What is perfect work in physics. Mechanical work and power of force

1. From the 7th grade physics course, you know that if a force acts on a body and it moves in the direction of the force, then the force does mechanical work A, equal to the product of the modulus of force and the modulus of displacement:

A=fs.

SI unit of work - joule (1 J).

[A] = [F][s] = 1 H 1 m = 1 N m = 1 J.

The unit of work is the work done by the force. 1 N on a way 1m.

It follows from the formula that mechanical work is not performed if the force is zero (the body is at rest or moves uniformly and rectilinearly) or the displacement is zero.

Suppose that the force vector acting on the body makes some angle a with the displacement vector (Fig. 65). Since the body does not move in the vertical direction, the force projection Fy per axle Y does not do work, but the projection of force Fx per axle X does work equal to A = F x s x.

Because the Fx = F cos a, and s x= s, then

In this way,

the work of a constant force is equal to the product of the modules of the vectors of force and displacement and the cosine of the angle between these vectors.

2. Let's analyze the resulting work formula.

If angle a = 0°, then cos 0° = 1 and A = fs. The work done is positive and its value is maximum if the direction of the force coincides with the direction of displacement.

If angle a = 90°, then cos 90° = 0 and A= 0. The force does not do work if it is perpendicular to the direction of movement of the body. Thus, the work of gravity is zero when a body moves along a horizontal plane. Zero is equal to the work of the force imparting centripetal acceleration to the body during its uniform motion in a circle, since this force at any point of the trajectory is perpendicular to the direction of motion of the body.

If angle a = 180°, then cos 180° = –1 and A = –fs. This case occurs when force and displacement are directed in opposite directions. Accordingly, the work done is negative and its value is maximum. Negative work is done, for example, by the force of sliding friction, since it is directed in the direction opposite to the direction of movement of the body.

If the angle a between the force and displacement vectors is acute, then the work is positive; if angle a is obtuse, then the work is negative.

3. We get the formula for calculating the work of gravity. Let the body mass m falls freely to the ground from a point A at the height h relative to the surface of the Earth, and after a while it turns out to be at a point B(Fig. 66, a). The work done by gravity is equal to

A = fs = mgh.

In this case, the direction of motion of the body coincides with the direction of the force acting on it, so the work of gravity in free fall is positive.

If a body moves vertically upward from a point B exactly A(Fig. 66, b), then its movement is directed in the direction opposite to gravity, and the work of gravity is negative:

A= –mgh

4. The work done by a force can be calculated using a force versus displacement graph.

Suppose a body moves under the influence of a constant force of gravity. Plot of the modulus of gravity F cord from the body movement module s is a straight line parallel to the x-axis (Fig. 67). Find the area of ​​the selected rectangle. It is equal to the product of its two sides: S = F heavy h = mgh. On the other hand, the same value is equal to the work of gravity A = mgh.

Thus, the work is numerically equal to the area of ​​the rectangle bounded by the graph, the coordinate axes and the perpendicular raised to the x-axis at the point h.

Consider now the case when the force acting on the body is directly proportional to the displacement. Such a force, as is known, is the force of elasticity. Its modulus is F extr = k D l, where D l- lengthening of the body.

Suppose a spring, the left end of which is fixed, was compressed (Fig. 68, a). At the same time, its right end shifted to D l 1. An elastic force has arisen in the spring F control 1, directed to the right.

If we now leave the spring to itself, then its right end will move to the right (Fig. 68, b), the elongation of the spring will be equal to D l 2, and the elastic force F exercise 2 .

Calculate the work of the elastic force when moving the end of the spring from the point with coordinate D l 1 to the point with coordinate D l 2. For this we use the dependency graph F control (D l) (Fig. 69). The work of the elastic force is numerically equal to the area of ​​the trapezoid ABCD. The area of ​​a trapezoid is equal to the product of half the sum of the bases and the height, i.e. S = AD. in a trapeze ABCD grounds AB = F ex 2 = k D l 2 , CD= F ex 1 = k D l 1 and the height AD=D l 1-D l 2. Substitute these quantities into the formula for the area of ​​a trapezoid:

S= (D l 1-D l 2) =– .

Thus, we have obtained that the work of the elastic force is equal to:

A =– .

