If a force acts on a body, then this force does work to move this body. Before giving a definition of work in the curvilinear motion of a material point, consider special cases:
In this case, mechanical work A is equal to:
A=
F s cos
=
,
or A=Fcos
×
s = F S ×
s ,
whereF S
– projection
strength 
to move. In this case F s =
const, and the geometric meaning of the work A is the area of the rectangle constructed in coordinates F S ,
,
s.
Let's build a graph of the projection of force on the direction of movement F S as a function of displacement s. We represent the total displacement as the sum of n small displacements 
. For small i
-th displacement 
work is 
or the area of the shaded trapezoid in the figure.
.
The value under the integral will represent the elementary work on an infinitesimal displacement 
:
- basic work.
and the work of the force 
by moving a material point from a point 1
exactly 2
defined as a curvilinear integral:
–work with curvilinear motion.
Example 1:
The work of gravity
during curvilinear motion of a material point.
.Further 
as a constant value can be taken out of the integral sign, and the integral 
according to the figure will represent a complete displacement 
.
.
If we denote the height of the point 1
from the earth's surface through 
, and the height of the point 2
through 
, then
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 in a closed path is zero: 
.
Forces whose work on a closed path is zero is calledconservative .
Example 2 : The work of the 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 of the friction force is always negative and cannot be equal to zero on a closed path. The work done per unit of time is called power. If in time 
work is done 
, then the power is
–mechanical power.
Taking 
as
,
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 watt: 1 W = 1 J / s.
mechanical energy.
Energy is a general quantitative measure of the movement of the 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 binds 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 at the expense of 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 the change in energy:
.
Energy as well as work in SI is measured in joules: [ E]=1 J.
Mechanical energy is of two types - kinetic and potential.
Kinetic energy
(or the energy of motion) is determined by the masses and velocities of the considered bodies. Consider a material point moving under the action of a force 
. The work of this force increases the kinetic energy of a material point 
. Let us calculate in this case a small increment (differential) of the kinetic energy:
When calculating 
using Newton's second law 
, as well as 
- velocity modulus of a material point. Then 
can be represented as:
-
- kinetic energy of a moving material point.
Multiplying and dividing this expression by 
, and taking into account that 
, we get
-
- relationship 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 of gravity 
with curvilinear motion of a material point 
can be represented as the difference between the values of the function 
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 way 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 is equal to the loss of potential energy.
Otherwise, we can say that the work is done due to the potential energy reserve.
the value 
, 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 mentioned above, is equal to:
.
Thus, if the power 
is a conservative force and there are no other external forces, then 
, i.e. in this case, the total mechanical energy of the body is conserved.
Do you know what work is? Without any doubt. What is work, every person knows, provided that he was born and lives on planet Earth. What is mechanical work?
This concept is also known to most people on the planet, although some individuals have a rather vague idea of \u200b\u200bthis process. But it's not about them now. Even fewer people have any idea what mechanical work from the point of view of physics. In physics, mechanical work is not the work of a person for the sake of food, it is a physical quantity that can be completely unrelated to either a person or any other living being. How so? Now let's figure it out.
Mechanical work in physics
Let's give two examples. In the first example, the waters of the river, colliding with the abyss, noisily fall down in the form of a waterfall. The second example is a man who holds a heavy object at outstretched arms, for example, keeps a broken roof over the porch of a country house from falling, while his wife and children are frantically looking for something to prop it up. When is mechanical work done?
Definition of mechanical work
Almost everyone, without hesitation, will answer: in the second. And they will be wrong. The case is just the opposite. In physics, mechanical work is described the following definitions: mechanical work is done when a force acts on a body and it moves. Mechanical work is directly proportional to the applied force and the distance traveled.
Mechanical work formula
The mechanical work is determined by the formula:
where A is work,
F - strength,
s - the distance traveled.
So, despite all the heroism of the tired roof holder, the work done by him is equal to zero, but the water, falling under the influence of gravity from a high cliff, does the most mechanical work. That is, if we push a heavy cabinet unsuccessfully, then the work we have done from the point of view of physics will be equal to zero, despite the fact that we are applying a lot of force. But if we move the cabinet a certain distance, then we will do work equal to the product of the applied force by the distance we moved the body.
The unit of work is 1 J. This is the work done by a force of 1 newton to move a body a distance of 1 m. If the direction of the applied force coincides with the direction of movement of the body, then this force does positive work. An example is when we push a body and it moves. And in the case when the force is applied in the direction opposite to the movement of the body, for example, friction force, then this force does negative work. If the applied force does not affect the motion of the body in any way, then the force produced by this work is equal to zero.
