Mechanical work definition. Laws of conservation in mechanics Law of conservation of momentum

If a force acts on a body, then this force does work to move the body. Before defining work during curvilinear motion of a material point, let us consider special cases:

In this case the mechanical work A is equal to:

A= F scos=
,

or A = Fcos× s = F S × s,

WhereF S – projection strength to move. In this case F s = const, And geometric meaning work A is the area of ​​the rectangle constructed in coordinates F S , , s.

Let's plot the projection of force on the direction of movement F S as a function of displacement s. Let us represent the total displacement as the sum of n small displacements
. For small i -th movement
work is equal

or the area of ​​the shaded trapezoid in the figure.

Complete mechanical work to move from a point 1 to the point 2 will be equal to:

.

The value under the integral will represent the elementary work of infinitesimal displacement
:

- basic work.

We divide the trajectory of a material point into infinitesimal movements and work of force by moving a material point from a point 1 to the point 2 defined as a curvilinear integral:

–work in curved motion.

Example 1: Work of gravity
during curvilinear motion of a material point.

.

Next as a constant value can be taken out of the integral sign, and the integral according to the figure will represent the full displacement . .

If we denote the height of a point 1 from the Earth's surface through , and the height of the point 2 through , That

We see that in this case the work is determined by the position of the material point at the initial and final moments of time and does not depend on the shape of the trajectory or path. The work done by gravity along a closed path is zero:
.

Forces whose work on a closed path is zero are calledconservative .

Example 2 : Work done by friction force.

This is an example of a non-conservative force. To show this, it is enough to consider the elementary work of the friction force:

,

those. The work done by the friction force is always a negative quantity and cannot be equal to zero on a closed path. The work done per unit time is called power. If during the time
work is being done
, then the power is equal

–mechanical power.

Taking
in the form

,

we get the expression for power:

.

The SI unit of work is the joule:
= 1 J = 1 N 1 m, and the unit of power is the watt: 1 W = 1 J/s.

Mechanical energy.

Energy is a general quantitative measure of the movement of interaction of all types of matter. Energy does not disappear and does not arise from nothing: it can only pass from one form to another. The concept of energy links together all phenomena in nature. In accordance with the various forms of motion of matter, different types of energy are considered - mechanical, internal, electromagnetic, nuclear, etc.

The concepts of energy and work are closely related to each other. It is known that work is done due to the energy reserve and, conversely, by doing work, it is possible to increase the energy reserve in any device. In other words, work is a quantitative measure of energy change:

.

Energy, like work in SI, is measured in joules: [ E]=1 J.

Mechanical energy is of two types - kinetic and potential.

Kinetic energy (or energy of motion) is determined by the masses and velocities of the bodies in question. Let's consider material point, moving under the influence of force . The work of this force increases the kinetic energy of a material point
. In this case, let us calculate the small increment (differential) kinetic energy:

When calculating
Newton's second law was used
, and also
- module of the velocity of the material point. Then
can be represented as:

-

- kinetic energy of a moving material point.

Multiplying and dividing this expression by
, and given that
, we get

-

- connection between momentum and kinetic energy of a moving material point.

Potential energy ( or the energy of the position of bodies) is determined by the action of conservative forces on the body and depends only on the position of the body .

We have seen that the work done by gravity
with curvilinear motion of a material point
can be represented as the difference in function values
, taken at the point 1 and at the point 2 :

.

It turns out that whenever the forces are conservative, the work of these forces on the path 1
2 can be represented as:

.

Function , which depends only on the position of the body is called potential energy.

Then for elementary work we get

–work equals loss potential energy .

Otherwise, we can say that work is done due to the reserve of potential energy.

Size , equal to the sum of the kinetic and potential energies of the particle, is called the total mechanical energy of the body:

–total mechanical energy of the body.

In conclusion, we note that using Newton’s second law
, kinetic energy differential
can be represented as:

.

Potential energy differential
, as indicated above, is equal to:

.

Thus, if the force – conservative force and there are no other external forces, then , i.e. in this case, the total mechanical energy of the body is conserved.

