For the quantitative characteristics of the energy exchange process between the interacting bodies in the mechanics, the concept of "work of force" is used.

With straightforward movement of the body and the action on it constant strength ($ \\ overline (f) $), which is a certain angle of $ \\ alpha $ with the direction of body movement ($ \\ overline (S) $), the operation of force ($ A $) is the value equal:

From formula (1) it follows that with $ \\ alpha \\ frac (\\ pi) (2) $ work force is a positive value, while the projection of force on the direction of movement coincides with the direction of the velocity vector of the body. With $ \\ alpha \u003d \\ frac (\\ pi) (2) $ work work is zero.

When exposed to the body, the force may vary both in magnitude and in the direction, therefore, for a general case, the expression (1) does not apply to the calculation of mechanical work. Enroll as follows. Consider the infinitely small movement of the body ($ D \\ Overline (S) $) at which the force can be considered a constant, and the movement of the point of the force application is straightforward. Then the elementary work ($ DA $) $ \\ overline (f) $ for the movement of $ D \\ Overline (S) $ is called scalar valueequal to:

where $ \\ alpha $ is the angle between the vectors of $ \\ overline (F \\) and \\ d \\ overline (s) $; $ \\ left | D \\ Overline (s) \\ Right | $ - elementary path. At the same time, the mechanical work of force on the trajectory site from one point to another is found as an algebraic amount elementary work In some small plots. In most cases, summation is replaced by integration:

In order to calculate the integral (3), it is necessary to know the dependence of force from the path along the trajectory from the first point to the second. If the dependence of force from the path is set graphically, then the mechanical work is equal to the area of \u200b\u200ba curvilinear trapezium, which is limited to the inside of the abscissa, at the top of the F (S) schedule, on the right and on the left of the ordinates of the extreme points.

Unit of measurement of work in the international system of units (SI) is Joule (J). One Joule is the work that the force is in one Newton on the way one meter.

\\ [\\ left \u003d 1n \\ Cdot 1m \u003d 1J. \\]

Work and kinetic body energy, work of conservative forces

Elementary mechanical work is equal to an infinitely small change in the kinetic energy of the body ($ de_k $):

The work of the force at the final portion of the path is equal to the change in the kinetic energy of the body:

$ E_ (K2) ;; E_ (K1) $ - the kinetic energy of the body in the final and starting points of the trajectory. The expression (5) is performed when the bodies move with any speeds.

The operation of the conservative forces is equal to the change in the potential energy ($ E_p $) of the system of interacting tel:

Formulas for calculating the work of some forces

The work of the strength of elasticity during stretching the spring can be found as:

where $ k $ is the coefficient of elasticity; $ \\ x_2-x_1 $ - the extension of the spring when it changes its length. When stretching the spring, the work of the force of elasticity is negative.

The work of the coulon for the movement of the charge from the point, which is determined by the $ radius (\\ Overline (R)) _ $ 1 to the point determined by the $ radius-vector (\\ Overline (R)) _ $ 2 is equal to:

$ R_1 $; $ \\ r_2 $ - the length of radius-vectors of the initial and endpoints of the trajectory of the movement of the point of the application of the work of work; $ Q_1, Q_2 $ - electric charges. With increasing distance between the charges of the repulsion force, there is a positive mechanical work, the attraction force is negative. The work of the power of the coulon does not depend on the trajectory of the body movement.

The work of the gravity forces is calculated using the formula:

$ m_1, m_2 $ - mass of interacting bodies; $ \\ gamma $ - gravitational constant. The work of the gravity forces does not depend on the trajectory of the movement of tel. It is defined only by radius-vectors of the initial and endpoint of the trajectory.

Examples of tasks with the solution

Example 1.

The task. The body has a mass equal to $ M $. It is raised with acceleration $ a $. What is the work of raising force if the body raised to the height of $ H $?

Decision. Make a drawing.

Using Newton's second law, relying on Fig. 1 we find the amount of force that performs mechanical work:

In the projection on the Y axis, equation (1.1) has the form:

express F from (1.2): \\

If the force when moving the body remains constant, then we will find work using the formula:

where, by the condition of the task of $ s \u003d h $. Figure 1 shows that the direction of force coincides with the direction of movement, so the final formula for work takes the form:

Answer. $ A \u003d m \\ left (a + g \\ right) h $

Example 2.

