Law of conservation of mechanical energy. Law of conservation of energy Law of conservation of kinetic energy formulation

Conservation Law 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 can occur kinetic energy into potential and back in equivalent quantities 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 movement 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 rigid 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:

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 expended heat 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.

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4.1. Loss of mechanical energy and work of non-potential forces. Efficiency Cars

If the law of conservation of mechanical energy were fulfilled in real installations (such as the Oberbeck machine), then many calculations could be done based on the equation:

T O + P O = T(t) + P(t) , (8)

Where: T O + P O = E O- mechanical energy at the initial moment of time;

T(t) + P(t) = E(t)- mechanical energy at some subsequent point in time t.

Let's apply formula (8) to the Oberbeck machine, where you can change the height of the load on the thread (the center of mass of the rod part of the installation does not change its position). We will lift the load to a height h from the lower level (where we consider P=0). Let the system with the lifted load initially be at rest, i.e. T O = 0, P O = mgh(m- mass of load on the thread). After releasing the load, movement begins in the system and its kinetic energy is equal to the sum of the energy of the translational motion of the load and the rotational motion of the rod part of the machine:

T= + , (9)

Where - speed of forward movement of the load;

, J- angular speed of rotation and moment of inertia of the rod part

For the moment of time when the load drops to the zero level, from formulas (4), (8) and (9) we obtain:

m gh=
, (10)

Where
,  0k - linear and angular velocities at the end of the descent.

Formula (10) is an equation from which (depending on the experimental conditions) the speeds can be determined And , mass m, moment of inertia J, or height h.

However, formula (10) describes ideal type installations in which there are no frictional or resistance forces during the movement of parts. If the work done by such forces is not zero, then the mechanical energy of the system is not conserved. Instead of equation (8), in this case one should write:

T O +P O = T(t) + P(t) + A s , (11)

Where A s- the total work of non-potential forces during the entire period of movement.

For the Oberbeck machine we get:

m gh =
, (12)

Where ,  k - linear and angular velocities at the end of the descent in the presence of energy losses.

In the installation studied here, friction forces act on the axis of the pulley and the additional block, as well as atmospheric resistance forces during the movement of the load and the rotation of the rods. The work of these non-potential forces noticeably reduces the speed of movement of machine parts.

As a result of the action of non-potential forces, part of the mechanical energy is converted into other forms of energy: internal energy and radiation energy. At the same time, work As is exactly equal to the total value of these other forms of energy, i.e. The fundamental, general physical law of conservation of energy is always fulfilled.

However, in installations where the movement of macroscopic bodies occurs, mechanical energy loss, determined by the amount of work As. This phenomenon exists in all real machines. For this reason, a special concept is introduced: coefficient useful action- efficiency. This coefficient determines the ratio useful work to stored (used) energy.

In Oberbeck's machine, useful work is equal to the total kinetic energy at the end of the descent of the load onto the thread, and efficiency. is determined by the formula:

efficiency.= (13)

Here P O = mgh- stored energy consumed (converted) into kinetic energy of the machine and into energy losses equal to As, T To- total kinetic energy at the end of the load descent (formula (9)).

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 transforms into other, non-mechanical forms of energy, for example, 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 . Magnitude total energy 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.

1. Consider the free fall of a body from a certain height h relative to the Earth's surface (Fig. 77). At the point A the body is motionless, therefore it has only potential energy. At the point B on high h 1 the body has both potential energy and kinetic energy, since the body at this point has a certain speed v 1 . At the moment of touching the surface of the Earth, the potential energy of the body is zero; it has only kinetic energy.

Thus, during the fall of a body, its potential energy decreases, and its kinetic energy increases.

Total mechanical energy E called the sum of potential and kinetic energies.

E = E n + E To.

2. Let us show that the total mechanical energy of a system of bodies is conserved. Let us consider once again the fall of a body onto the surface of the Earth from a point A exactly C(see Fig. 78). We will assume that the body and the Earth represent a closed system of bodies in which only conservative forces act, in in this case gravity.

At the point A the total mechanical energy of a body is equal to its potential energy

E = E n = mgh.