Basic theoretical information

Mechanical work

The energy characteristics of motion are introduced based on the concept mechanical work or force work. Work done by a constant force F, is called a physical quantity, equal to the product force and displacement modules multiplied by the cosine of the angle between the force vectors F and movements S:

Work is a scalar quantity. It can be either positive (0° ≤ α < 90°), так и отрицательна (90° < α ≤ 180°). At α = 90° the work done by the force is zero. In the SI system, work is measured in joules (J). A joule is equal to the work done by a force of 1 newton to move 1 meter in the direction of the force.

If the force changes over time, then to find the work, build a graph of the force versus displacement and find the area of ​​the figure under the graph - this is the work:

An example of a force whose modulus depends on the coordinate (displacement) is the elastic force of a spring, which obeys Hooke’s law ( F control = kx).

Power

The work done by a force per unit time is called power. Power P(sometimes denoted by the letter N) – physical quantity equal to the work ratio A to a period of time t during which this work was completed:

This formula calculates average power, i.e. power generally characterizing the process. So, work can also be expressed in terms of power: A = Pt(if, of course, the power and time of doing the work are known). The unit of power is called the watt (W) or 1 joule per second. If the motion is uniform, then:

Using this formula we can calculate instant power(power in at the moment time), if instead of speed we substitute the value of instantaneous speed into the formula. How do you know what power to count? If the problem asks for power at a moment in time or at some point in space, then instantaneous is considered. If they ask about power over a certain period of time or part of the route, then look for average power.

Efficiency – coefficient useful action , is equal to the ratio of useful work to expended, or useful power to expended:

Which work is useful and which is wasted is determined from the conditions of a specific task through logical reasoning. For example, if a crane does the work of lifting a load to a certain height, then the useful work will be the work of lifting the load (since it is for this purpose that the crane was created), and the expended work will be the work done by the crane’s electric motor.

So, useful and expended power do not have a strict definition, and are found by logical reasoning. In each task, we ourselves must determine what in this task was the purpose of doing the work ( useful work or power), and what was the mechanism or way of doing all the work (power expended or work).

In general, efficiency shows how efficiently a mechanism converts one type of energy into another. If the power changes over time, then the work is found as the area of ​​the figure under the graph of power versus time:

Kinetic energy

A physical quantity equal to half the product of a body’s mass and the square of its speed is called kinetic energy of the body (energy of movement):

That is, if a car weighing 2000 kg moves at a speed of 10 m/s, then it has kinetic energy equal to E k = 100 kJ and is capable of doing 100 kJ of work. This energy can be converted into heat (when a car brakes, the rubber of the wheels, the road and the brake discs heat up) or can be spent on deforming the car and the body that the car collided with (in an accident). When calculating kinetic energy, it does not matter where the car is moving, since energy, like work, is a scalar quantity.

A body has energy if it can do work. For example, a moving body has kinetic energy, i.e. energy of motion, and is capable of doing work to deform bodies or impart acceleration to bodies with which a collision occurs.

The physical meaning of kinetic energy: in order for a body at rest with a mass m began to move at speed v it is necessary to do work equal to the obtained value of kinetic energy. If the body has a mass m moves at speed v, then to stop it it is necessary to do work equal to its initial kinetic energy. When braking, kinetic energy is mainly (except for cases of impact, when the energy goes to deformation) “taken away” by the friction force.

Kinetic energy theorem: the work done by the resultant force is equal to the change in the kinetic energy of the body:

The theorem on kinetic energy is also valid in the general case, when a body moves under the influence of a changing force, the direction of which does not coincide with the direction of movement. It is convenient to apply this theorem in problems involving acceleration and deceleration of a body.

Potential energy

Along with kinetic energy or energy of motion, the concept plays an important role in physics potential energy or energy of interaction of bodies.

Potential energy is determined by the relative position of bodies (for example, the position of the body relative to the surface of the Earth). The concept of potential energy can be introduced only for forces whose work does not depend on the trajectory of the body and is determined only by the initial and final positions (the so-called conservative forces). The work done by such forces on a closed trajectory is zero. This property is possessed by gravity and elastic force. For these forces we can introduce the concept of potential energy.

Potential energy of a body in the Earth's gravity field calculated by the formula:

The physical meaning of the potential energy of a body: potential energy is equal to the work done by gravity when lowering the body to zero level ( h– distance from the center of gravity of the body to the zero level). If a body has potential energy, then it is capable of doing work when this body falls from a height h to zero level. The work done by gravity is equal to the change in the potential energy of the body, taken from opposite sign:

Often in energy problems one has to find the work of lifting (turning over, getting out of a hole) the body. In all these cases, it is necessary to consider the movement not of the body itself, but only of its center of gravity.

