What is mechanical work. School encyclopedia

« Physics - Grade 10 »

The law of conservation of energy is the fundamental law of nature, which allows to describe most occurring phenomena.

The description of the body movement is also possible with the help of the concepts of dynamics as work and energy.

Remember what work is and power in physics.

Do these concepts coincide with household ideas about them?

All our daily actions are reduced to the fact that we are using muscles or lead the surrounding bodies in motion and support this movement, or stop the moving bodies.

These bodies are tools of labor (hammer, handle, saw), in games - balls, washers, chess pieces. In production and agriculture, people also lead to the movement of the tools of labor.

The use of machines many times increases productivity due to the use of engines in them.

The purpose of any engine is to lead the body in motion and maintain this movement, despite the braking of both ordinary friction and the "working" resistance (the cutter should not just slide on the metal, and crashed into it, to shoot a chips; the plow must explode Earth, etc.). At the same time, the power should act on the moving body from the engine.

Work is performed in nature always when the strength (or a few forces) is acting on any body in the direction of his movement or against it (other bodies).

The strength of gravity makes a job when dropping raindrops or stone with a cliff. At the same time, the strength of the resistance acting on the falling drops or the stone from the air is performed. Makes the work and the power of elasticity, when the tree bent bent bent.

Definition of work.

Newton's second law in impulse form Δ \u003d Δt. Allows you to determine how the body's velocity is changed in the module and direction, if the force has been operating at it during it.

The impact on the body of the forces leading to the change in the module of their speed is characterized by the value depending both of the forces and the movements of the tel. This magnitude in the mechanics and call work of power.

The change in the velocity of the module is possible only when the projection of the force F R to the direction of movement of the body is different from zero. It is this projection that determines the effect of force that changes the body's velocity by module. She makes a job. Therefore, work can be viewed as a product of the projection of force F R on the movement module |Δ| (Fig. 5.1):

A \u003d f R | Δ |. (5.1)

If the angle between the force and movement is denoted by α, then F r \u003d fcosα.

Consequently, the work is equal to:

A \u003d | Δ | COSα. (5.2)

Our everyday idea of \u200b\u200bwork differs from the definition of work in physics. You keep a heavy suitcase, and it seems to you that you make a job. However, from the point of view of ishing, your work is zero.

The work of constant strength is equal to the product of the modules of force and moving the point of the application of the force and the cosine of the angle between them.

In the general case, when driving a solid body, moving its different points is different, but when determining the work of force, we Δ We understand the movement of her point of application. With a firmware progressive movement, the movement of all its points coincides with the movement of the point of the application of the force.

Work, in contrast to strength and movement, is not a vector, but scalar value. It can be positive, negative or equal to zero.

The work sign is determined by the cosine sign of the corner between the force and movement. If< 90°, то А > 0, as the cosine of sharp corners is positive. At α\u003e 90 °, the operation is negative, since the cosine of stupid angles is negative. At α \u003d 90 ° (the force perpendicular to the movement), the work is not performed.

If there are several forces on the body, the projection of the resulting force on the movement is equal to the amount of projections of individual forces:

F R \u003d F 1R + F 2R + ... .

Therefore, to work as a resulting force we get

A \u003d F 1R | Δ | + F 2R | Δ | + ... \u003d a 1 + and 2 + .... (5.3)

If there are several strength on the body, then complete work (the algebraic amount of work of all forces) is equal to the work of the resultant force.

Perfect work work can be represented graphically. I will explain this by depicting the dependence of the projection of the force from the coordinate of the body when it moves in a straight line.

Let the body move along the axis oh (Fig. 5.2), then

Fcosα \u003d f x, | Δ | \u003d Δ x..

To work forces we get

A \u003d f | δ | cosα \u003d f x Δx.

It is obvious that the area of \u200b\u200bthe rectangle shaded in Figure (5.3, a) is numerically equal to operation when moving the body from the point with the coordinate x1 to the point with the coordinate X2.

Formula (5.1) is valid when the projection of force on the movement is constant. In the case of a curvilinear trajectory, constant or variable force, we divide the trajectory for small segments that can be considered straightforward, and the projection of force on a small movement Δ - constant.

Then calculating the work on each movement Δ And then summing up these works, we determine the work of force on the final movement (Fig. 5.3, b).

Unit of work.

The unit of work can be established using the main formula (5.2). If, when moving the body per unit length, the force is valid for it, the module of which is equal to one, and the direction of force coincides with the direction of moving its point of application (α \u003d 0), then the work will be equal to one. In the international system (s), the unit of work is Joule (denotes J):

1 j \u003d 1 n 1 m \u003d 1 nm.

