The energy characteristics of motion are introduced on the basis of the concept of mechanical work or work of force.
Definition 1Work A performed by a constant force F → is a physical quantity equal to the product of the force and displacement modules multiplied by the cosine of the angle α , located between the force vectors F → and the displacement s →.
This definition discussed in Figure 1. 18 . 1 .
The work formula is written as,
A = F s cos α .
Work is scalar quantity. This makes it possible to be positive at (0° ≤ α< 90 °) , отрицательной при (90 ° < α ≤ 180 °) . Когда задается прямой угол α , тогда совершаемая сила равняется нулю. Единицы измерения работы по системе СИ - джоули (Д ж) .
A joule is equal to the work done by a force of 1 N to move 1 m in the direction of the force.
Picture 1 . 18 . 1 . Work of force F →: A = F s cos α = F s s
When projecting F s → force F → onto the direction of movement s → the force does not remain constant, and the calculation of work for small movements Δ s i is summed up and produced according to the formula:
A = ∑ ∆ A i = ∑ F s i ∆ s i .
This amount work is calculated from the limit (Δ s i → 0), after which it goes into the integral.
The graphical representation of the work is determined from the area of the curvilinear figure located under the graph F s (x) of Figure 1. 18 . 2.
Picture 1 . 18 . 2. Graphic definition of work Δ A i = F s i Δ s i .
An example of a force that depends on the coordinate is the elastic force of a spring, which obeys Hooke's law. To stretch a spring, it is necessary to apply a force F →, the modulus of which is proportional to the elongation of the spring. This can be seen in Figure 1. 18 . 3.
Picture 1 . 18 . 3. Stretched spring. Direction external force F → coincides with the direction of movement s →. F s = k x, where k denotes the spring stiffness.
F → y p = - F →
The dependence of the external force modulus on the x coordinates can be plotted using a straight line.
Picture 1 . 18 . 4 . Dependence of the external force modulus on the coordinate when the spring is stretched.
From the above figure it is possible to find work on external force the right free end of the spring, using the area of the triangle. The formula will take the form
This formula is applicable to express the work done by an external force when compressing a spring. Both cases show that the elastic force F → y p is equal to the work of the external force F → , but with the opposite sign.
Definition 2
If several forces act on a body, then the formula for the total work will look like the sum of all the work done on it. When a body moves translationally, the points of application of forces move equally, that is general work of all forces will be equal to the resultant work of the applied forces.
Picture 1 . 18 . 5 . Model of mechanical work.
Power determination
Definition 3Power is called the work done by a force per unit time.
Recording the physical quantity of power, denoted N, takes the form of the ratio of work A to the time period t of the work performed, that is:
Definition 4
The SI system uses the watt (W t) as a unit of power, equal to the power of the force that does 1 J of work in 1 s.
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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, in general, fields of physical quantities represented by vectors (field of fluid velocities, 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 electrical circuit theory, capacitance is the mutual capacitance between two conductors; parameter of a capacitive element of an 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 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, Ampère (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 of the magnetic field enters 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. 
Note that work and energy have the same units of measurement. This means that work can be converted into energy. For example, in order to raise a body to a certain height, then it will have potential energy, a force is needed that will do this work. The work done by the lifting force will turn into potential energy.
The rule for determining work according to the dependence graph F(r): the work is numerically equal to the area of the figure under the graph of force versus displacement.
Angle between force vector and displacement
1) Correctly determine the direction of the force that does the work; 2) We depict the displacement vector; 3) We transfer the vectors to one point and obtain the desired angle.
In the figure, the body is acted upon by the force of gravity (mg), the reaction of the support (N), the force of friction (Ftr) and the tension force of the rope F, under the influence of which the body moves r.
Work of gravity
Ground reaction work
Work of friction force
Work done by rope tension
Work done by resultant force
The work done by the resultant force can be found in two ways: 1st method - as the sum of the work (taking into account the “+” or “-” signs) of all forces acting on the body, in our example
Method 2 - first of all, find the resultant force, then directly its work, see figure
Work of elastic force
To find the work done by the elastic force, it is necessary to take into account that this force changes because it depends on the elongation of the spring. From Hooke's law it follows that as the absolute elongation increases, the force increases.
To calculate the work of the elastic force during the transition of a spring (body) from an undeformed state to a deformed state, use the formula
Power
A scalar quantity that characterizes the speed of work (an analogy can be drawn with acceleration, which characterizes the rate of change in speed). Determined by the formula
Efficiency
Efficiency is the ratio of the useful work done by a machine to all the work expended (energy supplied) during the same time
The efficiency is expressed as a percentage. The closer this number is to 100%, the higher the machine's performance. There cannot be an efficiency greater than 100, since it is impossible to do more work using less energy.