The potential energy Ep depends on the choice of the zero level, that is, on the choice of the origin of the OY axis. In each problem, the zero level is chosen for reasons of convenience. What has a physical meaning is not the potential energy itself, but its change when a body moves from one position to another. This change is independent of the choice of zero level.

Potential energy of a stretched spring calculated by the formula:

Where: k– spring stiffness. An extended (or compressed) spring can set a body attached to it in motion, that is, impart kinetic energy to this body. Consequently, such a spring has a reserve of energy. Tension or compression X must be calculated from the undeformed state of the body.

The potential energy of an elastically deformed body is equal to the work done by the elastic force during the transition from this state into a state with zero deformation. If in the initial state the spring was already deformed, and its elongation was equal to x 1, then upon transition to a new state with elongation x 2, the elastic force will do work equal to the change in potential energy, taken with the opposite sign (since the elastic force is always directed against the deformation of the body):

Potential energy during elastic deformation is the energy of interaction of individual parts of the body with each other by elastic forces.

The work of the friction force depends on the path traveled (this type of force, whose work depends on the trajectory and the path traveled is called: dissipative forces). The concept of potential energy for the friction force cannot be introduced.

Efficiency

Efficiency factor (efficiency)– characteristic of the efficiency of a system (device, machine) in relation to the conversion or transmission of energy. It is determined by the ratio of usefully used energy to the total amount of energy received by the system (the formula has already been given above).

Efficiency can be calculated both through work and through power. Useful and expended work (power) are always determined by simple logical reasoning.

In electrical engines efficiency– the ratio of the performed (useful) mechanical work to electrical energy, received from the source. In heat engines, the ratio of useful mechanical work to the amount of heat expended. In electrical transformers, the ratio of electromagnetic energy received in the secondary winding to the energy consumed by the primary winding.

Due to its generality, the concept of efficiency makes it possible to compare and evaluate such various systems, How nuclear reactors, electric generators and engines, thermal power plants, semiconductor devices, biological objects, etc.

Due to inevitable energy losses due to friction, heating of surrounding bodies, etc. Efficiency is always less than unity. Accordingly, efficiency is expressed as a fraction of the energy expended, that is, as a proper fraction or as a percentage, and is a dimensionless quantity. Efficiency characterizes how efficiently a machine or mechanism operates. The efficiency of thermal power plants reaches 35-40%, internal combustion engines with supercharging and pre-cooling - 40-50%, dynamos and high-power generators - 95%, transformers - 98%.

A problem in which you need to find the efficiency or it is known, you need to start with logical reasoning - which work is useful and which is wasted.

Law of conservation of mechanical energy

Total mechanical energy is called the sum of kinetic energy (i.e. the energy of motion) and potential (i.e. the energy of interaction of bodies by the forces of gravity and elasticity):

If mechanical energy does not transform into other forms, for example, into internal (thermal) energy, then the sum of kinetic and potential energy remains unchanged. If mechanical energy turns into thermal energy, then the change in mechanical energy is equal to the work of the friction force or energy losses, or the amount of heat released, and so on, in other words, the change in total mechanical energy is equal to the work of external forces:

The sum of the kinetic and potential energy of the bodies that make up a closed system (i.e. one in which there are no external forces acting, and their work is correspondingly zero) and the gravitational and elastic forces interacting with each other remains unchanged:

This statement expresses law of conservation of energy (LEC) in mechanical processes. It is a consequence of Newton's laws. The law of conservation of mechanical energy is satisfied only when bodies in a closed system interact with each other by forces of elasticity and gravity. In all problems on the law of conservation of energy there will always be at least two states of a system of bodies. The law states that the total energy of the first state will be equal to the total energy of the second state.

Algorithm for solving problems on the law of conservation of energy:

1. Find the points of the initial and final position of the body.
2. Write down what or what energies the body has at these points.
3. Equate the initial and final energy of the body.
4. Add other necessary equations from previous topics in physics.
5. Solve the resulting equation or system of equations using mathematical methods.

