Before revealing the topic “How work is measured”, it is necessary to make a small digression. Everything in this world obeys the laws of physics. Each process or phenomenon can be explained on the basis of certain laws of physics. For each measurable quantity, there is a unit in which it is customary to measure it. Units of measurement are fixed and have the same meaning throughout the world.

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System of International Units

The reason for this is the following. In 1960, at the eleventh general conference on weights and measures, a system of measurements was adopted, which is recognized throughout the world. This system was named Le Système International d'Unités, SI (SI System International). This system has become the basis for the definitions of units of measurement accepted throughout the world and their ratio.

Physical terms and terminology

In physics, the unit for measuring the work of a force is called J (Joule), in honor of the English physicist James Joule, who made a great contribution to the development of the section of thermodynamics in physics. One Joule is equal to the work done by a force of one N (Newton) when its application moves one M (meter) in the direction of the force. One N (Newton) is equal to a force with a mass of one kg (kilogram) at an acceleration of one m/s2 (meter per second) in the direction of the force.

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The formula for finding a job

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

Electric current, in its essence, is the ordered movement of electrons in a conductor per unit time. Electric current is based on positively or negatively charged particles. They are called electric charges. Denoted by the letters C, q, Kl (Pendant), named after the French scientist and inventor Charles Coulomb. In the SI system, it is a unit of measure for the number of charged electrons. 1 C is equal to the volume of charged particles flowing through the cross section of the conductor per unit time. The unit of time is one second. The formula for electric charge is shown below in the figure.

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The formula for finding electric charge

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

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The formula for finding the current strength

As mentioned above, electric current is the ordered movement of electrons in a conductor. But there is one caveat: for their movement, a certain impact is needed. This effect is created by creating a potential difference. The electrical charge can be positive or negative. Positive charges always tend to negative charges. This is necessary for the balance of the system. The difference between the number of positively and negatively charged particles is called electrical voltage.

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The formula for finding voltage

Power is the amount of energy expended to do work of one J (Joule) in a period of time of one second. The unit of measurement in physics is denoted as W (Watt), in the SI system W (Watt). Since electrical power is considered, here it is the value of the electrical energy expended to perform a certain action in a period of time.

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The formula for finding electrical power

In conclusion, it should be noted that the unit of measure of work is a scalar quantity, has a relationship with all sections of physics and can be considered from the side of not only electrodynamics or heat engineering, but also other sections. The article briefly considers the value that characterizes the unit of measurement of the work of force.

Video

Mechanical work is an energy characteristic of the movement of physical bodies, which has a scalar form. It is equal to the modulus of the force acting on the body, multiplied by the modulus of displacement caused by this force and the cosine of the angle between them.

Formula 1 - Mechanical work.

F - Force acting on the body.

s - body movement.

cosa - Cosine of the angle between force and displacement.

This formula has a general form. If the angle between the applied force and the displacement is zero, then the cosine is 1. Accordingly, the work will only be equal to the product of the force and the displacement. Simply put, if the body moves in the direction of application of the force, then the mechanical work is equal to the product of the force and the displacement.

The second special case is when the angle between the force acting on the body and its displacement is 90 degrees. In this case, the cosine of 90 degrees is equal to zero, respectively, the work will be equal to zero. And indeed, what happens is we apply force in one direction, and the body moves perpendicular to it. That is, the body is obviously not moving under the influence of our force. Thus, the work of our force to move the body is zero.

Figure 1 - The work of forces when moving the body.

If more than one force acts on the body, then the total force acting on the body is calculated. And then it is substituted into the formula as the only force. A body under the action of a force can move not only in a straight line, but also along an arbitrary trajectory. In this case, the work is calculated for a small section of movement, which can be considered straight and then summed up along the entire path.

Work can be both positive and negative. That is, if the displacement and force coincide in direction, then the work is positive. And if the force is applied in one direction, and the body moves in the other, then the work will be negative. An example of negative work is the work of the friction force. Since the friction force is directed against the movement. Imagine a body moving along a plane. A force applied to a body pushes it in a certain direction. This force does positive work to move the body. But at the same time, the friction force does negative work. It slows down the movement of the body and is directed towards its movement.

