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

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

What is the work of current called?

Electric current, as a physical quantity, in itself does not have practical significance. The most important factor is the effect of the current, characterized by the work it performs. The work itself represents certain actions during which one type of energy is transformed into another. For example, electrical energy is converted into mechanical energy by rotating the motor shaft. The work itself electric current consists in the movement of charges in a conductor under the influence of an electric field. In fact, all the work of moving charged particles is done by the electric field.

In order to perform calculations, a formula for the operation of electric current must be derived. To compile formulas, you will need parameters such as current strength and. Since the work done by an electric current and the work done by an electric field are the same thing, it will be expressed as the product of the voltage and the charge flowing in the conductor. That is: A = Uq. This formula was derived from the relationship that determines the voltage in the conductor: U = A/q. It follows that voltage represents the work done by the electric field A to transport 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 = It. In this formula, the relation for the current strength in the conductor was used: I = q/t. That is, it is the ratio of the charge to the period of time during which the charge passes through the cross section of the conductor. In its final form, the formula for the work of electric current will look like the product of known quantities: A = UIt.

In what units is the work of electric current measured?

Before directly addressing the question of how the work of electric current is measured, it is necessary to collect the units of measurement of all physical quantities, with the help of 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 amperes, and time is measured in seconds. This means the unit of measurement will look like this: 1 J = 1V x 1A x 1s.

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

Measurements are carried out using a voltmeter and a clock. These devices allow you to effectively solve the problem of how to find exact value this parameter. When connecting an ammeter and a voltmeter to a circuit, it is necessary to monitor their readings for a specified period of time. The obtained 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 electric current. Here another unit is used - 1 kW x h, which also means how much work was done during a unit of time.

You are already familiar with mechanical work (work of force) from your basic school physics course. Let us recall the definition of mechanical work given there 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 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 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 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 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 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. B starting moment the stretch 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 its initial 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 uniformly 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 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 weighing 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 from snowy mountain with an inclination angle α = 30º and travel 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.

When bodies interact pulse one body can be partially or completely transferred to another body. If a system of bodies is not acted upon by external forces from other bodies, such a system is called closed.

This fundamental law of nature is called law of conservation of momentum. It is a consequence of the second and third Newton's laws.

Let us consider any two interacting bodies that are part of a closed system. We denote the forces of interaction between these bodies by and According to Newton’s third law If these bodies interact during time t, then the impulses of the interaction forces are equal in magnitude and directed in opposite directions: Let us apply Newton’s second law to these bodies:

where and are the impulses of the bodies at the initial moment of time, and are the impulses of the bodies at the end of the interaction. From these relations it follows:

This equality means that as a result of the interaction of two bodies, their total momentum has not changed. Now considering all possible pair interactions of bodies included in a closed system, we can conclude that the internal forces of a closed system cannot change its total momentum, that is, the vector sum of the momentum of all bodies included in this system.

Mechanical work and power

The energy characteristics of motion are introduced based on the concept mechanical work or work of force.

Work A performed by a constant force is a physical quantity equal to the product of the force and displacement moduli multiplied by the cosine of the angle α between the force vectors and movements(Fig. 1.1.9):

Work is a scalar quantity. It can be either positive (0° ≤ α< 90°), так и отрицательна (90° < α ≤ 180°). При α = 90° работа, совершаемая силой, равна нулю. В системе СИ работа измеряется в joules (J).

A joule is equal to the work done by a force of 1 N to move 1 m in the direction of the force.

If the projection of the force on the direction of movement does not remain constant, the work should be calculated for small movements and sum the results:

An example of a force whose modulus depends on the coordinate is the elastic force of a spring, obeying Hooke's law. In order to stretch a spring, an external force must be applied to it, the modulus of which is proportional to the elongation of the spring (Fig. 1.1.11).

The dependence of the external force modulus on the x coordinate is depicted on the graph as a straight line (Fig. 1.1.12).

Based on the area of ​​the triangle in Fig. 1.18.4 you can determine the work done by an external force applied to the right free end of the spring:

The same formula expresses the work done by an external force when compressing a spring. In both cases, the work of the elastic force is equal in magnitude to the work of the external force and opposite in sign.