It is important to note that the law of conservation of mechanical energy made it possible to obtain a relationship between the coordinates and velocities of a body at two different points of the trajectory without analyzing the law of motion of the body at all intermediate points. The application of the law of conservation of mechanical energy can greatly simplify the solution of many problems.

IN real conditions Almost always, moving bodies, along with gravitational forces, elastic forces and other forces, are acted upon by frictional forces or environmental resistance forces. 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 bodies (heating). Thus, energy as a whole (i.e., not only mechanical) is conserved in any case.

During any physical interactions, energy neither appears nor disappears. It just changes from one form to another. This experimentally established fact expresses a fundamental law of nature - law of conservation and transformation of energy.

One of the consequences of the law of conservation and transformation of energy is the statement about the impossibility of creating “ perpetual motion machine"(perpetuum mobile) - a machine that could do work indefinitely without consuming energy.

Various tasks for work

If the problem requires finding mechanical work, then first choose how to find it:

1. A job can be found using the formula: A = FS∙cos α . Find the force that does the work and the amount of displacement of the body under the influence of this force in the chosen frame of reference. Note that the angle must be chosen between the force and displacement vectors.
2. The work done by an external force can be found as the difference in mechanical energy in the final and initial situations. Mechanical energy is equal to the sum of the kinetic and potential energies of the body.
3. The work done to lift a body at a constant speed can be found using the formula: A = mgh, Where h- height to which it rises body center of gravity.
4. Work can be found as the product of power and time, i.e. according to the formula: A = Pt.
5. The work can be found as the area of ​​the figure under the graph of force versus displacement or power versus time.

Law of conservation of energy and dynamics of rotational motion

The problems of this topic are quite complex mathematically, but if you know the approach, they can be solved using a completely standard algorithm. In all problems you will have to consider the rotation of the body in the vertical plane. The solution will come down to the following sequence of actions:

1. You need to determine the point you are interested in (the point at which you need to determine the speed of the body, the tension force of the thread, weight, and so on).
2. Write down Newton’s second law at this point, taking into account that the body rotates, that is, it has centripetal acceleration.
3. Write down the law of conservation of mechanical energy so that it contains the speed of the body in the same interesting point, as well as characteristics of the state of the body in some state about which something is known.
4. Depending on the condition, express the squared speed from one equation and substitute it into the other.
5. Carry out the remaining necessary mathematical operations to obtain the final result.

When solving problems, you need to remember that:

• The condition for passing the top point when rotating on a thread at a minimum speed is the support reaction force N at the top point is 0. The same condition is met when passing the top point of the dead loop.
• When rotating on a rod, the condition for passing the entire circle is: the minimum speed at the top point is 0.
• The condition for the separation of a body from the surface of the sphere is that the support reaction force at the separation point is zero.

Inelastic collisions

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 unknown active forces. An example of this type of problem is the impact interaction of bodies.

By impact (or collision) It is customary to call 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, it is impossible to consider the impact interaction 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.

We often have to deal with the impact interaction of bodies in everyday life, in technology and in physics (especially in the physics of the atom and elementary particles). In mechanics, two models of impact interaction are often used - absolutely elastic and absolutely inelastic impacts.

Absolutely inelastic impact called such an impact interaction in which bodies connect (stick together) with each other and move further as one body.

In a completely inelastic collision, mechanical energy is not conserved. It partially or completely turns into the internal energy of bodies (heating). To describe any impacts, you need to write down both the law of conservation of momentum and the law of conservation of mechanical energy, taking into account the heat released (it is highly advisable to make a drawing first).

Absolutely elastic impact

Absolutely elastic impact called a collision in which the mechanical energy of a system of bodies is conserved. In many cases, collisions of atoms, molecules and elementary particles obey the laws of absolutely elastic impact. With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is satisfied. A simple example A perfectly elastic collision can be a central impact of two billiard balls, one of which was at rest before the collision.

Central strike balls is called a collision in which the velocities of the balls before and after the impact are directed along the line of centers. Thus, using the laws of conservation of mechanical energy and momentum, it is possible to determine the velocities of the balls after a collision if their velocities before the collision are known. The central strike is very rarely implemented in practice, especially if we're talking about about collisions of atoms or molecules. In a non-central elastic collision, the velocities of particles (balls) before and after the collision are not directed in one straight line.