Figure 2 - Force of movement and friction.

Work in mechanics is measured in Joules. One Joule is the work done by a force of one Newton when a body moves one meter. In addition to the direction of movement of the body, the magnitude of the applied force can also change. For example, when a spring is compressed, the force applied to it will increase in proportion to the distance traveled. In this case, the work is calculated by the formula.

Formula 2 - Work of compression of a spring.

k is the stiffness of the spring.

x - move coordinate.

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.

Basic theoretical information

mechanical work

The energy characteristics of motion are introduced on the basis of the concept mechanical work or force work. Work done by a constant force F, is a physical quantity equal to the product of the modules of force and displacement, multiplied by the cosine of the angle between the force vectors F and displacement 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, they build a graph of the dependence of the force on the displacement and find the area of ​​\u200b\u200bthe 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 extr = kx).

Power

The work done by a force per unit of time is called power. Power P(sometimes referred to as N) is a physical quantity equal to the ratio of work A to the time span 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(unless, 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:

With 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 to know what power to count? If the task asks for power at a point in time or at some point in space, then it is considered instantaneous. If you are asking about power over a certain period of time or a section of the path, then look for the average power.

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

What work is useful and what is spent is determined from the condition of a particular task by logical reasoning. For example, if a crane does work to lift a load to a certain height, then the work of lifting the load will be useful (since the crane was created for it), and the work done by the crane's electric motor will be spent.

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 purpose of doing the work (useful work or power), and what was the mechanism or way of doing all the work (expended power or work).

In the general case, the efficiency shows how efficiently the 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 the body's mass and the square of its speed is called kinetic energy of the body (energy of motion):

That is, if a car with a mass of 2000 kg moves at a speed of 10 m/s, then it has a kinetic energy equal to E k \u003d 100 kJ and is capable of doing work of 100 kJ. This energy can be converted into heat (when the 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. the 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 mass m began to move at a speed v it is necessary to do work equal to the obtained value of kinetic energy. If the body mass m moving at a speed v, then to stop it, it is necessary to do work equal to its initial kinetic energy. During braking, the kinetic energy is mainly (except for cases of collision, when the energy is used for deformation) “taken away” by the friction force.

Kinetic energy theorem: the work of the resultant force is equal to the change in the kinetic energy of the body:

The kinetic energy theorem is also valid in the general case when the body moves under the action 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 of acceleration and deceleration of a body.

Potential energy

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

Potential energy is determined by the mutual position of the bodies (for example, the position of the body relative to the Earth's surface). 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 of such forces on a closed trajectory is zero. This property is possessed by the force of gravity and the force of elasticity. 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 the body: the potential energy is equal to the work done by the force of gravity when lowering the body to the zero level ( h is the 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 down to zero. The work of gravity is equal to the change in the potential energy of the body, taken with the opposite sign:

Often in tasks for energy, you have to find work to lift (turn over, get out of the pit) 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. It is not the potential energy itself that has physical meaning, but its change when the body moves from one position to another. This change does not depend on the choice of the zero level.

Potential energy of a stretched spring calculated by the formula:

where: k- spring stiffness. A stretched (or compressed) spring is capable of setting in motion a body attached to it, that is, imparting kinetic energy to this body. Therefore, such a spring has a reserve of energy. Stretch 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 of 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 distance traveled (this type of force whose work depends on the trajectory and the distance traveled is called: dissipative forces). The concept of potential energy for the friction force cannot be introduced.

Efficiency

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

Efficiency can be calculated both in terms of work and in terms of power. Useful and expended work (power) is 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 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 the inevitable energy losses due to friction, heating of surrounding bodies, etc. The efficiency is always less than unity. Accordingly, the 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 works. 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%.

The task in which you need to find the efficiency or it is known, you need to start with a logical reasoning - what work is useful and what is spent.

