The physical quantity of mechanical work. mechanical work

AT Everyday life We often come across the concept of work. What does this word mean in physics and how to determine the work of an elastic force? You will find 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 by the following formula:

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

The force that acts on the body does not always do work. For example, the work of 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 - angle between force and displacement vectors.

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

mechanical work sign

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

• positive (0°≤ α<90°);
• negative (90°<α≤180°);
• 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. For A<0 сила препятствует ускорению тела. Например, действие силы трения скольжения.

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

The work of the elastic force

The work of a force can also be determined graphically. For this, the area of ​​the curvilinear figure under the graph F s (x) is calculated.

So, according to the graph of the dependence of the elastic force on the elongation of the spring, it is possible to 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 acts on a body, which leads to the displacement 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 elastic force as an example, you learned about a graphical way to determine work.

Content:

Electric current is generated in order to further use it for certain purposes, to perform any work. Thanks to electricity, all devices, devices and equipment function. The work itself is a certain effort applied to move an electric charge a certain distance. Conventionally, such work within the circuit section will be equal to the numerical value of the voltage in this section.

To perform the necessary calculations, it is necessary to know how the work of the current is measured. All calculations are carried out on the basis of the initial data obtained using measuring instruments. The larger the charge, the more effort is required to move it, the more work will be done.

What is called the work of the current

Electric current, as a physical quantity, in itself has no practical significance. The most important factor is the action of the current, which is characterized by the work performed by it. The work itself is a certain action in the process of which one type of energy is converted into another. For example, electrical energy is converted into mechanical energy by rotating the motor shaft. The work of the electric current itself consists in the movement of charges in the conductor under the influence of an electric field. In fact, all the work of moving charged particles is done by an electric field.

In order to perform calculations, the formula for the work of an electric current must be derived. To draw up formulas, you will need parameters such as current strength and. Since the work of an electric current and the work of an electric field are the same thing, it will be expressed as the product of voltage and charge flowing in a conductor. That is: A = Uq. This formula was derived from the ratio that determines the voltage in the conductor: U = A/q. It follows that the voltage is the work of the electric field A on the transfer of a charged particle q.

The charged particle or charge itself is displayed as the product of the current strength and the time spent on the movement of this charge along the conductor: q \u003d It. In this formula, the ratio for the current strength in the conductor was used: I \u003d q / t. That is, is the ratio of the charge to the time interval for which the charge passes through the cross section of the conductor. In its final form, the formula for the work of an electric current will look like a product of known quantities: A \u003d UIt.

In what units is the work of electric current measured?

Before directly solving the question of what the work of the electric current is measured in, it is necessary to collect the units of measurement of all physical quantities with which this parameter is calculated. Any work, therefore, the unit of measurement of this quantity will be 1 Joule (1 J). Voltage is measured in volts, current is measured in amps, and time is measured in seconds. So the unit of measurement will look like this: 1 J = 1V x 1A x 1s.

Based on the units of measurement obtained, the work of the electric current will be determined as the product of the current strength in the circuit section, the voltage at the ends of the section and the time interval during which the current flows through the conductor.

The measurement is carried out using a voltmeter and a watch. These devices allow you to effectively solve the problem of how to find the exact value of a given parameter. When you turn on the ammeter and voltmeter in the circuit, it is necessary to monitor their readings for a specified period of time. The resulting data is inserted into the formula, after which the final result is displayed.

The functions of all three devices are combined in electric meters that take into account the energy consumed, and in fact the work done by the electric current. Here, another unit is used - 1 kWh, which also means how much work was done during a unit of time.

You are already familiar with mechanical work (work of force) from the basic school physics course. Recall the definition of mechanical work given there for the following cases.

If the force is directed in the same direction as the displacement 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 is

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 of the force is zero:

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

1 J = 1 N * m.

1. A bar weighing 0.5 kg was moved along the table by 2 m, applying an elastic force equal to 4 N to it (Fig. 28.1). The coefficient of friction between the bar and the table is 0.2. What is the work done on the bar:
a) gravity m?
b) normal reaction forces ?
c) elastic force?
d) forces of sliding friction tr?

The total work of several forces acting on a body can be found in two ways:
1. Find the work of each force and add 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 verify this, return to the previous task and answer the questions of task 2.

2. What is equal to:
a) the sum of the work of all the forces acting on the block?
b) the resultant of all forces acting on the bar?
c) the work of the 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 modulus of force F times the modulus of displacement 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 that the general definition of work leads to the conclusions shown in the following diagram. Formulate them verbally and write them down in your notebook.

4. A force is applied to the bar 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 the bar moves 60 cm across the table, this force does the work: a) 3 J; b) –3 J; c) –3 J; d) -6 J? Make explanatory drawings.

2. The 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 down (h n > h k, 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 done by gravity

A \u003d 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 bar? 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 of gravity when the bar moves up along the entire same plane?

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

But then formula (5) for the work of gravity is valid when the 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 of gravity during movement but any trajectory is expressed by the formula

A t \u003d mg (h n - h k),

where h n - the initial height of the body, h to - its final height.
The work of gravity does not depend on the shape of the trajectory.

For example, the work of 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 work 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, is deflected by 90º, keeping the thread taut, and released without a push.
a) What is the work of gravity during the time during which the ball moves to the equilibrium position (Fig. 28.6)?
b) What is the work of the elastic force of the thread in the same time?
c) What is the work of the resultant forces applied to the ball in the same time?

3. The work of the force of elasticity

When the spring returns to its undeformed state, the elastic force always does positive work: its direction coincides with the direction of movement (Fig. 28.7).

Find the work of the elastic force.
The modulus of this force is related to the modulus of deformation x by the relation (see § 15)

The work of such a force can be found graphically.

Note first that the work of 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 plot of F(x) for the elastic force. Let us mentally divide the entire displacement of the body into such small intervals that the force on 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 the work is equal to the sum of the work in these areas.

Consequently, in this case, the work is also numerically equal to the area of ​​the figure under the F(x) dependence graph.

7. Using Figure 28.10, prove that

the work of the elastic force when the spring returns to the undeformed state is expressed by the formula

A = (kx 2)/2. (7)

8. Using the graph in Figure 28.11, prove that when the deformation of the spring 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 it returns to its initial state, then the work of the elastic force is zero. Recall that the work of gravity has the same property.

9. At the initial moment, the tension of the spring with a stiffness of 400 N / m is 3 cm. The spring is stretched 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 of the elastic force of the spring?

4. The work of the friction force

Let the body slide on 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 for any direction of movement (Fig. 28.12).

Therefore, if the bar is moved to the right, and with a peg the same distance to the left, then, although it returns to its initial position, the total work of the sliding friction force will not be equal to zero. This is the most important difference between the work of the sliding friction force and the work of the force of gravity and the force of elasticity. Recall that the work of these forces when moving the body along a closed trajectory is equal to zero.

11. A bar 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) Did the block return to its starting point?
b) What is the total work of the friction force acting on the bar? The coefficient of friction between the bar and the table is 0.3.

5. Power

Often, not only the work done is important, but also the speed of the work. It is characterized by power.

The power P is the ratio of the work done A to the time interval t during which this work is 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 of power.)

The unit of power is the watt (denoted: 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 in terms of work and time, but in terms of force and speed.

Consider the case when the force is directed along the displacement. Then the work of the force A = Fs. Substituting this expression into formula (9) for power, we obtain:

P = (Fs)/t = F(s/t) = Fv. (ten)

13. A car is driving along 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 is moving along a horizontal road at a constant speed, the traction force is equal in absolute value to the drag force of the car.

14. How long will it take to evenly lift a 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 crane motor is 75%?

Clue. The efficiency of the 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 of mass 200 g is thrown from a balcony 10 high 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 done by gravity in 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 extra data in the condition?

16. A ball weighing 0.5 kg is suspended from a spring with a stiffness of 250 N/m and is in equilibrium. The ball is lifted so that the spring becomes undeformed and released without a push.
a) To what height was the ball raised?
b) What is the work of gravity during the time during which the ball moves to the equilibrium position?
c) What is the work of the elastic force during the time during which the ball moves to the equilibrium position?
d) What is the work of 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 α = 30º without initial speed and travels some 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 l = 15 m.

a) What is the modulus of the friction force when the sled moves on a horizontal surface?
b) What is the work of the friction force when the sled moves along a horizontal surface on a path of 20 m?
c) What is the modulus of the friction force when the sled moves up the mountain?
d) What is the work done by the friction force during the descent of the sled?
e) What is the work done by gravity during the descent of the sled?
f) What is the work of 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 of combustion is 45 MJ/kg. What is the engine efficiency? Is there extra data in the condition?
Clue. The efficiency of a heat engine is equal to the ratio of the work done by the engine to the amount of heat released during the combustion of fuel.

Almost everyone, without hesitation, will answer: in the second. And they will be wrong. The case is just the opposite. In physics, mechanical work is described the following definitions: mechanical work is done when a force acts on a body and it moves. Mechanical work is directly proportional to the applied force and the distance traveled.

Mechanical work formula

The mechanical work is determined by the formula:

where A is work, F is force, s is the distance traveled.