A special case of an off-central elastic impact can be the collision of two billiard balls of the same mass, one of which was motionless before the collision, and the speed of the second was not directed along the line of the centers of the balls. In this case, the velocity vectors of the balls after an elastic collision are always directed perpendicular to each other.

Conservation laws. Complex tasks

Multiple bodies

In some problems on the law of conservation of energy, the cables with which certain objects are moved can have mass (that is, not be weightless, as you might already be used to). In this case, the work of moving such cables (namely their centers of gravity) also needs to be taken into account.

If two bodies connected by a weightless rod rotate in a vertical plane, then:

1. choose a zero level to calculate potential energy, for example at the level of the axis of rotation or at the level of the lowest point finding one of the loads and making a drawing;
2. write down the law of conservation of mechanical energy, in which on the left side we write the sum of the kinetic and potential energy of both bodies in the initial situation, and on the right side we write the sum of the kinetic and potential energy of both bodies in the final situation;
3. take into account that the angular velocities of the bodies are the same, then the linear velocities of the bodies are proportional to the radii of rotation;
4. if necessary, write down Newton's second law for each of the bodies separately.

Shell burst

When a projectile explodes, explosive energy is released. To find this energy, it is necessary to subtract the mechanical energy of the projectile before the explosion from the sum of the mechanical energies of the fragments after the explosion. We will also use the law of conservation of momentum, written in the form of the cosine theorem (vector method) or in the form of projections onto selected axes.

Collisions with a heavy plate

Let us meet a heavy plate that moves at speed v, a light ball of mass moves m at speed u n. Since the momentum of the ball is much less than the momentum of the plate, after the impact the speed of the plate will not change, and it will continue to move at the same speed and in the same direction. As a result of the elastic impact, the ball will fly away from the plate. It is important to understand here that the speed of the ball relative to the plate will not change. In this case, for the final speed of the ball we obtain:

Thus, the speed of the ball after impact increases by twice the speed of the wall. Similar reasoning for the case when before the impact the ball and the plate were moving in the same direction leads to the result that the speed of the ball decreases by twice the speed of the wall:

In physics and mathematics, among other things, three most important conditions must be met:

1. Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to solve quickly and without failures large number tasks for different topics And of varying complexity. The latter can only be learned by solving thousands of problems.
2. Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do, necessary formulas in physics there are only about 200 pieces, and in mathematics even a little less. Each of these subjects has about a dozen standard methods for solving problems basic level difficulties that can also be learned, and thus solved completely automatically and without difficulty right moment most of the DH. After this, you will only have to think about the most difficult tasks.
3. Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, fill out the answer form correctly, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

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In our everyday experience, the word “work” appears very often. But one should distinguish between physiological work and work from the point of view of the science of physics. When you come home from class, you say: “Oh, I’m so tired!” This is physiological work. Or, for example, the work of a team in folk tale"Turnip".

Figure 1. Work in the everyday sense of the word

We will talk here about work from the point of view of physics.

Mechanical work is performed if a body moves under the influence of a force. The work is indicated Latin letter A. A more strict definition of work sounds like this.

The work of a force is a physical quantity equal to the product of the magnitude of the force and the distance traveled by the body in the direction of the force.

Figure 2. Work is a physical quantity

The formula is valid when a constant force acts on the body.

IN international system SI units of work are measured in joules.

This means that if under the influence of a force of 1 newton a body moves 1 meter, then 1 joule of work is done by this force.

The unit of work is named after the English scientist James Prescott Joule.

Fig 3. James Prescott Joule (1818 - 1889)

From the formula for calculating work it follows that there are three possible cases when work is equal to zero.

The first case is when a force acts on a body, but the body does not move. For example, a house is subject to a huge force of gravity. But she does not do any work because the house is motionless.

The second case is when the body moves by inertia, that is, no forces act on it. For example, spacecraft moves in intergalactic space.

The third case is when a force acts on the body perpendicular to the direction of movement of the body. In this case, although the body moves and a force acts on it, there is no movement of the body in the direction of the force.

Figure 4. Three cases when work is zero

It should also be said that the work done by a force can be negative. This will happen if the body moves against the direction of the force. For example, when a crane lifts a load above the ground using a cable, the work done by gravity is negative (and the work done by the elastic force of the cable directed upward, on the contrary, is positive).