If several forces are applied to a body, then the total work of all forces is equal to the algebraic sum of the work done by individual forces, and is equal to the work resultant of the applied forces.

The work done by a force per unit time is called power. Power N is a physical quantity equal to the ratio of work A to the time period t during which this work was performed.

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.

Basic theoretical information

Mechanical work

The energy characteristics of motion are introduced based on the concept mechanical work or force work. Work done by a constant force F, is a physical quantity equal to the product of the force and displacement moduli multiplied by the cosine of the angle between the force vectors F and movements S:

Work is a scalar quantity. It can be either positive (0° ≤ α < 90°), так и отрицательна (90° < α ≤ 180°). At α = 90° the work done by the force is zero. In the SI system, work is measured in joules (J). A joule is equal to the work done by a force of 1 newton to move 1 meter in the direction of the force.

If the force changes over time, then to find the work, build a graph of the force versus displacement and find the area of ​​the figure under the graph - this is the work:

An example of a force whose modulus depends on the coordinate (displacement) is the elastic force of a spring, which obeys Hooke’s law ( F control = kx).

Power

The work done by a force per unit time is called power. Power P(sometimes denoted by the letter N) – physical quantity equal to the work ratio A to a period of time t during which this work was completed:

This formula calculates average power, i.e. power generally characterizing the process. So, work can also be expressed in terms of power: A = Pt(if, of course, the power and time of doing the work are known). The unit of power is called the watt (W) or 1 joule per second. If the motion is uniform, then:

Using this formula we can calculate instant power(power at a given time), if instead of speed we substitute the value of instantaneous speed into the formula. How do you know what power to count? If the problem asks for power at a moment in time or at some point in space, then instantaneous is considered. If they ask about power over a certain period of time or part of the route, then look for average power.

Efficiency - efficiency factor, is equal to the ratio of useful work to expended, or useful power to expended:

Which work is useful and which is wasted is determined from the conditions of a specific task through logical reasoning. For example, if a crane does the work of lifting a load to a certain height, then the useful work will be the work of lifting the load (since it is for this purpose that the crane was created), and the expended work will be the work done by the crane’s electric motor.

So, useful and expended power do not have a strict definition, and are found by logical reasoning. In each task, we ourselves must determine what in this task was the goal of doing work (useful work or power), and what was the mechanism or way of doing all the work (expended power or work).

In general, efficiency shows how efficiently a mechanism converts one type of energy into another. If the power changes over time, then the work is found as the area of ​​the figure under the graph of power versus time:

Kinetic energy

A physical quantity equal to half the product of a body’s mass and the square of its speed is called kinetic energy of the body (energy of movement):

That is, if a car weighing 2000 kg moves at a speed of 10 m/s, then it has kinetic energy equal to E k = 100 kJ and is capable of doing 100 kJ of work. This energy can turn into heat (when a car brakes, the tires of the wheels, the road and the brake discs heat up) or can be spent on deforming the car and the body that the car collided with (in an accident). When calculating kinetic energy, it does not matter where the car is moving, since energy, like work, is a scalar quantity.

A body has energy if it can do work. For example, a moving body has kinetic energy, i.e. energy of motion, and is capable of doing work to deform bodies or impart acceleration to bodies with which a collision occurs.

The physical meaning of kinetic energy: in order for a body at rest with a mass m began to move at speed v it is necessary to do work equal to the obtained value of kinetic energy. If the body has a mass m moves at speed v, then to stop it it is necessary to do work equal to its initial kinetic energy. When braking, kinetic energy is mainly (except for cases of impact, when the energy goes to deformation) “taken away” by the friction force.

Theorem on kinetic energy: the work of the resultant force is equal to the change in the kinetic energy of the body:

The theorem on kinetic energy is also valid in the general case, when a body moves under the influence of a changing force, the direction of which does not coincide with the direction of movement. It is convenient to apply this theorem in problems involving acceleration and deceleration of a body.

Potential energy

Along with kinetic energy or energy of motion, the concept plays an important role in physics potential energy or energy of interaction of bodies.

Potential energy is determined by the relative position of bodies (for example, the position of the body relative to the surface of the Earth). The concept of potential energy can be introduced only for forces whose work does not depend on the trajectory of the body and is determined only by the initial and final positions (the so-called conservative forces). The work done by such forces on a closed trajectory is zero. This property is possessed by gravity and elastic force. For these forces we can introduce the concept of potential energy.

Potential energy of a body in the Earth's gravity field calculated by the formula:

The physical meaning of the potential energy of a body: potential energy is equal to the work done by gravity when lowering the body to zero level ( h– distance from the center of gravity of the body to the zero level). If a body has potential energy, then it is capable of doing work when this body falls from a height h to zero level. The work done by gravity is equal to the change in the potential energy of the body, taken with the opposite sign:

Often in energy problems one has to find the work of lifting (turning over, getting out of a hole) the body. In all these cases, it is necessary to consider the movement not of the body itself, but only of its center of gravity.

The potential energy Ep depends on the choice of the zero level, that is, on the choice of the origin of the OY axis. In each problem, the zero level is chosen for reasons of convenience. What has a physical meaning is not the potential energy itself, but its change when a body moves from one position to another. This change is independent of the choice of zero level.

Potential energy of a stretched spring calculated by the formula:

Where: k– spring stiffness. An extended (or compressed) spring can set a body attached to it in motion, that is, impart kinetic energy to this body. Consequently, such a spring has a reserve of energy. Tension or compression X must be calculated from the undeformed state of the body.

The potential energy of an elastically deformed body is equal to the work done by the elastic force during the transition from a given state to a state with zero deformation. If in the initial state the spring was already deformed, and its elongation was equal to x 1, then upon transition to a new state with elongation x 2, the elastic force will do work equal to the change in potential energy, taken with the opposite sign (since the elastic force is always directed against the deformation of the body):

Potential energy during elastic deformation is the energy of interaction of individual parts of the body with each other by elastic forces.

The work of the friction force depends on the path traveled (this type of force, whose work depends on the trajectory and the path traveled is called: dissipative forces). The concept of potential energy for the friction force cannot be introduced.

Efficiency

Efficiency factor (efficiency)– characteristic of the efficiency of a system (device, machine) in relation to the conversion or transmission of energy. It is determined by the ratio of usefully used energy to the total amount of energy received by the system (the formula has already been given above).

Efficiency can be calculated both through work and through power. Useful and expended work (power) are always determined by simple logical reasoning.

In electric motors, efficiency is the ratio of the performed (useful) mechanical work to the electrical energy received from the source. In heat engines, the ratio of useful mechanical work to the amount of heat expended. In electrical transformers, the ratio of the electromagnetic energy received in the secondary winding to the energy consumed by the primary winding.

Due to its generality, the concept of efficiency makes it possible to compare and evaluate from a unified point of view such different systems as nuclear reactors, electric generators and engines, thermal power plants, semiconductor devices, biological objects, etc.

Due to inevitable energy losses due to friction, heating of surrounding bodies, etc. Efficiency is always less than unity. Accordingly, efficiency is expressed as a fraction of the energy expended, that is, as a proper fraction or as a percentage, and is a dimensionless quantity. Efficiency characterizes how efficiently a machine or mechanism operates. The efficiency of thermal power plants reaches 35–40%, internal combustion engines with supercharging and pre-cooling – 40–50%, dynamos and high-power generators – 95%, transformers – 98%.

A problem in which you need to find the efficiency or it is known, you need to start with logical reasoning - which work is useful and which is wasted.