Joule - This is the work performed by force 1 H on moving 1 If the direction of force and movement coincide.

Often use multiple units of work - Kilodzhoule and mega Joule:

1 kJ \u003d 1000 J,
1 MJ \u003d 1000000 J.

Work can be done both in a large period of time and for very small. In practice, however, it is not indifferent, work quickly or slowly work. The time during which work is performed, the performance of any engine is determined. A tiny electric motor can make a very big job, but it will take a lot of time. Therefore, along with work, the size of which characterizes the speed with which it is produced is power.

Power is the ratio of work A at the time interval Δt, for which this work is done, i.e. power is the speed of work:

Substituting in formula (5.4) instead of work and its expression (5.2), we get

Thus, if the power and speed of the body are constant, the power is equal to the product of the power vector module on the velocity vector module and the corner module between the directions of these vectors. If these values \u200b\u200bare variables, then according to formula (5.4), you can determine the average power is similar to the definition of the average body movement.

The concept of power is introduced to evaluate work per unit of time performed by any mechanism (pump, lifting crane, motor motor, etc.). Therefore, in formulas (5.4) and (5.5), the force of thrust is always meant.

In the power, power is expressed in watts (W).

Power is 1 W if the work equal to 1 J is performed for 1 s.

Along with Watt, larger (multiple) power units are used:

1 kW (kilowatt) \u003d 1000 W,
1 MW (Megawatt) \u003d 1 000 000 W.

With mechanical work (work of force) you are already familiar from the course of physics of the main school. We will remind the definition of mechanical work there for the following cases.

If the force is directed just like the movement of the body, then the work of force

In this case, the work of the force is positive.

If the force is directed opposite to the movement of the body, then the work of force

In this case, the work of the force is negative.

If the force F_VEC is directed perpendicular to the movement of s_vec of the body, then the operation of the force is zero:

Work is a scalar value. The unit of work is called Joule (indicate: J) in honor of the English scientist James Joule, who played an important role in the opening of the law of energy conservation. From formula (1) follows:

1 j \u003d 1 n * m.

1. The bar weighing 0.5 kg moved across the table by 2 m, applying the force of elasticity equal to 4 H (Fig. 28.1). The friction coefficient between the bar and the table is 0.2. What is equal to the work of acting on the bar:
a) gravity m?
b) the forces of normal reaction?
c) the strength of elasticity?
d) friction force slip TR?

The total work of several forces acting on the body can be found in two ways:
1. Find the work of each strength and fold these works with the signs of signs.
2. Find the equal to all the forces attached to the body and calculate the work of the resultant.

Both methods lead to the same result. To make sure that you return to the previous task and answer questions about Questions 2.

2. What is equal to:
a) Am the work of all the forces acting on the bar?
b) the resultant of all forces acting on the bar?
c) work as an equal? In general, (when the force F_VEC is directed at an arbitrary angle to the movement S_VEC) determination of the operation of the force is such.

The operation A permanent strength is equal to the product of the force module F per module of movement S and on the cosine of the angle α between the direction of force and the direction of movement:

A \u003d fs cos α (4)

3. Show that out of the general definition of work follow the conclusions shown in the following scheme. Word them verbally and write down in a notebook.

4. A force is applied to the Bruck on the table, the module of which is 10 N. What is the angle between this force and the movement of the bar, if, when moving the bar on the table, this force made this force: a) 3 J; b) -3 j; c) -3 j; d) -6 J? Make explanatory drawings.

2. Work of gravity

Let the body mass M move vertically from the initial height H H to the ultimate height H to.

If the body moves down (h n\u003e h to, Fig. 28.2, a), the direction of movement coincides with the direction of gravity, so the work of gravity is positive. If the body moves up (H n< h к, рис. 28.2, б), то работа силы тяжести отрицательна.

In both cases, the work of gravity

A \u003d mg (h n - h to). (five)

We will now find the work of gravity when driving at an angle to the vertical.

5. The small lump mass M slipped along the inclined plane of the length S and the height H (Fig. 28.3). The inclined plane is the angle α with a vertical.

a) What is the angle between the direction of gravity and the direction of moving the bar? Make an explanatory drawing.
b) Express the work of gravity through M, G, S, α.
c) Express s via H and α.
d) Express the work of gravity through M, G, H.
e) What is the work of gravity strength when driving along the entire same plane up?

After completing this task, you made sure that the work of gravity is expressed by formula (5) and then when the body moves at an angle to the vertical - both down and up.

But then formula (5) for the operation of gravity is valid when the body moves along any trajectory, because any trajectory (Fig. 28.4, a) can be represented as a combination of small "inclined planes" (Fig. 28.4, b).