The efficiency of an inclined plane is the ratio of the work done by gravity to the work expended in moving along the inclined plane.
The main thing to remember
1) Formulas and units of measurement;
2) The work is performed by force;
3) Be able to determine the angle between the force and displacement vectors
If the work done by a force when moving a body along a closed path is zero, then such forces are called conservative or potential. The work done by the friction force when moving a body along a closed path is never equal to zero. The friction force, unlike the force of gravity or elastic force, is non-conservative or non-potential.
There are conditions under which the formula cannot be used 
If the force is variable, if the trajectory of movement is a curved line. In this case, the path is divided into small sections for which these conditions are met, and the elementary work on each of these sections is calculated. The total work in this case is equal to the algebraic sum of the elementary works: 
The value of the work done by a certain force depends on the choice of reference system.
Before revealing the topic “How work is measured,” it is necessary to make a small digression. Everything in this world obeys the laws of physics. Each process or phenomenon can be explained on the basis of certain laws of physics. For each measured quantity there is a unit in which it is usually measured. Units of measurement are constant and have the same meaning throughout the world.
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System of international units
The reason for this is the following. In nineteen sixty, at the Eleventh General Conference on Weights and Measures, a system of measurements was adopted that is recognized throughout the world. This system was named Le Système International d’Unités, SI (SI System International). This system has become the basis for determining units of measurement accepted throughout the world and their relationships.
Physical terms and terminology
In physics, the unit of measurement of the work of force is called J (Joule), in honor of the English physicist James Joule, who made a great contribution to the development of the branch of thermodynamics in physics. One Joule is equal to the work done by a force of one N (Newton) when its application moves one M (meter) in the direction of the force. One N (Newton) is equal to a force of one kg (kilogram) mass with an acceleration of one m/s2 (meter per second) in the direction of the force.
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Formula for finding a job
For your information. In physics, everything is interconnected; performing any work involves performing additional actions. As an example we can take household fan. When the fan is plugged in, the fan blades begin to rotate. The rotating blades influence the air flow, giving it directional movement. This is the result of the work. But to perform the work, the influence of other external forces is necessary, without which the action is impossible. These include electric current, power, voltage and many other related values.
Electric current, at its core, is the ordered movement of electrons in a conductor per unit time. Electric current is based on positively or negatively charged particles. They are called electric charges. Denoted by the letters C, q, Kl (Coulomb), named after the French scientist and inventor Charles Coulomb. In the SI system, it is a unit of measurement for the number of charged electrons. 1 C is equal to the volume of charged particles flowing through cross section conductor per unit time. The unit of time is one second. The formula for electric charge is shown in the figure below.
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Formula for finding electric charge
The strength of electric current is indicated by the letter A (ampere). Ampere is a unit in physics that characterizes the measurement of the work of force that is expended to move charges along a conductor. At its core, electricity is the ordered movement of electrons in a conductor under the influence of an electromagnetic field. A conductor is a material or molten salt (electrolyte) that has little resistance to the passage of electrons. The strength of electric current is affected by two physical quantities: voltage and resistance. They will be discussed below. Current strength is always directly proportional to voltage and inversely proportional to resistance.
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Formula for finding current strength
As mentioned above, electric current is the ordered movement of electrons in a conductor. But there is one caveat: they need a certain impact to move. This effect is created by creating a potential difference. Electric charge may be positive or negative. Positive charges always tend towards negative charges. This is necessary for the balance of the system. The difference between the number of positively and negatively charged particles is called electrical voltage.
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Formula for finding voltage
Power is the amount of energy expended to do one J (Joule) of work in a period of time of one second. The unit of measurement in physics is designated as W (Watt), in the SI system W (Watt). Since electrical power is considered, here it is the value of the electrical energy expended to perform a certain action in a period of time.
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Formula for finding electrical power
In conclusion, it should be noted that the unit of measurement of work is a scalar quantity, has a relationship with all branches of physics and can be considered from the perspective of not only electrodynamics or thermal engineering, but also other sections. The article briefly examines the value characterizing the unit of measurement of the work of force.
Video
You are already familiar with mechanical work (work of force) from the basic school physics course. Let us recall the definition of mechanical work given there for the following cases.
If the force is directed in the same direction as the movement of the body, then the work done by the force
In this case, the work done by the force is positive.
If the force is directed opposite to the movement of the body, then the work done by the force
In this case, the work done by the force is negative.