The task. Some body weighing $ m $ raise vertically up from the surface of the Earth, acting on it by the force of $ \\ overline (f) $. The force varies depending on the height by law: $ \\ overline (f) \u003d - 2m \\ Overline (G) (1-CY) $, where $ c \u003d const\u003e $ 0 Considering the field of gravity homogeneous define, what work does the force on the first third of the rise? The initial body speed is zero.

Decision. Find the height of the lifting of the body. From the law of change of force with a height:

\\ [\\ Overline (f) \u003d - 2m \\ Overline (G) (1-CY) (2.1) \\]

obviously, the body will rise until the force becomes equal to zero. From this condition we will find the height of the lift:

\\ [- 2m \\ Overline (G) \\ Left (1-Cy \\ Right) \u003d 0 \\ to 2m \\ Overline (G) \\ Ne 0 \\ to 1-Cy \u003d 0 \\ To y \u003d \\ FRAC (1) (C) . \\]

We will look for work using its definition:

where $ ds \u003d dy $ Since the movement occurs along the Y axis; From the equation $ \\ overline (f) (y) $ it follows that $ \\ overline (f) \\ Uparrow \\ Uparrow D \\ Overline (S) $, formula (2.2) will be presenting as:

\\) \u003d \\ FRAC (5MG) (9C).) \\]

Answer. $ A \u003d \\ FRAC (5mg) (9C) $

To be able to characterize energy characteristics Movements, the concept of mechanical work was introduced. And it was her in her various manifestations an article is devoted. To understand the topic at the same time and light, and quite complicated. The author sincerely tried to make it more understandable and accessible to understand, and remains only to hope that the goal is achieved.

What is called mechanical work?

What is so called? If there is some force over the body, and as a result of the action, the body moves, then this is called mechanical work. In approach from the point of view of scientific philosophy, several additional aspects can be distinguished here, but the article will be disclosed in terms of physics. Mechanical work - It is not difficult if the words written here are well. But the word "mechanical" is usually not written, and everything is reduced to the word "work". But not every work is mechanical. Here is sitting a man and thinks. Does he work? Mentally yes! But does this work mechanical? Not. And if a person goes? If the body moves under the action of force, then this is a mechanical work. Everything is simple. In other words, the force acting on the body performs (mechanical) work. And more: it is the work that you can characterize the result of the action of a certain force. So the man goes, then certain forces (friction, gravity, etc.) make a mechanical work over a person, and as a result of their actions, a person changes the point of his stay, in other words moves.

Work as a physical value equals strength, which acts on the body that is multiplier by the path, which made the body under the influence of this force and in the direction indicated by it. It can be said that the mechanical work was made if 2 conditions were observed at the same time: the strength acted on the body, and it moved into its direction. But it was not committed or not accomplished if the strength 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 catch up with a huge boulder in order to move it, but there is no power. The force acts on the stone, and it does not move, and the work does not happen.
2. The body moves in the coordinate system, and the force equals zero or they are all compensated. This can be observed while inertia movement.
3. When the direction in which the body moves perpendicular to the action of force. When the train moves along the horizontal line, the strength of gravity does not make its work.

Depending on certain conditions, mechanical work is negative and positive. So, if the directions and forces, and the movement of the body are the same, then there is a positive work. An example of positive work is the effect of gravity on a falling water drop. But if the power and direction of movement are opposite, then negative mechanical work is happening. An example of such an option is the upward air ball and gravity, which makes negative work. When the body is influenced by several forces, such a job is called the "work of the resulting force."

Features of practical application (kinetic energy)

Go from the theory to the practical part. Separately, you should talk about mechanical work and its use in physics. As many probably remembered, all body energy is divided into kinetic and potential. When the object is in the equilibrium position and does not move anywhere, its potential energy equals total energy, and kinetic equals zero. When the movement begins, the potential energy begins to decrease, kinetic grow, but in sum they are equal to the total energy of the object. For a material point, kinetic energy is defined as the work of the force, which accelerated the point from zero to the value of H, and in the formula of the kinetics of the body is ½ * m * n, where M is a mass. To learn the kinetic energy of the object, which consists of a variety of particles, it is necessary to find the amount of the entire kinetic energy of the particles, and it will be kinetic energy 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 general one, then the work is not performed. 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 that gas performs during expansion is calculated on the integral of multiplication of pressure on the volume. This approach is applicable not only in cases where there is an accurate volume function, but also to all processes that can be displayed in the pressure plane / volume. Also applies knowledge of mechanical work not only to gases, but also to everything that can put pressure.