In this way,
work of gravity when driving but any trajectory is expressed by the formula

A T \u003d Mg (H H - H to),

where H H is the initial height of the body, H to - its final height.
The work of gravity does not depend on the form of the trajectory.

For example, the work of gravity when moving the body from point A to point B (Fig. 28.5) along the trajectory 1, 2 or 3 is the same. From here, in particular, it follows that the ribot of gravity when moving along a closed trajectory (when the body returns to the starting point) is zero.

6. The ball mass M, hanging on the thread Length L, rejected 90º, holding a strained thread, and released without a push.
a) what is the work of gravity for the time during which the ball moves to the position of equilibrium (Fig. 28.6)?
B) What is the work of the force of the elasticity of the thread for the same time?
c) What is the work of the equal forces attached to the ball, during the same time?

3. Work of elasticity

When the spring returns to an undeformed state, the strength of elasticity is always positive: its direction coincides with the direction of movement (Fig. 28.7).

We find the work of the force of elasticity.
The module of this force is associated with the deformation module X by the ratio (see § 15)

The work of this force can be found graphically.

We first note that the work of constant strength is numerically equal to the area of \u200b\u200bthe rectangle under the chart of the dependence of the force from movement (Fig. 28.8).

Figure 28.9 shows a graph of the dependence f (x) for the force of elasticity. We break mentally all the movement of the body to such small gaps so that on each of them the force can be considered constant.

Then work on each of these gaps is numerically equal to the area of \u200b\u200bthe figure under the appropriate section of the chart. All the work is equal to the amount of work in these areas.

Therefore, in this case, the work is numerically equal to the area of \u200b\u200bthe figure under the graph of the dependence f (x).

7. Using Figure 28.10, prove that

the work of the force of elasticity at the return of the spring to the undeformed state is expressed by the formula

A \u003d (KX 2) / 2. (7)

8. Using a graph in Figure 28.11, prove that when the springs deformation changes from X N to x to the work of the force of elasticity is expressed by the formula

From formula (8) we see that the work of the force of elasticity depends only on the initial and final deformation of the spring, so if the body is first deformed, and then it returns to the initial state, then the work of the force of elasticity is zero. Recall that the work of gravity is also possessed as the same property.

9. In the initial moment, the stretching of the spring with stiffness of 400 N / m is 3 cm. The spring was stretched for another 2 cm.
a) What is the final deformation of the spring?
b) What is the work of the Spring's elasticity force?

10. At the initial moment of the spring, the rigidity of 200 n / m is stretched by 2 cm, and at the end moment it is compressed by 1 cm. What is the operation of the force of elasticity of the spring?

4. Work of friction force

Let the body slides on a fixed support. The gliding friction force acting on the body is always directed opposite to movement and, therefore, the work of the slide friction force is negative at any direction of movement (Fig. 28.12).

Therefore, if you move the bar to the right, and Peg to the same distance left, then, although it will return to the initial position, the total work of the slip of the slip will not be zero. This consists of the most important difference in the work of the friction force of sliding from the work of gravity and the strength of elasticity. Recall that the work of these forces when moving the body on a closed trajectory is zero.

11. Laming weighing 1 kg moved on the table so that its trajectory was the square with a side of 50 cm.
a) whether the bar returned to the starting point?
B) What is the total work of the friction force acting on the bar? The friction coefficient between the bar and the table is 0.3.

5. Power

Not only work performed, but also the speed of work is often important. It is characterized by power.

The power of P is called the ratio of perfect work A by the time interval T, for which this work is done:

(Sometimes the power in mechanics is denoted by the letter n, and in the electrodynamics - the letter P. We consider the same power designation more convenient.)

The power unit is Watt (denotes: W), named after the English inventor James Watt. From formula (9) it follows that

1 W \u003d 1 j / c.

12. What power develops a person evenly raising a water bucket weighing 10 kg to a height of 1 m within 2 s?

Often, the power is convenient to express not through work and time, but through force and speed.

Consider the case when the force is directed along the movement. Then the work of force A \u003d FS. Substituting this expression in formula (9) for power, we get:

P \u003d (FS) / T \u003d F (S / T) \u003d FV. (10)

13. The car rides horizontal road at a speed of 72 km / h. At the same time, its engine develops the power of 20 kW. What is the resistance force of the car's movement?

Prompt. When the car moves along a horizontal road at a constant speed, the thrust force is equal to the module by the power of resistance to the movement of the car.

14. How long is it necessary for a uniform lifting of the concrete block weighing 4 tons to a height of 30 m, if the power of the lifting crane engine is 20 kW, and the efficiency of the electric motor of the lifting crane is 75%?

Prompt. The efficiency of the electric motor is equal to the cost of working on the lifting of the engine to the operation of the engine.

Additional questions and tasks

15. The ball weighing 200 g was thrown from a balcony with a height of 10 and at an angle of 45º to the horizon. Having reached a maximum height of 15 m in flight, the ball fell to the ground.
a) what is the work of gravity when lifting the ball?
b) What is the work of gravity, when the ball is descending?
B) What is the work of gravity for all the time of the ball?
d) Is there any extra data on the condition?

16. The ball with a mass of 0.5 kg is suspended to the rigidity of 250 N / m and is in equilibrium. The ball is raised so that the spring becomes undeformed, and they are released without a push.
a) What height raised the ball?
b) What is the work of gravity for the time during which the ball is moving towards the position of equilibrium?
c) What is the work of the force of elasticity for the time during which the ball moves to the position of equilibrium?
d) What is the work equal to the equality of all the forces attached to the ball during the time during which the ball moves to the position of equilibrium?

17. Sanki weighing 10 kg moves without initial speed from a snowy mountain with an angle of inclination α \u003d 30º and some distance along the horizontal surface (Fig. 28.13). The friction coefficient between sledding and snow 0.1. Mountain base length L \u003d 15 m.

a) What is the friction force module when the skew is moving along the horizontal surface?
b) What is the work of the friction force when the Sanok moves along the horizontal surface on the way 20 m?
c) What is the friction force module when the sled is moving on the mountain?
d) What is the work of the friction force during the descent of the Sanok?
e) What is the work of gravity for the descent of the Sanok?
e) What is the work of the automatic forces acting on Sanki, when they are descending from the mountain?

18. The car weighing 1 t is moving 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 of 45 MJ / kg. What is the engine kpd? Is there any extra data in the condition?
Prompt. The efficiency of the thermal engine is equal to the ratio of the engine performed by the engine to the amount of heat, which was separated during the combustion of the fuel.

Each body performing movement can be characterized by the work. In other words, it characterizes the action of forces.

Work is defined as:
The product of the power module and the path of the body passed, multiplied by the cosine of the angle between the direction of force and movement.

Work is measured in Joules:
1 [J] \u003d \u003d [kg * m2 / c2]

For example, the body A under the action of Power in 5 N, 10 m passed. Determine the work perfect.

Since the direction of movement and the action of the force coincide, the angle between the strength vector and the movement vector will be 0 °. The formula is simplified, because the cosine of an angle of 0 ° is equal to 1.

Substituting the initial parameters in the formula, we find:
A \u003d 15 J.

Consider another example, a body weighing 2 kg, moving with an acceleration of 6 m / s2, was 10 m. Determine the work done by the body, if it moved along the inclined plane up at an angle of 60 °.

To begin with, we calculate what force you need to apply to inform the body to accelerate 6 m / s2.

F \u003d 2 kg * 6 m / s2 \u003d 12 H.
Under the influence of 12H, the body passed 10 m. The work can be calculated by the already known formula:

Where, as is 30 °. Substituting the initial data in the formula we get:
A \u003d 103, 2 J.

Power

Many machinery machines perform the same operation for a variety of time. To compare them, the concept of power is introduced.
Power is a value showing the amount of work performed per unit of time.

Power is measured in Watt, in honor of the Scottish engineer James Watta.
1 [watt] \u003d 1 [J / C].

For example, a large crane raised the cargo weighing 10 tons at a height of 30 m per 1 min. A small crane on the same height of 1 min raised 2 tons of bricks. Compare cranes power.
We define the work performed by the cranes. The load rises to 30m, while overcoming gravity, therefore the force spent on lifting the goods will be equal to the power of the interaction of the Earth and the cargo (F \u003d M * G). And work - the work of forces for the distance passed by cargo, that is, height.

For a large crane A1 \u003d 10 000 kg * 30 m * 10 m / s2 \u003d 3 000 000 J, and for small A2 \u003d 2 000 kg * 30 m * 10 m / s2 \u003d 600 000 J.
Power can be calculated by dividing operation at the time. Both cranes raised the cargo per 1 min (60 seconds).

From here:
N1 \u003d 3 000 000 J / 60 C \u003d 50 000 W \u003d 50 kW.
N2 \u003d 600 000 J / 60 C \u003d 10 000 W \u003d 10 K W.
Of the above data, it is clearly seen that the first crane is 5 times more powerful.

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 the work, F is the power, S is the path traveled.

POTENTIAL (Potential function), a notion characterizing a wide class of physical power points (electrical, gravitational, etc.) and in general the fields of physical quantities represented by the presents (fluid velocity field, etc.). In the general case, the potential of the vector field A ( x.,y.,z.) - suchcare function u.(x.,y.,z.) that a \u003d grad

35. Conductors in the electric field. Electrical capacity.Conductors in the electric field.Conductors are substances characterized by the presence of a large number of free chargers in them capable of moving under the action of an electric field. Conductors include metals, electrolytes, coal. In metals, the carriers of free charges are electrons of the outer shells of atoms, which in the interaction of atoms fully lose relations with "their" atoms and become the property of the entire conductor as a whole. Free electrons are involved in thermal motion like gas molecules and can be moved along the metal in any direction. Electrical Capacity - Characteristics of the conductor, the measure of its ability to accumulate an electrical charge. In the theory of electrical circuits, the container is called the mutual capacity between the two conductors; The parameter of the capacitive element of the electrical circuit represented in the form of a two-pole. Such a container is defined as the ratio of the amount of electric charge to the potential difference between these conductors.

36. Capacity of a flat condenser.

Capacity of a flat capacitor.

So The container of a flat capacitor depends only on its size, shape and dielectric constant. To create a large capacitance condenser, it is necessary to increase the area of \u200b\u200bthe plates and reduce the thickness of the dielectric layer.

37. Magnetic interaction of currents in vacuum. Ampere Law.Ampere Law. In 1820, Ampere (French scientist (1775-1836)) established an experimentally law on which one can calculate force acting on the element of the conductor length with current.

where - vector magnetic induction, - vector of the element of the length of the conductor spent in the current direction.

The power module, where the angle between the current direction in the conductor and the direction of the induction of the magnetic field. For rectilinear conductor long with a toxav homogeneous field

The direction of the current force can be determined by rules of the left hand:

If the palm of the left hand is to position so that the normal (to the current) component of the magnetic field is in the palm, and the four elongated fingers are directed along the current, the thumb indicates the direction in which the ampere power is valid.

38. Digitivity of the magnetic field. Bio-Savara Laplace LawMagnetic field tension (Standard designation N. ) - vector physical quantityequal to the difference of the vector magnetic induction B. and vector magnetization J. .

IN International Unit (SI): where- magnetic constant.

Law BSL.Law defining a magnetic field of a separate current element

39. Applications of the Bio-Savara Laplace Law.For direct current field

For a circular turn.

And for Solenoid

40. Magnetic field inductionThe magnetic field is characterized by a vector value that is called the induction of the magnetic field (vector magnitude, which is the power characteristic of the magnetic field at this point of space). Mi. (B) This is not the force acting on the conductors, this is a value that is through this force according to the following formula: B \u003d F / (I * L) (verbel: Mel module. (B) equals the ratio of the power module F, with which the magnetic field acts on the conductor with a current perpendicular to the magnetic lines, to the current in the conductor I and the length of the conductor L.Magnetic induction depends only on the magnetic field. In connection with this, induction can be considered a quantitative characteristic of the magnetic field. It defines, with what force (Lorentz power) magnetic field applies the table moving at speeds. Measured in Teslas (1 TL). At the same time, 1 TL \u003d 1 N / (A * M). Mi has a direction. Graphically it can be sketched in the form of lines. In homogeneous magnetic fields parallel, and the vector will be directed as well at all points. In the case of an inhomogeneous magnetic field, for example, the fields around the conductor with a current, the magnetic induction vector will change at each point of the space around the conductor, and the tangents of this vector will create concentric circles around the conductor.

41. Movement of particles in a magnetic field. Lorentz power.a) - if the particle flies into the area of \u200b\u200ba homogeneous magnetic field, and the vector V is perpendicular to the vector B, then it moves around the circle of the radius R \u003d MV / QB, since the Lorentz force Fl \u003d MV ^ 2 / R plays the role of centripetal force. The handling period is t \u003d 2pir / v \u003d 2pm / qb and it does not depend on the particle speed (this is true only at 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.

L. Power is determined by the relation: Fl \u003d q · v · b · sina (q - the magnitude of the moving charge; V is the module of its speed; b - the module of the magnetic field induction vector; angle between the vector V and the vector c) Lorentz power perpendicular to the speed And therefore it does not make work, does not change the charge rate module and its kinetic energy. But the direction of speed changes continuously. Lorentz's power perpendicular to vectors in and v, and its direction is determined using the same rule of the left hand as the direction of the ampere force: if the left hand is located so that the component of the magnetic induction in, perpendicular to the charge speed, was in the palm, and four fingers were Aimed by the movement of a positive charge (against the movement of the negative), then a thumb stranded at 90 degrees will show the direction of the Lorentz f l.