5 * . Let us assume that a body of mass m moving from point A exactly B(Fig. 70), moving first without friction along an inclined plane from the point A exactly C, and then without friction along the horizontal plane from the point C exactly B. The work of gravity on the site CB is zero because the force of gravity is perpendicular to the displacement. When moving on an inclined plane, the work done by gravity is:

A AC = F heavy l sin a. Because l sin a = h, then A AC = ft heavy h = mgh.

The work of gravity when a body moves along a trajectory ACB is equal to A ACB = A AC + A CB = mgh + 0.

In this way, A ACB = mgh.

The result obtained shows that the work of gravity does not depend on the shape of the trajectory. It depends only on the initial and final positions of the body.

Let us now assume that the body moves along a closed trajectory ABCA(see fig. 70). When moving a body from a point A exactly B along the trajectory ACB the work done by gravity is A ACB = mgh. When moving a body from a point B exactly A gravity does negative work, which is equal to A BA = –mgh. Then the work of gravity on a closed trajectory A = A ACB + A BA = 0.

The work of the elastic force on a closed trajectory is also equal to zero. Indeed, suppose that a spring that was not deformed at the beginning was stretched and its length increased by D l. The elastic force does work A 1 = . When returning to a state of equilibrium, the elastic force does work A 2 = . The total work of the elastic force during the stretching of the spring and its return to the undeformed state is zero.

6. The work of the force of gravity and the force of elasticity on a closed trajectory is equal to zero.

Forces whose work on any closed trajectory is equal to zero (or does not depend on the shape of the trajectory) are called conservative.

Forces whose work depends on the shape of the trajectory are called non-conservative.

Friction force is non-conservative. For example, a body moves from a point 1 exactly 2 straight ahead first 12 (Fig. 71), and then along a broken line 132 . On each section of the trajectory, the friction force is the same. In the first case, the work of the friction force

A 12 = –F tr l 1 ,

and in the second -

A 132 = A 13 + A 32, A 132 = –F tr l 2 – F tr l 3 .

From here A 12A 132.

7. From the 7th grade physics course, you know that an important characteristic of devices that do work is power.

Power is a physical quantity equal to the ratio of work to the period of time for which it is done:

Power characterizes the speed of doing work.

Unit of power in SI - watt (1 W).

[N] === 1 W.

The unit of power is the power at which the work 1 J committed for 1 s .

Questions for self-examination

1. What is called work? What is the unit of work?

2. When does a force do negative work? positive work?

3. What is the formula for calculating the work of gravity? elastic force?

5. What forces are called conservative; non-conservative?

6 * . Prove that the work done by the force of gravity and the force of elasticity does not depend on the shape of the trajectory.

7. What is called power? What is the unit of power?

Task 18

1. A boy weighing 20 kg is pulled evenly on a sled, applying a force of 20 N. The rope, by which the sled is pulled, makes an angle of 30 ° with the horizon. What is the work of the elastic force arising in the rope if the sled moved 100 m?

2. An athlete weighing 65 kg jumps into the water from a tower located at a height of 3 m above the surface of the water. What work is done by the force of gravity acting on the athlete as he moves to the surface of the water?

3. Under the action of an elastic force, the length of a deformed spring with a stiffness of 200 N / m decreased by 4 cm. What is the work of the elastic force?

4 * . Prove that the work of a variable force is numerically equal to the area of ​​the figure bounded by the force-coordinate graph and the coordinate axes.

5. What is the traction force of a car engine if, at a constant speed of 108 km/h, it develops a power of 55 kW?

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.

–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.

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:

1. 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.
2. 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.
3. 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. 1 J = 1 kg m²/s² = 1 Nm
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 a few separate examples that will allow you to consider it from many, but not all, sides:

1. When a person lifts a stone with his hands, then 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 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;
4. There is also mechanical work when the friction force acts on the 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 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.

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 measurable quantity, there is a unit in which it is customary to measure it. Units of measurement are fixed and have the same meaning throughout the world.

The reason for this is the following. In 1960, 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 the definitions of units of measurement accepted throughout the world and their ratio.