Let's assume that when performing construction work, the pit needs to be filled with sand. It would take a few minutes for an excavator to do this, but a worker with a shovel would have to work for several hours. But both the excavator and the worker would have completed the same job.

Fig 5. The same work can be completed in different times

To characterize the speed of performing work in physics, a quantity called power is used.

Power is a physical quantity equal to the ratio of work to the time it is performed.

Power is indicated by a Latin letter N.

The SI unit of power is the watt.

One watt is the power at which one joule of work is done in one second.

The power unit is named after the English scientist, inventor of the steam engine, James Watt.

Fig 6. James Watt (1736 - 1819)

Let's combine the formula for calculating work with the formula for calculating power.

Let us now remember that the ratio of the path traveled by the body is S, by the time of movement t represents the speed of movement of the body v.

Thus, power is equal to the product of the numerical value of the force and the speed of the body in the direction of the force.

This formula is convenient to use when solving problems in which a force acts on a body moving with a known speed.

References

1. Lukashik V.I., Ivanova E.V. Collection of problems in physics for grades 7-9 of general education institutions. - 17th ed. - M.: Education, 2004.
2. Peryshkin A.V. Physics. 7th grade - 14th ed., stereotype. - M.: Bustard, 2010.
3. Peryshkin A.V. Collection of problems in physics, grades 7-9: 5th ed., stereotype. - M: Publishing House “Exam”, 2010.

1. Internet portal Physics.ru ().
2. Internet portal Festival.1september.ru ().
3. Internet portal Fizportal.ru ().
4. Internet portal Elkin52.narod.ru ().

Homework

1. In what cases is work equal to zero?
2. How is the work done along the path traveled in the direction of the force? In the opposite direction?
3. How much work is done by the frictional force acting on the brick when it moves 0.4 m? The friction force is 5 N.

Almost everyone, without hesitation, will answer: in the second. And they will be wrong. The opposite is true. In physics, mechanical work is described with the following definitions: Mechanical work is performed when a force acts on a body and it moves. Mechanical work is directly proportional to the force applied and the distance traveled.

Mechanical work formula

Mechanical work is determined by the formula:

where A is work, F is force, s is the distance traveled.

POTENTIAL(potential function), a concept that characterizes a wide class of physical force fields (electric, gravitational, etc.) and fields in general physical quantities, represented by vectors (fluid velocity field, etc.). In the general case, the vector field potential a( x,y,z) is such a scalar function u(x,y,z) that a=grad

35. Conductors in an electric field. Electrical capacity.Conductors in an electric field. Conductors are substances characterized by the presence in them of a large number of free charge carriers that can move under the influence of an electric field. Conductors include metals, electrolytes, and coal. In metals, the carriers of free charges are the electrons of the outer shells of atoms, which, when the atoms interact, completely lose connections with “their” atoms and become the property of the entire conductor as a whole. Free electrons participate in thermal motion like gas molecules and can move through the metal in any direction. Electrical capacity- characteristic of a conductor, a measure of its ability to accumulate electrical charge. In the theory of electrical circuits, capacitance is called mutual capacitance between two conductors; parameter of a capacitive element of an electrical circuit, presented in the form of a two-terminal network. This capacity is defined as the ratio of the quantity electric charge to the potential difference between these conductors

36. Capacitance of a parallel-plate capacitor.

Capacitance of a parallel plate capacitor.

That. The capacitance of a flat capacitor depends only on its size, shape and dielectric constant. To create a high-capacity capacitor, it is necessary to increase the area of ​​the plates and reduce the thickness of the dielectric layer.

37. Magnetic interaction of currents in a vacuum. Ampere's law.Ampere's law. In 1820, Ampere (French scientist (1775-1836)) experimentally established a law by which one can calculate force acting on a conductor element of length carrying current.

where is the vector of magnetic induction, is the vector of the element of the length of the conductor drawn in the direction of the current.

Force modulus , where is the angle between the direction of the current in the conductor and the direction of the magnetic field induction. For a straight conductor of length carrying current in a uniform field

The direction of the acting force can be determined using left hand rules:

If the palm of the left hand is positioned so that the normal (to the current) component magnetic field entered the palm, and the four extended fingers are directed along the current, then the thumb will indicate the direction in which the Ampere force acts.