POTENTIAL(potential function), a concept that characterizes a wide class of physical force fields (electric, gravitational, etc.) and, in general, fields of physical quantities represented by vectors (fluid velocity field, etc.). In the general case, the potential of the vector field a( x,y,z) is such a scalar function u(x,y,z) that a=grad

35. Conductors in an electric field. Electrical capacity.conductors in an electric field. Conductors are substances characterized by the presence in them of a large number of free charge carriers that can move under the influence of an electric field. Conductors include metals, electrolytes, coal. In metals, the carriers of free charges are the electrons of the outer shells of atoms, which, when atoms interact, completely lose their connection with “their” atoms and become the property of the entire conductor as a whole. Free electrons participate in thermal motion like gas molecules and can move through the metal in any direction. Electric capacity- a characteristic of a conductor, a measure of its ability to accumulate an electric charge. In the theory of electrical circuits, capacitance is the mutual capacitance between two conductors; parameter of the capacitive element of the electrical circuit, presented in the form of a two-terminal network. Such capacitance is defined as the ratio of the magnitude of the electric charge to the potential difference between these conductors

36. Capacitance of a flat capacitor.

Capacitance of a flat capacitor.

That. the capacitance of a flat capacitor depends only on its size, shape and dielectric constant. To create a high-capacity capacitor, it is necessary to increase the area of ​​the plates and reduce the thickness of the dielectric layer.

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

where is the vector of magnetic induction, is the vector of the length element of the conductor drawn in the direction of the current.

Force modulus , where is the angle between the direction of the current in the conductor and the direction of the magnetic field. For a straight conductor with current in a uniform field

The direction of the acting force can be determined using left hand rules:

If the palm of the left hand is positioned so that the normal (to the current) component of the magnetic field enters the palm, and four outstretched fingers are directed along the current, then the thumb will indicate the direction in which the Ampère force acts.

38. Magnetic field strength. Biot-Savart-Laplace lawMagnetic field strength(standard designation H ) - vector physical quantity, equal to the difference of the vector magnetic induction B and magnetization vector J .

AT International System of Units (SI): where- magnetic constant.

BSL law. The law that determines the magnetic field of an individual current element

39. Applications of the Biot-Savart-Laplace law. For direct current field

For a circular loop.

And for the solenoid

40. Magnetic field induction The magnetic field is characterized by a vector quantity, which is called the magnetic field induction (a vector quantity, which is the force characteristic of the magnetic field at a given point in space). MI. (B) this is not a force acting on conductors, it is a quantity that is found through a given force according to the following formula: B \u003d F / (I * l) (Verbally: MI vector modulus. (B) is equal to the ratio of the modulus of force F, with which the magnetic field acts on a current-carrying conductor located perpendicular to the magnetic lines, to the current strength in the conductor I and the length of the conductor l. Magnetic induction depends only on the magnetic field. In this regard, induction can be considered a quantitative characteristic of the magnetic field. It determines with what force (Lorentz Force) the magnetic field acts on a charge moving with speed. MI is measured in Tesla (1 T). In this case, 1 Tl \u003d 1 N / (A * m). MI has direction. Graphically, it can be drawn as lines. In a uniform magnetic field, the MIs are parallel, and the MI vector will be directed in the same way at all points. In the case of a non-uniform magnetic field, for example, a field around a conductor with current, the magnetic induction vector will change at each point in space around the conductor, and tangents to this vector will create concentric circles around the conductor.

41. Motion of a particle in a magnetic field. Lorentz force. a) - If a particle flies into a region of a uniform magnetic field, and the vector V is perpendicular to the vector B, then it moves along a circle of radius R=mV/qB, since the Lorentz force Fl=mV^2/R plays the role of a centripetal force. The period of revolution is T=2piR/V=2pim/qB and it does not depend on the speed of the particle (This is true only for 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.

The force of L. is determined by the relation: Fl = q V B sina (q is the value of the moving charge; V is the modulus of its velocity; B is the modulus of the magnetic field induction vector; alpha is the angle between the vector V and the vector B) The Lorentz force is perpendicular to the velocity and therefore it does not do work, does not change the modulus of the speed of the charge and its kinetic energy. But the direction of the speed changes continuously. The Lorentz force is perpendicular to the vectors B and v, and its direction is determined using the same rule of the left hand as the direction of the Ampère force: if the left hand is positioned so that the magnetic induction component B, perpendicular to the charge velocity, enters the palm, and four fingers are are directed along the movement of a positive charge (against the movement of a negative one), then the thumb bent 90 degrees will show the direction of the Lorentz force acting on the charge F l.