To be able to characterize the energy characteristics of motion, the concept of mechanical work was introduced. And it is to her in her various manifestations that the article is devoted. To understand the topic is both easy and quite complex. The author sincerely tried to make it more understandable and understandable, and one can only hope that the goal has been achieved.
What is mechanical work?
What is it called? If some force works on the body, and as a result of the action of this force, the body moves, then this is called mechanical work. When approached from the point of view of scientific philosophy, several additional aspects can be distinguished 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 reduced to the word "work". But not every job is mechanical. Here a man sits and thinks. Does it work? Mentally yes! But is it mechanical work? No. What if the person is walking? If the body moves under the influence of a force, then this is mechanical work. Everything is simple. In other words, the force acting on the 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 a person, and as a result of their action, a person changes his point of location, in other words, he moves.
Work as a physical quantity is equal to the force that acts on the body, multiplied by the path that the body 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: the force acted on the body, and it moved in the direction of its action. But it was not performed or is not performed if the force acted, and the body did not change its location in the coordinate system. Here are small examples where mechanical work is not done:
- So a person can fall 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 work does not occur.
- The body moves in the coordinate system, and the force is equal to zero or they are all compensated. This can be observed during inertial motion.
- When the direction in which the body moves is perpendicular to the force. When the train moves along a horizontal line, the force of gravity does not do its work.
Depending on certain conditions, mechanical work can be negative and positive. So, if the directions and 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 up and gravity, which does negative work. When a body is subjected to the influence of several forces, such work is called "resultant force work".
Features of practical application (kinetic energy)
We pass from theory to practical part. Separately, we should talk about mechanical work and its use in physics. As many probably remembered, 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 is equal to the total energy, and its kinetic energy is zero. When the movement begins, the potential energy begins to decrease, the kinetic energy 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 the force that accelerated the point from zero to the value H, and in formula form, the kinetics of the body is ½ * M * H, where M is the 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 in the 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 multiplied by volume. This approach is applicable not only in cases where there is an exact function of volume, but also to all processes that can be displayed in the pressure/volume plane. The knowledge of mechanical work is also applied not only to gases, but to everything 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, these are projections. She also gives her own definition for various formulas of mechanical work (an example of the definition for the Rimmer integral): the limit to which the sum of all the forces of elementary work tends when the fineness of the partition tends to zero is called the work of the force along the curve. Probably difficult? But nothing, with theoretical mechanics everything. Yes, and all the mechanical work, physics and other difficulties are over. Further there will be only examples and a conclusion.
Mechanical work units
The SI uses joules to measure work, while the GHS uses ergs:
- 1 J = 1 kg m²/s² = 1 Nm
- 1 erg = 1 g cm²/s² = 1 dyn cm
- 1 erg = 10 −7 J
Examples of mechanical work
In order to finally understand such a concept as mechanical work, you should study a few separate examples that will allow you to consider it from many, but not all, sides:
- When a person lifts a stone with his hands, then mechanical work occurs with the help of the muscular strength of the hands;
- When a train travels along the rails, it is pulled by the traction force of the tractor (electric locomotive, diesel locomotive, etc.);
- If you take a gun and shoot from it, then thanks to the pressure force that the powder gases will create, 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;
- There is also mechanical work when the friction force acts on the body, forcing it to reduce the speed of its movement;
- 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 want 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 such 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 watt. A watt is equal to the power that does the work of one joule in one second: 1 W = 1J \ 1s.
Almost everyone, without hesitation, will answer: in the second. And they will be wrong. The case is just the opposite. In physics, mechanical work is described the following definitions: mechanical work is done when a force acts on a body and it moves. Mechanical work is directly proportional to the applied force and the distance traveled.
Mechanical work formula
The 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, in general, fields of physical quantities represented by vectors (fluid velocity field, etc.). In the general case, the potential of the vector field 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, coal. In metals, the carriers of free charges are the electrons of the outer shells of atoms, which, when atoms interact, completely lose their bonds 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. Electric capacity- a characteristic of a conductor, a measure of its ability to accumulate an electric charge. In the theory of electrical circuits, capacitance is the mutual capacitance between two conductors; parameter of the capacitive element of the electrical circuit, presented in the form of a two-terminal network. Such capacitance is defined as the ratio of the magnitude of the electric charge to the potential difference between these conductors
36. Capacitance of a flat capacitor.
Capacitance of a flat 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 vacuum. Ampere's law.Ampere's law. In 1820, Ampère (a French scientist (1775-1836)) established experimentally a law by which one can calculate force acting on a conductor element of length with current.
where is the vector of magnetic induction, is the vector of the length element 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. For a straight conductor with 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 of the magnetic field enters the palm, and four outstretched fingers are directed along the current, then the thumb will indicate the direction in which the Ampère force acts.
38. Magnetic field strength. Biot-Savart-Laplace lawMagnetic field strength(standard designation H ) - vector physical quantity, equal to the difference of the vector magnetic induction B and magnetization vector J .
AT International System of Units (SI): 
where- magnetic constant.
BSL law. The law that determines the magnetic field of an individual current element 
39. Applications of the Biot-Savart-Laplace law. For direct current field
For a circular loop.
And for the solenoid
40. Magnetic field induction The magnetic field is characterized by a vector quantity, which is called the magnetic field induction (a vector quantity, which is the force characteristic of the magnetic field at a given point in space). MI. (B) this is not a force acting on conductors, it is a quantity that is found through a given force according to the following formula: B \u003d F / (I * l) (Verbally: MI vector modulus. (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 the magnetic field. It determines with what force (Lorentz Force) the magnetic field acts on a charge moving with speed. 
MI is measured in Tesla (1 T). In this case, 1 Tl \u003d 1 N / (A * m). MI has direction. Graphically, it can be drawn as lines. In a uniform magnetic field, the MIs 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 conductor with current, the magnetic induction vector will change at each 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 along 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 T=2piR/V=2pim/qB and it does not depend on the speed of the particle (This is true only 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 L. 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 velocity; B is the modulus of the magnetic field induction vector; alpha is the angle between the vector V and the vector B) The Lorentz force is perpendicular to the velocity and therefore it does not do work, does not change the modulus of the speed of the charge and its kinetic energy. But the direction of the speed changes continuously. The Lorentz force is perpendicular to the vectors B and v, and its direction is determined using the same rule of the left hand as the direction of the Ampère force: if the left hand is positioned so that the magnetic induction component B, perpendicular to the charge velocity, enters the palm, and four fingers are directed along the movement of a positive charge (against the movement of a negative one), then the thumb bent 90 degrees will show the direction of the Lorentz force acting on the charge F l. 
What does it mean?
In physics, "mechanical work" is the work of some force (gravity, elasticity, friction, etc.) on the body, as a result of which the body moves.
Often the word "mechanical" is simply not spelled.
Sometimes you can find the expression "the body has done the work", which basically means "the force acting on the body has done the work."
I think - I'm working.
I go - I also work.
Where is the mechanical work here?
If a body moves under the action of a force, then mechanical work is done.
The body is said to do work.
More precisely, it will be like this: the work is done by the force acting on the body.
Work characterizes the result of the action of a force.
The forces acting on a person do mechanical work on him, and as a result of the action of these forces, the person moves.
Work is a physical quantity equal to the product of the force acting on the body and the path taken by the body under the action of the force in the direction of this force.
A - mechanical work,
F - strength,
S - the distance traveled.
Work is done, if 2 conditions are met simultaneously: a force acts on the body and it
moves in the direction of the force.
Work is not done(i.e. equal to 0) if:
1. The force acts, but the body does not move.
For example: we act with force on a stone, but we cannot move it.
2. The body moves, and the force is equal to zero, or all forces are compensated (ie, the resultant of these forces is equal to 0).
For example: when moving by inertia, no work is done.
3. The direction of the force and the direction of motion of the body are mutually perpendicular.
For example: when a train moves horizontally, gravity does no work.
Work can be positive or negative.
1. If the direction of the force and the direction of motion of the body are the same, positive work is done.
For example: gravity, acting on a drop of water falling down, does positive work.
2. If the direction of the force and the movement of the body are opposite, negative work is done.
For example: the force of gravity acting on a rising balloon does negative work.
If several forces act on a body, then the total work of all forces is equal to the work of the resulting force.
Units of work
In honor of the English scientist D. Joule, the unit of work was named 1 Joule.
In the international system of units (SI):
[A] = J = N m
1J = 1N 1m
Mechanical work is equal to 1 J if, under the influence of a force of 1 N, the body moves 1 m in the direction of this force.
When flying from the thumb of a person to the index
a mosquito does work - 0,000,000,000,000,000,000,000,000,001 J.
The human heart performs approximately 1 J of work in one contraction, which corresponds to the work done when lifting a load of 10 kg to a height of 1 cm.
TO WORK, FRIENDS!