If the force f_vec is directed perpendicular to the displacement s_vec of the body, then the work done by the force is zero:
Work is a scalar quantity. The unit of work is called the joule (symbol: J) in honor of the English scientist James Joule, who played an important role in the discovery of the law of conservation of energy. From formula (1) it follows:
1 J = 1 N * m.
1. A block weighing 0.5 kg was moved along the table 2 m, applying an elastic force of 4 N to it (Fig. 28.1). The coefficient of friction between the block and the table is 0.2. What is the work acting on the block?
a) gravity m?
b) normal reaction forces?
c) elastic forces?
d) sliding friction forces tr?
The total work done by several forces acting on a body can be found in two ways:
1. Find the work of each force and add up these works, taking into account the signs.
2. Find the resultant of all forces applied to the body and calculate the work of the resultant.
Both methods lead to the same result. To make sure of this, go back to the previous task and answer the questions in task 2.
2. What is it equal to:
a) the sum of the work done by all forces acting on the block?
b) the resultant of all forces acting on the block?
c) work resultant? In the general case (when the force f_vec is directed at an arbitrary angle to the displacement s_vec) the definition of the work of the force is as follows.
The work A of a constant force is equal to the product of the force modulus F by the displacement modulus s and the cosine of the angle α between the direction of the force and the direction of displacement:
A = Fs cos α (4)
3. Show what general definition The work follows to the conclusions shown in the following diagram. Formulate them verbally and write them down in your notebook.
4. A force is applied to a block located on the table, the modulus of which is 10 N. Why equal to the angle between this force and the movement of the block, if when moving the block along the table by 60 cm, this force did the work: a) 3 J; b) –3 J; c) –3 J; d) –6 J? Make explanatory drawings.
2. Work of gravity
Let a body of mass m move vertically from the initial height h n to the final height h k.
If the body moves downwards (h n > h k, Fig. 28.2, a), the direction of movement coincides with the direction of gravity, therefore the work of gravity is positive. If the body moves upward (h n< h к, рис. 28.2, б), то работа силы тяжести отрицательна.
In both cases, the work done by gravity
A = mg(h n – h k). (5)
Let us now find the work done by gravity when moving at an angle to the vertical.
5. A small block of mass m slid along an inclined plane of length s and height h (Fig. 28.3). The inclined plane makes an angle α with the vertical.
a) What is the angle between the direction of gravity and the direction of movement of the block? Make an explanatory drawing.
b) Express the work of gravity in terms of m, g, s, α.
c) Express s in terms of h and α.
d) Express the work of gravity in terms of m, g, h.
e) What is the work done by gravity when the block moves upward along the entire same plane?
Having completed this task, you are convinced that the work of gravity is expressed by formula (5) even when the body moves at an angle to the vertical - both down and up.
But then formula (5) for the work of gravity is valid when a body moves along any trajectory, because any trajectory (Fig. 28.4, a) can be represented as a set of small “inclined planes” (Fig. 28.4, b). 
Thus,
the work done by gravity when moving along any trajectory is expressed by the formula
A t = mg(h n – h k),
where h n is the initial height of the body, h k is its final height.
The work done by gravity does not depend on the shape of the trajectory.
For example, the work of gravity when moving a body from point A to point B (Fig. 28.5) along trajectory 1, 2 or 3 is the same. From here, in particular, it follows that the force of gravity when moving along a closed trajectory (when the body returns to the starting point) is equal to zero. 
6. A ball of mass m, hanging on a thread of length l, was deflected by 90º, keeping the thread taut, and released without a push.
a) What is the work done by gravity during the time during which the ball moves to the equilibrium position (Fig. 28.6)?
b) What is the work done by the elastic force of the thread during the same time?
c) What is the work done by the resultant forces applied to the ball during the same time?
3. Work of elastic force
When the spring returns to an undeformed state, the elastic force always does positive work: its direction coincides with the direction of movement (Fig. 28.7). 
Let's find the work done by the elastic force.
The modulus of this force is related to the modulus of deformation x by the relation (see § 15)
The work done by such a force can be found graphically.
Let us first note that the work done by a constant force is numerically equal to the area of the rectangle under the graph of force versus displacement (Fig. 28.8). 
Figure 28.9 shows a graph of F(x) for the elastic force. Let us mentally divide the entire movement of the body into such small intervals that the force at each of them can be considered constant. 
Then the work on each of these intervals is numerically equal to the area of the figure under the corresponding section of the graph. All work is equal to the sum of work in these areas.
Consequently, in this case, the work is numerically equal to the area of the figure under the graph of the dependence F(x). 
7. Using Figure 28.10, prove that
the work done by the elastic force when the spring returns to its undeformed state is expressed by the formula
A = (kx 2)/2. (7)
8. Using the graph in Figure 28.11, prove that when the spring deformation changes from x n to x k, the work of the elastic force is expressed by the formula
From formula (8) we see that the work of the elastic force depends only on the initial and final deformation of the spring. Therefore, if the body is first deformed and then returns to its initial state, then the work of the elastic force is zero. Let us recall that the work of gravity has the same property.
