

Law of conservation of mechanical energy can be formulated as follows: in conservative systems, the total mechanical energy is conserved.

The law of conservation of mechanical energy is associated with the uniformity of time. The homogeneity of time is manifested in the fact that physical laws are invariant with respect to the choice of the time reference point.

There is another type of system - dissipative systems, in which mechanical energy is gradually reduced by conversion to other (non-mechanical) forms of energy. This process is called dissipation (or scattering) of energy.

In conservative systems, the total mechanical energy remains constant. Only transformations of kinetic energy into potential energy and back in equivalent quantities can occur so that the total energy remains unchanged.

This law is not just a law quantitative conservation of energy, and the law of conservation and transformation of energy, expressing and high-quality side of the mutual transformation of various forms of movement into each other.

The law of conservation and transformation of energy - fundamental law of nature, it is valid both for systems of macroscopic bodies and for systems of microbodies.

In a system in which they also operate non-conservative forces, for example, friction forces, total mechanical energy of the system not saved. However, when mechanical energy “disappears,” an equivalent amount of another type of energy always appears.

14. Moment of inertia of a rigid body. Moment of impulse. Steiner's theorem.

moment of inertia system (body) relative to a given axis is a physical quantity equal to the sum of the products of the masses of n material points of the system by the squares of their distance to the axis in question:

The summation is performed over all elementary masses m into which the body is divided.

In the case of a continuous distribution of masses, this sum is reduced to an integral: where integration is carried out over the entire volume of the body.

The value r in this case is a function of the position of the point with coordinates x, y, z. Moment of inertia- magnitude additive: the moment of inertia of a body relative to a certain axis is equal to the sum of the moments of inertia of parts of the body relative to the same axis.

If the moment of inertia of a body relative to an axis passing through its center of mass is known, then the moment of inertia relative to any other parallel axis is determined Steiner's theorem:

the moment of inertia of a body J relative to an arbitrary axis is equal to the moment of its inertia Jc relative to a parallel axis passing through the center of mass C of the body, added to the product of the body mass and the square of the distance a between the axes:

Examples of moments of inertia of some bodies (bodies are considered homogeneous, m is the mass of the body):

Momentum (momentum) material point A relative to a fixed point O is a physical quantity determined by the vector product:

where r is the radius vector drawn from point O to point A;

p = mv - momentum of a material point;

L is a pseudo-vector, its direction coincides with the direction of translational motion of the right propeller as it rotates from to.

Modulus of the angular momentum vector:

where a is the angle between vectors r and p;

l - arm of vector p relative to point O.

Momentum relative to the fixed axis z is called a scalar quantity Lz equal to the projection onto this axis of the angular momentum vector defined relative to an arbitrary point O of this axis. The angular momentum Lz does not depend on the position of point O on the z axis.

When rotating absolutely solid around a fixed axis z, each individual point of the body moves in a circle of constant radius r, with a certain speed Vi. The velocity Vi and momentum mV are perpendicular to this radius, i.e. the radius is an arm of the vector. Therefore, the angular momentum of an individual particle is equal to:

Momentum of a rigid body relative to the axis is the sum of the angular momentum of individual particles:

Using the formula, we find that the angular momentum of a solid body relative to an axis is equal to the product of the moment of inertia of the body relative to the same axis and the angular velocity: