Formula for useful work in physics. Definition of mechanical work

In our everyday experience, the word “work” appears very often. But one should distinguish between physiological work and work from the point of view of the science of physics. When you come home from class, you say: “Oh, I’m so tired!” This is physiological work. Or, for example, the work of a team in folk tale"Turnip".

Figure 1. Work in the everyday sense of the word

We will talk here about work from the point of view of physics.

Mechanical work is performed if a body moves under the influence of a force. Work is designated by the Latin letter A. A more strict definition of work sounds like this.

The work of a force is a physical quantity equal to the product of the magnitude of the force and the distance traveled by the body in the direction of the force.

Figure 2. Work is a physical quantity

The formula is valid when a constant force acts on the body.

In the international system of SI units, work is measured in joules.

This means that if under the influence of a force of 1 newton a body moves 1 meter, then 1 joule of work is done by this force.

The unit of work is named after the English scientist James Prescott Joule.

Fig 3. James Prescott Joule (1818 - 1889)

From the formula for calculating work it follows that there are three possible cases when work is equal to zero.

The first case is when a force acts on a body, but the body does not move. For example, a house is subject to a huge force of gravity. But she does not do any work because the house is motionless.

The second case is when the body moves by inertia, that is, no forces act on it. For example, spaceship moves in intergalactic space.

The third case is when a force acts on the body perpendicular to the direction of movement of the body. In this case, although the body moves and a force acts on it, there is no movement of the body in the direction of the force.

Figure 4. Three cases when work is zero

It should also be said that the work done by a force can be negative. This will happen if the body moves against the direction of the force. For example, when crane with the help of a cable, lifts a load above the ground, the work of gravity is negative (and the work of the elastic force of the cable directed upward, on the contrary, is positive).

Suppose, when executing construction work the pit must be filled with sand. It would take a few minutes for an excavator to do this, but a worker with a shovel would have to work for several hours. But both the excavator and the worker would have completed the same job.

Fig 5. The same work can be completed in different time

To characterize the speed of work done in physics, a quantity called power is used.

Power is a physical quantity equal to the ratio of work to the time it is performed.

Power is indicated by a Latin letter N.

The SI unit of power is the watt.

One watt is the power at which one joule of work is done in one second.

The power unit is named after the English scientist and inventor steam engine James Watt.

Fig 6. James Watt (1736 - 1819)

Let's combine the formula for calculating work with the formula for calculating power.

Let us now remember that the ratio of the path traveled by the body is S, by the time of movement t represents the speed of movement of the body v.

Thus, power is equal to the product of the numerical value of the force and the speed of the body in the direction of the force.

This formula is convenient to use when solving problems in which a force acts on a body moving with a known speed.

Bibliography

  1. Lukashik V.I., Ivanova E.V. Collection of physics problems for grades 7-9 educational institutions. - 17th ed. - M.: Education, 2004.
  2. Peryshkin A.V. Physics. 7th grade - 14th ed., stereotype. - M.: Bustard, 2010.
  3. Peryshkin A.V. Collection of problems in physics, grades 7-9: 5th ed., stereotype. - M: Publishing House “Exam”, 2010.
  1. Internet portal Physics.ru ().
  2. Internet portal Festival.1september.ru ().
  3. Internet portal Fizportal.ru ().
  4. Internet portal Elkin52.narod.ru ().

Homework

  1. In what cases is work equal to zero?
  2. How is the work done along the path traveled in the direction of the force? In the opposite direction?
  3. How much work is done by the frictional force acting on the brick when it moves 0.4 m? The friction force is 5 N.

IN Everyday life Often we 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.

« Physics - 10th grade"

The law of conservation of energy is a fundamental law of nature that allows us to describe most occurring phenomena.

Description of the movement of bodies is also possible using such concepts of dynamics as work and energy.

Remember what work and power are in physics.

Do these concepts coincide with everyday ideas about them?

All our daily actions come down to the fact that we, with the help of muscles, either set the surrounding bodies in motion and maintain this movement, or stop the moving bodies.

These bodies are tools (hammer, pen, saw), in games - balls, pucks, chess pieces. In production and agriculture, people also set tools in motion.

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

The purpose of any engine is to set bodies in motion and maintain this movement, despite braking by both ordinary friction and “working” resistance (the cutter should not just slide along the metal, but, cutting into it, remove chips; the plow should loosen land, etc.). In this case, a force must act on the moving body from the side of the engine.

Work is performed in nature whenever a force (or several forces) from another body (other bodies) acts on a body in the direction of its movement or against it.

The force of gravity does work when raindrops or stones fall from a cliff. At the same time, work is also done by the resistance force acting on the falling drops or on the stone from the air. The elastic force also performs work when a tree bent by the wind straightens.

Definition of work.


Newton's second law in impulse form Δ = Δt allows you to determine how the speed of a body changes in magnitude and direction if a force acts on it during a time Δt.

The influence of forces on bodies that lead to a change in the modulus of their velocity is characterized by a value that depends on both the forces and the movements of the bodies. In mechanics this quantity is called work of force.

A change in speed in absolute value is possible only in the case when the projection of the force F r on the direction of movement of the body is different from zero. It is this projection that determines the action of the force that changes the velocity of the body modulo. She does the work. Therefore, work can be considered as the product of the projection of force F r by the displacement modulus |Δ| (Fig. 5.1):

A = F r |Δ|. (5.1)

If the angle between force and displacement is denoted by α, then Fr = Fcosα.

Therefore, the work is equal to:

A = |Δ|cosα. (5.2)

Our everyday idea of ​​work differs from the definition of work in physics. You are holding a heavy suitcase, and it seems to you that you are doing work. However, from a physical point of view, your work is zero.

The work of a constant force is equal to the product of the moduli of the force and the displacement of the point of application of the force and the cosine of the angle between them.