9. At the initial moment, the tension of a spring with a stiffness of 400 N/m is 3 cm. The spring is stretched by another 2 cm.
a) What is the final deformation of the spring?
b) What is the work done by the elastic force of the spring?
10. At the initial moment, a spring with a stiffness of 200 N/m is stretched by 2 cm, and at the final moment it is compressed by 1 cm. What is the work done by the elastic force of the spring?
4. Work of friction force
Let the body slide along a fixed support. The sliding friction force acting on the body is always directed opposite to the movement and, therefore, the work of the sliding friction force is negative in any direction of movement (Fig. 28.12). 
Therefore, if you move the block to the right, and the peg the same distance to the left, then, although it will return to its initial position, the total work done by the sliding friction force will not be equal to zero. This is the most important difference between the work of sliding friction and the work of gravity and elasticity. Let us recall that the work done by these forces when moving a body along a closed trajectory is zero.
11. A block with a mass of 1 kg was moved along the table so that its trajectory turned out to be a square with a side of 50 cm.
a) Has the block returned to its starting point?
b) What is the total work done by the frictional force acting on the block? The coefficient of friction between the block and the table is 0.3.
5.Power
Often it is not only the work being done that is important, but also the speed at which the work is being done. It is characterized by power.
Power P is the ratio of the work done A to the time period t during which this work was done:
(Sometimes power in mechanics is denoted by the letter N, and in electrodynamics by the letter P. We find it more convenient to use the same designation for power.)
The unit of power is the watt (symbol: W), named after the English inventor James Watt. From formula (9) it follows that
1 W = 1 J/s.
12. What power does a person develop by uniformly lifting a bucket of water weighing 10 kg to a height of 1 m for 2 s?
It is often convenient to express power not through work and time, but through force and speed.
Let's consider the case when the force is directed along the displacement. Then the work done by the force A = Fs. Substituting this expression into formula (9) for power, we obtain:
P = (Fs)/t = F(s/t) = Fv. (10)
13. A car is traveling on a horizontal road at a speed of 72 km/h. At the same time, its engine develops a power of 20 kW. What is the force of resistance to the movement of the car?
Clue. When a car moves along a horizontal road at a constant speed, the traction force is equal in magnitude to the resistance force to the movement of the car.
14. How long will it take to rise evenly? concrete block weighing 4 tons to a height of 30 m, if the power of the crane motor is 20 kW, and the efficiency of the electric motor of the crane is 75%?
Clue. The efficiency of an electric motor is equal to the ratio of the work of lifting the load to the work of the engine. 
Additional questions and tasks
15. A ball weighing 200 g was thrown from a balcony with a height of 10 and an angle of 45º to the horizontal. Reaching in flight maximum height 15 m, the ball fell to the ground.
a) What is the work done by gravity when lifting the ball?
b) What is the work done by gravity when the ball is lowered?
c) What is the work done by gravity during the entire flight of the ball?
d) Is there any extra data in the condition?
16. A ball with a mass of 0.5 kg is suspended from a spring with a stiffness of 250 N/m and is in equilibrium. The ball is raised so that the spring becomes undeformed and released without a push.
a) To what height was the ball raised?
b) What is the work done by gravity during the time during which the ball moves to the equilibrium position?
c) What is the work done by the elastic force during the time during which the ball moves to the equilibrium position?
d) What is the work done by the resultant of all forces applied to the ball during the time during which the ball moves to the equilibrium position?
17. A sled weighing 10 kg slides down a snowy mountain with an inclination angle of α = 30º without initial speed and travels a certain distance along a horizontal surface (Fig. 28.13). The coefficient of friction between the sled and snow is 0.1. The length of the base of the mountain is l = 15 m. 
a) What is the magnitude of the friction force when the sled moves on a horizontal surface?
b) What is the work done by the friction force when the sled moves along a horizontal surface over a distance of 20 m?
c) What is the magnitude of the friction force when the sled moves along the mountain?
d) What is the work done by the friction force when lowering the sled?
e) What is the work done by gravity when lowering the sled?
f) What is the work done by the resultant forces acting on the sled as it descends from the mountain?
18. A car weighing 1 ton moves at a speed of 50 km/h. The engine develops a power of 10 kW. Gasoline consumption is 8 liters per 100 km. The density of gasoline is 750 kg/m 3, and its specific heat combustion 45 MJ/kg. What is equal to Engine efficiency? Is there any extra data in the condition?
Clue. The efficiency of a heat engine is equal to the ratio of the work performed by the engine to the amount of heat released during fuel combustion.
