IN Everyday life I often come across such a concept as work. What does this word mean in physics and how to determine the work of the elastic force? You will find out the answers to these questions in the article.

Mechanical work

Work is a scalar algebraic quantity that characterizes the relationship between force and displacement. If the direction of these two variables coincides, it is calculated using the following formula:

• F- module of the force vector that does the work;
• S- displacement vector module.

A force that acts on a body does not always do work. For example, the work done by gravity is zero if its direction is perpendicular to the movement of the body.

If the force vector forms a non-zero angle with the displacement vector, then another formula should be used to determine the work:

A=FScosα

α - the angle between the force and displacement vectors.

Means, mechanical work is the product of the projection of force on the direction of displacement and the module of displacement, or the product of the projection of displacement on the direction of force and the module of this force.

Mechanical work sign

Depending on the direction of the force relative to the movement of the body, the work A can be:

• positive (0°≤ α<90°);
• negative (90°<α≤180°);
• equal to zero (α=90°).

If A>0, then the speed of the body increases. An example is an apple falling from a tree to the ground. At A<0 сила препятствует ускорению тела. Например, действие силы трения скольжения.

The SI (International System of Units) unit of work is Joule (1N*1m=J). A joule is the work done by a force, the value of which is 1 Newton, when a body moves 1 meter in the direction of the force.

Work of elastic force

The work of force can also be determined graphically. To do this, calculate the area of ​​the curvilinear figure under the graph F s (x).

Thus, from the graph of the dependence of the elastic force on the elongation of the spring, one can derive the formula for the work of the elastic force.

It is equal to:

A=kx 2 /2

• k- rigidity;
• x- absolute elongation.

What have we learned?

Mechanical work is performed when a force is applied to a body, which leads to movement of the body. Depending on the angle that occurs between the force and the displacement, the work can be zero or have a negative or positive sign. Using the example of elastic force, you learned about a graphical method for determining work.

Do you know what work is? Without any doubt. Every person knows what work is, 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 understanding of this process. But we are not talking about them now. Even fewer people have any idea what it is mechanical work from the point of view of physics. In physics, mechanical work is not human labor for food, it is a physical quantity that may be completely unrelated to either a person or any other living creature. How so? Let's figure it out now.

Mechanical work in physics

Let's give two examples. In the first example, the waters of the river, faced with an abyss, noisily fall down in the form of a waterfall. The second example is a man who holds a heavy object in his outstretched arms, for example, holding the broken roof over the porch of a country house from falling, while his wife and children are frantically looking for something to support it with. When is mechanical work performed?

Definition of mechanical work

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 - strength,
s is the distance traveled.

So, despite all the heroism of the tired roof holder, the work he has done is 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 apply 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 and the distance over which 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 over 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 a 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 movement of the body in any way, then the force performed by this work is equal to zero.

The horse pulls the cart with some force, let's denote it F traction. Grandfather, sitting on the cart, presses on it with some force. Let's denote it F pressure The cart moves along the direction of the horse's traction force (to the right), but in the direction of the grandfather's pressure force (downward) the cart does not move. That's why in physics they say that F traction does work on the cart, and F the pressure does not do work on the cart.

So, work of force on the body or mechanical work– a physical quantity whose modulus is equal to the product of the force and the path traveled by the body along the direction of action of this force s:

In honor of the English scientist D. Joule, the unit of mechanical work was named 1 joule(according to the formula, 1 J = 1 N m).

If a certain force acts on the body in question, then some body acts on it. That's why the work of force on the body and the work of the body on the body are complete synonyms. However, the work of the first body on the second and the work of the second body on the first are partial synonyms, since the moduli of these works are always equal, and their signs are always opposite. That is why there is a “±” sign in the formula. Let's discuss the signs of work in more detail.

Numerical values ​​of force and path are always non-negative quantities. In contrast, mechanical work can have both positive and negative signs. If the direction of the force coincides with the direction of motion of the body, then the work done by the force is considered positive. If the direction of force is opposite to the direction of motion of the body, the work done by a force is considered negative(we take “–” from the “±” formula). If the direction of motion of the body is perpendicular to the direction of the force, then such a force does not do any work, that is, A = 0.

Consider three illustrations of three aspects of mechanical work.

Doing work by force may look different from the perspective of different observers. Let's consider an example: a girl rides up in an elevator. Does it perform mechanical work? A girl can do work only on those bodies that are acted upon by force. There is only one such body - the elevator cabin, since the girl presses on its floor with her weight. Now we need to find out whether the cabin goes a certain way. Let's consider two options: with a stationary and moving observer.

Let the observer boy sit on the ground first. In relation to it, the elevator car moves upward and passes a certain distance. The girl’s weight is directed in the opposite direction - down, therefore, the girl performs negative mechanical work on the cabin: A dev< 0. Вообразим, что мальчик-наблюдатель пересел внутрь кабины движущегося лифта. Как и ранее, вес девочки действует на пол кабины. Но теперь по отношению к такому наблюдателю кабина лифта не движется. Поэтому с точки зрения наблюдателя в кабине лифта девочка не совершает механическую работу: A dev = 0.

Mechanical work is an energy characteristic of the movement of physical bodies, which has a scalar form. It is equal to the modulus of the force acting on the body, multiplied by the modulus of the displacement caused by this force and by the cosine of the angle between them.

Formula 1 - Mechanical work.

F - Force acting on the body.

s - Body movement.

cosa - Cosine of the angle between force and displacement.

This formula has a general form. If the angle between the applied force and the displacement is zero, then the cosine is equal to 1. Accordingly, the work will be equal only to the product of the force and the displacement. Simply put, if a body moves in the direction of application of force, then mechanical work is equal to the product of force and displacement.

The second special case is when the angle between the force acting on the body and its displacement is 90 degrees. In this case, the cosine of 90 degrees is equal to zero, so the work will be equal to zero. And indeed, what happens is that we apply force in one direction, and the body moves perpendicular to it. That is, the body clearly does not move under the influence of our force. Thus, the work done by our force to move the body is zero.

Figure 1 - Work of forces when moving a body.

If more than one force acts on a body, then the total force acting on the body is calculated. And then it is substituted into the formula as the only force. A body under the influence of force can move not only rectilinearly, but also along an arbitrary trajectory. In this case, the work is calculated for a small section of movement, which can be considered rectilinear, and then summed up along the entire path.

Work can be both positive and negative. That is, if the displacement and force coincide in direction, then the work is positive. And if a force is applied in one direction, and the body moves in another, then the work will be negative. An example of negative work is the work of a frictional force. Since the friction force is directed counter to the movement. Imagine a body moving along a plane. A force applied to a body pushes it in a certain direction. This force does positive work to move the body. But at the same time, the friction force does negative work. It slows down the movement of the body and is directed towards its movement.

Figure 2 - Force of motion and friction.

Mechanical work is measured in Joules. One Joule is the work done by a force of one Newton when moving a body one meter. In addition to the direction of movement of the body, the magnitude of the applied force can also change. For example, when a spring is compressed, the force applied to it will increase in proportion to the distance traveled. In this case, the work is calculated using the formula.

Formula 2 - Work of compression of a spring.

k is the spring stiffness.

x - moving coordinate.

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 the geometric meaning of the 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.

.

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

- basic work.

–work in curved motion.

Example 1: 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 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
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 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. Consider a material point moving under the influence of a force . The work of this force increases the kinetic energy of a material point
. In this case, let us calculate the small increment (differential) of kinetic energy:

When calculating
Newton's second law was used
, and
- 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 of 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.