Features of practical application in practice (theoretical mechanics)

In theoretical mechanics, all of the above properties and formulas are considered in more detail, in particular these projections. It gives its definition for various formulas for mechanical work (an example of determining for the integral of the RMERMER): the limit to which the sum of all elementary work forces seeks when the partness of the partition tends to zero value, is called the operation of force along the curve. Probably difficult? But nothing, with theoretical mechanics, everything. Yes, and all mechanical work, physics and other difficulties ended. Then there will be only examples and conclusion.

Mechanical Operation Units

To measure work in si, Jouley uses, and the GHS uses ERG:

1. 1 j \u003d 1 kg · m² / s² \u003d 1 n · m
2. 1 erg \u003d 1 g · cm² / s² \u003d 1 din · cm
3. 1 erg \u003d 10 -7 j

Examples of mechanical work

In order to figure it out completely with such a concept as a mechanical work, several separate examples should be examined, which will consider it from the set, but not all sides:

1. When a person raises the stone with his hands, then there is a mechanical work with the help of muscular strength of the hands;
2. When the train is driving along the rails, it pulls the force of the traction (electric locomotive, diesel locomotive, etc.);
3. If you take a gun and shoot it, then thanks to the power of the pressure that the powder gases will create, work will be made: the bullet moves along the rifle barrel simultaneously with the increase in the bullet itself;
4. There is a mechanical work and then when the friction force acts on the body, forcing it to reduce the speed of its movement;
5. The example described with the balls when they rise in the opposite direction relative to the direction of gravity, is also an example of mechanical work, but the force of the Archimedes is also valid, when the strength of the Archimedes is also applied when everything is lifted.

What is power?

Finally, I want to touch the topic of power. The work of the force that is performed in one unit of time, and is called power. In fact, the power is such a physical value that is the display of the work attitude towards a certain period of time, during which this work was performed: M \u003d P / B, where M is the power, p - work, in-time. The power unit in C is denoted in 1 W. Watt equals the power that makes a job in one joule in one second: 1 W \u003d 1J \\ 1C.

Do you know what work is? Without any doubt. What is work, knows every person, provided that he is born and lives on the planet Earth. What is mechanical work?

This concept is also known to most people on the planet, although some individual personalities have a rather vague idea of \u200b\u200bthis process. But it's not about them now. An even fewer people have an idea what mechanical work in terms of physics. In physics, mechanical work is not a person's work for food for food, this is a physical value that can not be completely connected with anyone with any other living being. How so? We'll see now.

Mechanical work in physics

We give two examples. In the first example of the water of the river, encountered with the precipice, falling down in the form of a waterfall. The second example is a person who holds a heavy item on the elongated hands, for example, holds the roof over the roof over the porch of the country's house from falling, while his wife and children seek convulsively than to rest. In which case is the mechanical work?

Definition of mechanical work

Almost everything, without thinking, will answer: in the second. And they will be wrong. The situation is just the opposite. In physics, mechanical work is described the following definitions: Mechanical work is performed when the power acts on the body, and it moves. The mechanical work is directly proportional to the applied strength and the path traveled.

Formula of mechanical work

The mechanical work is determined by the formula:

where a is work
F - force,
s - traveled path.

So, despite all the heroism of the tired roof holder, the work done by them is zero, but water falling under the action of gravity from a high rock, makes the most that neither there is mechanical work. That is, if we push the heavy wardrobe unsuccessfully, then the work that we did from the point of view of physics will be zero, despite the fact that we make a lot of strength. But if we slide the closet for a while, then we will do the work equal to the product of the applied force at a distance that we moved the body.

The unit of work is 1 J. This is a work performed by force in 1 Newton, on the movement of the body for a distance of 1 m. If the direction of the applied force coincides with the direction of the body movement, then this force makes positive operation. An example is when we pushing any body, and it moves. And in the case when the force is applied to the opposite body movement, for example, the friction force, this force makes a negative work. If the attached force does not affect the movement of the body, the force performed by this work is zero.