At the point B the total mechanical energy of the body is equal to

E = E p1 + E k1.
E n1 = mgh 1 , E k1 = .

Then

E = mgh 1 + .

Body speed v 1 can be found using the kinematics formula. Since the movement of a body from a point A exactly B equals

s = h – h 1 = , then = 2 g(h – h 1).

Substituting this expression into the formula for total mechanical energy, we get

E = mgh 1 + mg(h – h 1) = mgh.

Thus, at the point B

E = mgh.

At the moment of touching the surface of the Earth (point C) the body has only kinetic energy, therefore, its total mechanical energy

E = E k2 = .

The speed of the body at this point can be found using the formula = 2 gh, taking into account that the initial speed of the body is zero. After substituting the expression for speed into the formula for total mechanical energy, we obtain E = mgh.

Thus, we obtained that at the three considered points of the trajectory, the total mechanical energy of the body is equal to the same value: E = mgh. We will arrive at the same result by considering other points of the body’s trajectory.

The total mechanical energy of a closed system of bodies, in which only conservative forces act, remains unchanged during any interactions of the bodies of the system.

This statement is the law of conservation of mechanical energy.

3. In real systems, friction forces act. Thus, when a body falls freely in the example considered (see Fig. 78), the force of air resistance acts, therefore the potential energy at the point A more total mechanical energy at a point B and at the point C by the amount of work done by the force of air resistance: D E = A. In this case, the energy does not disappear; part of the mechanical energy is converted into the internal energy of the body and air.

4. As you already know from the 7th grade physics course, to facilitate human labor, various machines and mechanisms are used, which, having energy, perform mechanical work. Such mechanisms include, for example, levers, blocks, cranes, etc. When work is performed, energy is converted.

Thus, any machine is characterized by a quantity that shows what part of the energy transferred to it is used usefully or what part of the perfect (total) work is useful. This quantity is called efficiency(efficiency).

The efficiency h is a value equal to the ratio of useful work A n to full work A.

Efficiency is usually expressed as a percentage.

h = 100%.

5. Example of problem solution

A parachutist weighing 70 kg separated from the motionless hanging helicopter and, having flown 150 m before the parachute opened, acquired a speed of 40 m/s. What is the work done by air resistance?

ical energy at a given altitude is zero. Having flown the distance s= h, the parachutist acquired kinetic energy, and his potential energy at this level became zero. Thus, in the second position, the total mechanical energy of the paratrooper is equal to his kinetic energy:

E = E k = .

Potential energy of a skydiver E n when separated from the helicopter is not equal to the kinetic E k, since the force of air resistance does work. Hence,

A = E To - E P;

A =– mgh.

A=– 70 kg 10 m/s 2,150 m = –16,100 J.

The work has a minus sign because it is equal to the loss of total mechanical energy.

Answer: A= –16,100 J.

Self-test questions

1. What is called total mechanical energy?

2. Formulate the law of conservation of mechanical energy.

3. Is the law of conservation of mechanical energy satisfied if a friction force acts on the bodies of the system? Explain your answer.

4. What does efficiency show?

Task 21

1. A ball of mass 0.5 kg is thrown vertically upward at a speed of 10 m/s. What is the potential energy of the ball at its highest point?

2. An athlete weighing 60 kg jumps from a 10-meter platform into the water. What is equal to: the potential energy of the athlete relative to the surface of the water before the jump; its kinetic energy upon entering water; its potential and kinetic energy at a height of 5 m relative to the surface of the water? Neglect air resistance.

3. Determine the efficiency of an inclined plane 1 m high and 2 m long when a load weighing 4 kg moves along it under the influence of a force of 40 N.

Chapter 1 highlights

1. Types of mechanical movement.

2. Basic kinematic quantities (Table 2).

table 2

3. Basic equations of motion (Table 3).

Table 3

4. Basic traffic schedules.

Table 4

5. Basic dynamic quantities.

Table 5

6. Basic laws of mechanics

Table 6

7. Forces in mechanics.

8. Basic energy quantities.

Table 7