38. Magnetic field strength. Biot-Savart-Laplace LawMagnetic field strength(standard designation N ) - vector physical quantity, equal to the difference of the vector magnetic induction B And magnetization vector J .

IN International System of Units (SI): Where- magnetic constant.

BSL Law. The law determining the magnetic field of an individual current element

39. Applications of the Bio-Savart-Laplace law. For direct current field

For a circular turn.

And for the solenoid

40. Magnetic field induction A magnetic field is characterized by a vector quantity, which is called magnetic field induction (a vector quantity that is a force characteristic of the magnetic field at a given point in space). MI. (B) this is not a force acting on the conductors, it is a quantity that is found through this force using the following formula: B=F / (I*l) (Verbally: MI vector module. (B) is equal to the ratio of the modulus of force F, with which the magnetic field acts on a current-carrying conductor located perpendicular to the magnetic lines, to the current strength in the conductor I and the length of the conductor l. Magnetic induction depends only on the magnetic field. In this regard, induction can be considered a quantitative characteristic of a magnetic field. It determines with what force (Lorentz force) the magnetic field acts on a charge moving at speed. MI is measured in teslas (1 Tesla). In this case, 1 T=1 N/(A*m). MI has a direction. Graphically it can be sketched in the form of lines. In a uniform magnetic field, the MI lines are parallel, and the MI vector will be directed in the same way at all points. In the case of a non-uniform magnetic field, for example, a field around a current-carrying conductor, the magnetic induction vector will change at every point in space around the conductor, and tangents to this vector will create concentric circles around the conductor.

41. Motion of a particle in a magnetic field. Lorentz force. a) - If a particle flies into a region of a uniform magnetic field, and the vector V is perpendicular to the vector B, then it moves in a circle of radius R=mV/qB, since the Lorentz force Fl=mV^2/R plays the role of a centripetal force. The period of revolution is equal to T=2piR/V=2pim/qB and it does not depend on the particle speed (This is only true for V<<скорости света) - Если угол между векторами V и B не равен 0 и 90 градусов, то частица в однородном магнитном поле движется по винтовой линии. - Если вектор V параллелен B, то частица движется по прямой линии (Fл=0). б) Силу, действующую со стороны магнитного поля на движущиеся в нем заряды, называют силой Лоренца.

The magnetic force is determined by the relation: Fl = q·V·B·sina (q is the magnitude of the moving charge; V is the modulus of its speed; B is the modulus of the magnetic field induction vector; alpha is the angle between vector V and vector B) The Lorentz force is perpendicular to the speed and therefore it does not do work, does not change the modulus of the charge speed and its kinetic energy. But the direction of speed changes continuously. The Lorentz force is perpendicular to the vectors B and v, and its direction is determined using the same left-hand rule as the direction of the Ampere force: if the left hand is positioned so that the component of magnetic induction B, perpendicular to the speed of the charge, enters the palm, and the four fingers are are directed along the movement of the positive charge (against the movement of the negative), then the thumb bent 90 degrees will show the direction of the Lorentz force F l acting on the charge.

In order to be able to characterize the energy characteristics of movement, the concept of mechanical work was introduced. And the article is dedicated to it in its various manifestations. The topic is both easy and quite difficult to understand. The author sincerely tried to make it more understandable and accessible to understanding, and one can only hope that the goal has been achieved.

What is mechanical work called?

What is it called? If some force works on a body, and as a result of its action the body moves, then this is called mechanical work. When approaching from the point of view of scientific philosophy, several additional aspects can be highlighted here, but the article will cover the topic from the point of view of physics. Mechanical work is not difficult if you think carefully about the words written here. But the word “mechanical” is usually not written, and everything is shortened to the word “work.” But not every job is mechanical. Here is a man sitting and thinking. Does it work? Mentally yes! But is this mechanical work? No. What if a person walks? If a body moves under the influence of force, then this is mechanical work. It's simple. In other words, a force acting on a body does (mechanical) work. And one more thing: it is work that can characterize the result of the action of a certain force. So, if a person walks, then certain forces (friction, gravity, etc.) perform mechanical work on the person, and as a result of their action, the person changes his point of location, in other words, moves.

Work as a physical quantity is equal to the force that acts on the body, multiplied by the path that the body has made under the influence of this force and in the direction indicated by it. We can say that mechanical work was done if 2 conditions were simultaneously met: a force acted on the body, and it moved in the direction of its action. But it did not occur or does not occur if the force acted and the body did not change its location in the coordinate system. Here are small examples when mechanical work is not performed:

1. So a person can lean on a huge boulder in order to move it, but there is not enough strength. The force acts on the stone, but it does not move, and no work occurs.
2. The body moves in the coordinate system, and the force is equal to zero or they have all been compensated. This can be observed while moving by inertia.
3. When the direction in which a body moves is perpendicular to the action of the force. When a train moves along a horizontal line, gravity does not do its work.

Depending on certain conditions, mechanical work can be negative and positive. So, if the directions of both the forces and the movements of the body are the same, then positive work occurs. An example of positive work is the effect of gravity on a falling drop of water. But if the force and direction of movement are opposite, then negative mechanical work occurs. An example of such an option is a balloon rising upward and the force of gravity, which does negative work. When a body is subject to the influence of several forces, such work is called “resultant force work.”

Features of practical application (kinetic energy)

Let's move from theory to practical part. Separately, we should talk about mechanical work and its use in physics. As many probably remember, all the energy of the body is divided into kinetic and potential. When an object is in equilibrium and not moving anywhere, its potential energy equals its total energy and its kinetic energy equals zero. When movement begins, potential energy begins to decrease, kinetic energy begins to increase, but in total they are equal to the total energy of the object. For a material point, kinetic energy is defined as the work of a force that accelerates the point from zero to the value H, and in formula form the kinetics of a body is equal to ½*M*N, where M is mass. To find out the kinetic energy of an object that consists of many particles, you need to find the sum of all the kinetic energy of the particles, and this will be the kinetic energy of the body.

Features of practical application (potential energy)

In the case when all the forces acting on the body are conservative, and the potential energy is equal to the total, then no work is done. This postulate is known as the law of conservation of mechanical energy. Mechanical energy in a closed system is constant over a time interval. The conservation law is widely used to solve problems from classical mechanics.

Features of practical application (thermodynamics)

In thermodynamics, the work done by a gas during expansion is calculated by the integral of pressure times volume. This approach is applicable not only in cases where there is an exact volume function, but also to all processes that can be displayed in the pressure/volume plane. It also applies knowledge of mechanical work not only to gases, but to anything that can exert pressure.

Features of practical application in practice (theoretical mechanics)

In theoretical mechanics, all the properties and formulas described above are considered in more detail, in particular projections. It also gives its definition for various formulas of mechanical work (an example of a definition for the Rimmer integral): the limit to which the sum of all forces of elementary work tends, when the fineness of the partition tends to zero, is called the work of force along the curve. Probably difficult? But nothing, everything is fine with theoretical mechanics. Yes, all the mechanical work, physics and other difficulties are over. Further there will be only examples and a conclusion.

Units of measurement of mechanical work

The SI uses joules to measure work, while the GHS uses ergs:

1. 1 J = 1 kg m²/s² = 1 N m
2. 1 erg = 1 g cm²/s² = 1 dyne cm
3. 1 erg = 10 −7 J

Examples of mechanical work

In order to finally understand such a concept as mechanical work, you should study several individual examples that will allow you to consider it from many, but not all, sides:

1. When a person lifts a stone with his hands, mechanical work occurs with the help of the muscular strength of the hands;
2. When a train travels along the rails, it is pulled by the traction force of the tractor (electric locomotive, diesel locomotive, etc.);
3. If you take a gun and fire from it, then thanks to the pressure force created by the powder gases, work will be done: the bullet is moved along the barrel of the gun at the same time as the speed of the bullet itself increases;
4. Mechanical work also exists when the friction force acts on a body, forcing it to reduce the speed of its movement;
5. The above example with balls, when they rise in the opposite direction relative to the direction of gravity, is also an example of mechanical work, but in addition to gravity, the Archimedes force also acts, when everything that is lighter than air rises up.

What is power?

Finally, I would like to touch on the topic of power. The work done by a force in one unit of time is called power. In fact, power is a physical quantity that is a reflection of the ratio of work to a certain period of time during which this work was done: M=P/B, where M is power, P is work, B is time. The SI unit of power is 1 W. A watt is equal to the power that does one joule of work in one second: 1 W=1J\1s.