Law of conservation of mechanical energy

Total mechanical energy is called the sum of kinetic energy (i.e. the energy of motion) and potential (i.e. the energy of interaction of bodies by the forces of gravity and elasticity):

If mechanical energy does not transform into other forms, for example, into internal (thermal) energy, then the sum of kinetic and potential energy remains unchanged. If mechanical energy turns into thermal energy, then the change in mechanical energy is equal to the work of the friction force or energy losses, or the amount of heat released, and so on, in other words, the change in total mechanical energy is equal to the work of external forces:

The sum of the kinetic and potential energy of the bodies that make up a closed system (i.e. one in which there are no external forces acting, and their work is correspondingly zero) and the gravitational and elastic forces interacting with each other remains unchanged:

This statement expresses law of conservation of energy (LEC) in mechanical processes. It is a consequence of Newton's laws. The law of conservation of mechanical energy is satisfied only when bodies in a closed system interact with each other by forces of elasticity and gravity. In all problems on the law of conservation of energy there will always be at least two states of a system of bodies. The law states that the total energy of the first state will be equal to the total energy of the second state.

Algorithm for solving problems on the law of conservation of energy:

1. Find the points of the initial and final position of the body.
2. Write down what or what energies the body has at these points.
3. Equate the initial and final energy of the body.
4. Add other necessary equations from previous physics topics.
5. Solve the resulting equation or system of equations using mathematical methods.

It is important to note that the law of conservation of mechanical energy made it possible to obtain a relationship between the coordinates and velocities of a body at two different points of the trajectory without analyzing the law of motion of the body at all intermediate points. The application of the law of conservation of mechanical energy can greatly simplify the solution of many problems.

In real conditions, moving bodies are almost always acted upon, along with gravitational forces, elastic forces and other forces, by friction forces or environmental resistance forces. The work done by the friction force depends on the length of the path.

If friction forces act between the bodies that make up a closed system, then mechanical energy is not conserved. Part of the mechanical energy is converted into internal energy of bodies (heating). Thus, energy as a whole (i.e., not only mechanical) is conserved in any case.

During any physical interactions, energy neither appears nor disappears. It just changes from one form to another. This experimentally established fact expresses a fundamental law of nature - law of conservation and transformation of energy.

One of the consequences of the law of conservation and transformation of energy is the statement about the impossibility of creating a “perpetual motion machine” (perpetuum mobile) - a machine that could do work indefinitely without consuming energy.

Various tasks for work

If the problem requires finding mechanical work, then first select a method for finding it:

1. A job can be found using the formula: A = FS∙cos α . Find the force that does the work and the amount of displacement of the body under the influence of this force in the chosen frame of reference. Note that the angle must be chosen between the force and displacement vectors.
2. The work done by an external force can be found as the difference in mechanical energy in the final and initial situations. Mechanical energy is equal to the sum of the kinetic and potential energies of the body.
3. The work done to lift a body at a constant speed can be found using the formula: A = mgh, Where h- height to which it rises body center of gravity.
4. Work can be found as the product of power and time, i.e. according to the formula: A = Pt.
5. The work can be found as the area of ​​the figure under the graph of force versus displacement or power versus time.

Law of conservation of energy and dynamics of rotational motion

The problems of this topic are quite complex mathematically, but if you know the approach, they can be solved using a completely standard algorithm. In all problems you will have to consider the rotation of the body in the vertical plane. The solution will come down to the following sequence of actions:

1. You need to determine the point you are interested in (the point at which you need to determine the speed of the body, the tension force of the thread, weight, and so on).
2. Write down Newton’s second law at this point, taking into account that the body rotates, that is, it has centripetal acceleration.
3. Write down the law of conservation of mechanical energy so that it contains the speed of the body at that very interesting point, as well as the characteristics of the state of the body in some state about which something is known.
4. Depending on the condition, express the squared speed from one equation and substitute it into the other.
5. Carry out the remaining necessary mathematical operations to obtain the final result.

When solving problems, you need to remember that:

• The condition for passing the top point when rotating on a thread at a minimum speed is the support reaction force N at the top point is 0. The same condition is met when passing the top point of the dead loop.
• When rotating on a rod, the condition for passing the entire circle is: the minimum speed at the top point is 0.
• The condition for the separation of a body from the surface of the sphere is that the support reaction force at the separation point is zero.

Inelastic collisions

The law of conservation of mechanical energy and the law of conservation of momentum make it possible to find solutions to mechanical problems in cases where the acting forces are unknown. An example of this type of problem is the impact interaction of bodies.

By impact (or collision) It is customary to call a short-term interaction of bodies, as a result of which their speeds experience significant changes. During a collision of bodies, short-term impact forces act between them, the magnitude of which, as a rule, is unknown. Therefore, it is impossible to consider the impact interaction directly using Newton's laws. The application of the laws of conservation of energy and momentum in many cases makes it possible to exclude the collision process itself from consideration and obtain a connection between the velocities of bodies before and after the collision, bypassing all intermediate values ​​of these quantities.

We often have to deal with the impact interaction of bodies in everyday life, in technology and in physics (especially in the physics of the atom and elementary particles). In mechanics, two models of impact interaction are often used - absolutely elastic and absolutely inelastic impacts.

Absolutely inelastic impact They call this impact interaction in which bodies connect (stick together) with each other and move on as one body.

In a completely inelastic collision, mechanical energy is not conserved. It partially or completely turns into the internal energy of bodies (heating). To describe any impacts, you need to write down both the law of conservation of momentum and the law of conservation of mechanical energy, taking into account the heat released (it is highly advisable to make a drawing first).

Absolutely elastic impact

Absolutely elastic impact called a collision in which the mechanical energy of a system of bodies is conserved. In many cases, collisions of atoms, molecules and elementary particles obey the laws of absolutely elastic impact. With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is satisfied. A simple example of a perfectly elastic collision would be the central impact of two billiard balls, one of which was at rest before the collision.

Central strike balls is called a collision in which the velocities of the balls before and after the impact are directed along the line of centers. Thus, using the laws of conservation of mechanical energy and momentum, it is possible to determine the velocities of the balls after a collision if their velocities before the collision are known. Central impact is very rarely implemented in practice, especially when it comes to collisions of atoms or molecules. In a non-central elastic collision, the velocities of particles (balls) before and after the collision are not directed in one straight line.

A special case of an off-central elastic impact can be the collision of two billiard balls of the same mass, one of which was motionless before the collision, and the speed of the second was not directed along the line of the centers of the balls. In this case, the velocity vectors of the balls after an elastic collision are always directed perpendicular to each other.

Conservation laws. Complex tasks

Multiple bodies

In some problems on the law of conservation of energy, the cables with which certain objects are moved can have mass (that is, not be weightless, as you might already be used to). In this case, the work of moving such cables (namely their centers of gravity) also needs to be taken into account.

If two bodies connected by a weightless rod rotate in a vertical plane, then:

1. choose a zero level to calculate potential energy, for example at the level of the axis of rotation or at the level of the lowest point of one of the weights and be sure to make a drawing;
2. write down the law of conservation of mechanical energy, in which on the left side we write the sum of the kinetic and potential energy of both bodies in the initial situation, and on the right side we write the sum of the kinetic and potential energy of both bodies in the final situation;
3. take into account that the angular velocities of the bodies are the same, then the linear velocities of the bodies are proportional to the radii of rotation;
4. if necessary, write down Newton's second law for each of the bodies separately.

Shell burst

When a projectile explodes, explosive energy is released. To find this energy, it is necessary to subtract the mechanical energy of the projectile before the explosion from the sum of the mechanical energies of the fragments after the explosion. We will also use the law of conservation of momentum, written in the form of the cosine theorem (vector method) or in the form of projections onto selected axes.

Collisions with a heavy plate

Let us meet a heavy plate that moves at speed v, a light ball of mass moves m with speed u n. Since the momentum of the ball is much less than the momentum of the plate, after the impact the speed of the plate will not change, and it will continue to move at the same speed and in the same direction. As a result of the elastic impact, the ball will fly away from the plate. It is important to understand here that the speed of the ball relative to the plate will not change. In this case, for the final speed of the ball we obtain:

Thus, the speed of the ball after impact increases by twice the speed of the wall. Similar reasoning for the case when before the impact the ball and the plate were moving in the same direction leads to the result that the speed of the ball decreases by twice the speed of the wall:

In physics and mathematics, among other things, three most important conditions must be met:

1. Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on different topics and of varying complexity. The latter can only be learned by solving thousands of problems.
2. Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty solving most of the CT at the right time. After this, you will only have to think about the most difficult tasks.
3. Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

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