Physical terms and terminology

In physics, the unit for measuring the work of a force is called J (Joule), in honor of the English physicist James Joule, who made a great contribution to the development of the section 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 with a mass of one kg (kilogram) at an acceleration of one m/s2 (meter per second) in the direction of the force.

Note. In physics, everything is interconnected, the performance of any work is associated with the performance of additional actions. An example is a household fan. When the fan is switched on, the fan blades begin to rotate. Rotating blades act on the air flow, giving it a directional movement. This is the result of work. But to perform the work, the influence of other external forces is necessary, without which the performance of the action is impossible. These include the strength of the electric current, power, voltage and many other interrelated values.

Electric current, in its essence, 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 (Pendant), named after the French scientist and inventor Charles Coulomb. In the SI system, it is a unit of measure for the number of charged electrons. 1 C is equal to the volume of charged particles flowing through the cross section of the conductor per unit time. The unit of time is one second. The formula for electric charge is shown below in the figure.

The strength of the electric current is denoted by the letter A (ampere). An ampere is a unit in physics that characterizes the measurement of the work of a force that is expended to move charges along a conductor. At its core, an electric current is an ordered movement of electrons in a conductor under the influence of an electromagnetic field. By conductor is meant a material or molten salt (electrolyte) that has little resistance to the passage of electrons. Two physical quantities affect the strength of an electric current: voltage and resistance. They will be discussed below. Current is always directly proportional to voltage and inversely proportional to resistance.

As mentioned above, electric current is the ordered movement of electrons in a conductor. But there is one caveat: for their movement, a certain impact is needed. This effect is created by creating a potential difference. The electrical charge can be positive or negative. Positive charges always tend to 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.

Power is the amount of energy expended to do work of one J (Joule) in a period of time of one second. The unit of measurement in physics is denoted 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.

Let the body, on which the force acts, pass, moving along a certain trajectory, the path s. In this case, the force either changes the speed of the body, imparting acceleration to it, or compensates for the action of another force (or forces) that opposes the movement. The action on the path s is characterized by a quantity called work.

Mechanical work is a scalar quantity equal to the product of the projection of the force on the direction of movement Fs and the path s traversed by the point of application of the force (Fig. 22):

A = Fs*s.(56)

Expression (56) is valid if the value of the projection of the force Fs on the direction of movement (i.e., on the direction of speed) remains unchanged all the time. In particular, this takes place when the body moves in a straight line and a force of constant magnitude forms a constant angle α with the direction of motion. Since Fs = F * cos(α), expression (47) can be given the following form:

A = F*s*cos(α).

If is a displacement vector, then the work is calculated as the scalar product of two vectors and :

. (57)

Work is an algebraic quantity. If the force and direction of movement form an acute angle (cos(α) > 0), the work is positive. If the angle α is obtuse (cos(α)< 0), работа отрицательна. При α = π/2 работа равна нулю. Последнее обстоятельство особенно отчетливо показывает, что понятие работы в механике существенно отличается от обыденного представления о работе. В обыденном понимании всякое усилие, в частности и мускульное напряжение, всегда сопровождается совершением работы. Например, для того чтобы держать тяжелый груз, стоя неподвижно, а тем более для того, чтобы перенести этот груз по горизонтальному пути, носильщик затрачивает много усилий, т. е. «совершает работу». Однако это – «физиологическая» работа. Механическая работа в этих случаях равна нулю.

Work when moving under the influence of force

If the magnitude of the projection of the force on the direction of movement does not remain constant during movement, then the work is expressed as an integral:

. (58)

An integral of this kind in mathematics is called a curvilinear integral along the trajectory S. The argument here is a vector variable , which can vary both in absolute value and in direction. Under the integral sign is the scalar product of the force vector and the elementary displacement vector.

A unit of work is the work done by a force equal to one and acting in the direction of movement, on a path equal to one. in SI The unit of work is the joule (J), which is equal to the work done by a force of 1 newton in a path of 1 meter:

1J = 1N * 1m.

In the CGS, the unit of work is the erg, which is equal to the work done by a force of 1 dyne in a path of 1 centimeter. 1J = 10 7 erg.

Sometimes a non-systemic unit kilogrammeter (kg * m) is used. This is the work done by a force of 1 kg on a path of 1 meter. 1kg*m = 9.81 J.