In the general case, when a rigid body moves, the displacements of its different points are different, but when determining the work of a force, we are under Δ we understand the movement of its point of application. During the translational motion of a rigid body, the movement of all its points coincides with the movement of the point of application of the force.

Work, unlike force and displacement, is not a vector, but a scalar quantity. It can be positive, negative or zero.

The sign of the work is determined by the sign of the cosine of the angle between force and displacement. If α< 90°, то А >0, since the cosine of acute angles is positive. For α > 90°, the work is negative, since the cosine of obtuse angles is negative. At α = 90° (force perpendicular to displacement) no work is done.

If several forces act on a body, then the projection of the resultant force on the displacement is equal to the sum of the projections of the individual forces:

F r = F 1r + F 2r + ... .

Therefore, for the work of the resultant force we obtain

A = F 1r |Δ| + F 2r |Δ| + ... = A 1 + A 2 + .... (5.3)

If several forces act on a body, then full time job(algebraic sum of the work of all forces) is equal to the work of the resultant force.

The work done by a force can be represented graphically. Let us explain this by depicting in the figure the dependence of the projection of force on the coordinates of the body when it moves in a straight line.

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

Fcosα = F x , |Δ| = Δ x.

For the work of force we get

A = F|Δ|cosα = F x Δx.

Obviously, the area of ​​the rectangle shaded in Figure (5.3, a) is numerically equal to the work done when moving a body from a point with coordinate x1 to a point with coordinate x2.

Formula (5.1) is valid in the case when the projection of the force onto the displacement is constant. In the case of a curvilinear trajectory, constant or variable force, we divide the trajectory into small segments, which can be considered rectilinear, and the projection of the force at a small displacement Δ - constant.

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

Unit of work.


The unit of work can be established using the basic formula (5.2). If, when moving a body per unit length, it is acted upon by a force whose modulus is equal to one, and the direction of the force coincides with the direction of movement of its point of application (α = 0), then the work will be equal to one. The International System (SI) unit of work is the joule (denoted by J):

1 J = 1 N 1 m = 1 N m.

Joule- this is the work done by a force of 1 N on displacement 1 if the directions of force and displacement coincide.

Multiple units of work are often used: kilojoule and megajoule:

1 kJ = 1000 J,
1 MJ = 1000000 J.



Work can be completed either in a large period of time or in a very short one. In practice, however, it is far from indifferent whether work can be done quickly or slowly. The time during which work is performed determines the performance of any engine. Very great job A tiny electric motor can do this, but it will take a lot of time. Therefore, along with work, a quantity is introduced that characterizes the speed with which it is produced - power.

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

Substituting into formula (5.4) instead of work A its expression (5.2), we obtain

Thus, if the force and speed of a body are constant, then the power is equal to the product of the magnitude of the force vector by the magnitude of the velocity vector and the cosine of the angle between the directions of these vectors. If these quantities are variable, then using formula (5.4) we can determine the average power similar to the definition average speed body movements.

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

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

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

Along with the watt, larger (multiple) units of power are used:

1 kW (kilowatt) = 1000 W,
1 MW (megawatt) = 1,000,000 W.

You are already familiar with mechanical work (work of force) from the basic school physics course. Let us recall the definition given there mechanical work 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 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 done by 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 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 starting 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 with a mass of 200 g was thrown from a balcony with a height of 10 and an angle of 45º to the horizontal. Having reached a maximum height of 15 m in flight, 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 the efficiency of the engine? 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.

One of the most important concepts in mechanics is work of force .

Work of force

All physical bodies in the world around us are set in motion with the help of force. If a moving body in the same or opposite direction is acted upon by a force or several forces from one or more bodies, then it is said that work is being done .

That is, mechanical work is performed by a force acting on the body. Thus, the traction force of an electric locomotive sets the entire train in motion, thereby performing mechanical work. The bicycle is driven by the muscular power of the cyclist's legs. Consequently, this force also does mechanical work.

In physics work of force called physical quantity, equal to the product of the force modulus, the displacement modulus of the point of application of the force and the cosine of the angle between the force and displacement vectors.

A = F s cos (F, s) ,

Where F force module,

s – travel module .

Work is always done if the angle between the winds of force and displacement is not zero. If the force acts in the direction opposite to the direction of motion, the amount of work is negative.

No work is done if no forces act on the body, or if the angle between the applied force and the direction of movement is 90 o (cos 90 o = 0).

If a horse pulls a cart, then the horse's muscular force, or the traction force directed along the direction of the cart's movement, does work. But the force of gravity with which the driver presses on the cart does not do any work, since it is directed downward, perpendicular to the direction of movement.

The work of force is a scalar quantity.

Unit of work in the SI measurement system - joule. 1 joule is the work done by a force of 1 newton at a distance of 1 m if the directions of the force and displacement coincide.

If on the body or material point If several forces act, we talk about the work done by their resultant force.

If the applied force is not constant, then its work is calculated as an integral:

Power

The force that sets a body in motion does mechanical work. But how this work is done, quickly or slowly, is sometimes very important to know in practice. After all, the same work can be completed in different times. The work that a large electric motor does can be done by small motor. But he will need much more time for this.

In mechanics, there is a quantity that characterizes the speed of work. This quantity is called power.

Power is the ratio of work performed in a certain period of time to the value of this period.

N= A /∆ t

A-priory A = F s cos α , A s/∆ t = v , hence

N= F v cos α = F v ,

Where F - force, v speed, α – the angle between the direction of force and the direction of speed.

That is power – This scalar product force vector to the body velocity vector.

In the international SI system, power is measured in watts (W).

1 watt of power is 1 joule (J) of work done in 1 second (s).

Power can be increased by increasing the force doing work or the rate at which this work is done.