Law of conservation of mechanical energy

full mechanical energy 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) is called:

If mechanical energy does not pass into other forms, for example, into internal (thermal) energy, then the sum of kinetic and potential energy remains unchanged. If mechanical energy is converted 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 energies of the bodies that make up a closed system (i.e., one in which no external forces act, and their work is equal to zero, respectively) and interacting with each other by gravitational forces and elastic forces, remains unchanged:

This statement expresses law of conservation of energy (LSE) in mechanical processes. It is a consequence of Newton's laws. The law of conservation of mechanical energy is fulfilled only when the 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 the system of bodies. The law says 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 connection between the coordinates and velocities of the 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, almost always moving bodies, along with gravitational forces, elastic forces and other forces, are acted upon by friction forces or resistance forces of the medium. The work of 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, the energy as a whole (i.e. not only mechanical energy) is conserved in any case.

In any physical interactions, energy does not arise and does not disappear. It only changes from one form to another. This experimentally established fact expresses the 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 assertion that it is impossible to create a “perpetual motion machine” (perpetuum mobile) - a machine that could do work indefinitely without consuming energy.

Miscellaneous work tasks

If you need to find mechanical work in the problem, then first select the method for finding it:

1. Jobs 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 action of this force in the selected reference frame. Note that the angle must be chosen between the force and displacement vectors.
2. The work of an external force can be found as the difference between the 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 by the formula: A = mgh, where h- the height to which it rises center of gravity of the body.
4. Work can be found as the product of power and time, i.e. according to the formula: A = Pt.
5. Work can be found as the area of ​​a figure under a graph of force versus displacement or power versus time.

The law of conservation of energy and the dynamics of rotational motion

The tasks of this topic are quite complex mathematically, but with knowledge of the approach they are solved according to a completely standard algorithm. In all problems you will have to consider the rotation of the body in the vertical plane. The solution will be reduced to the following sequence of actions:

1. It is necessary to determine the point of interest to you (the point at which it is necessary to determine the speed of the body, the force of the thread tension, weight, and so on).
2. Write down Newton's second law at this point, given 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 speed squared from one equation and substitute it into another.
5. Carry out the rest of the necessary mathematical operations to obtain the final result.

When solving problems, remember that:

• The condition for passing the upper point during rotation on the threads at a minimum speed is the reaction force of the support N at the top point is 0. The same condition is met when passing through 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 the body from the surface of the sphere is that the reaction force of the support 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 such problems is the impact interaction of bodies.

Impact (or collision) It is customary to call the short-term interaction of bodies, as a result of which their velocities experience significant changes. During the 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 with the help of Newton's laws. The application of the laws of conservation of energy and momentum in many cases makes it possible to exclude the process of collision from consideration and obtain a relationship between the velocities of bodies before and after the collision, bypassing all intermediate values ​​of these quantities.

One often has 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 Such a shock interaction is called, in which the bodies are connected (stick together) with each other and move on as one body.

In a perfectly inelastic impact, mechanical energy is not conserved. It partially or completely passes 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 released heat (it is highly desirable to draw a drawing beforehand).

Absolutely elastic impact

Absolutely elastic impact is 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 fulfilled. 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.

center punch balls is called a collision, in which the speeds 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 the collision, if their velocities before the collision are known. The central impact is very rarely realized in practice, especially when it comes to collisions of atoms or molecules. In non-central elastic collision, the velocities of particles (balls) before and after the collision are not directed along the same straight line.

A special case of a non-central elastic impact is the collision of two billiard balls of the same mass, one of which was stationary 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 elastic collision are always directed perpendicular to each other.

Conservation laws. Difficult tasks

Multiple bodies

In some tasks on the law of conservation of energy, the cables with which some objects move 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) must also be taken into account.

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

1. choose a zero level for calculating potential energy, for example, at the level of the axis of rotation or at the level of the lowest point where one of the loads is located and make a drawing;
2. the law of conservation of mechanical energy is written, in which the sum of the kinetic and potential energies of both bodies in the initial situation is written on the left side, and the sum of the kinetic and potential energies of both bodies in the final situation is written on the right side;
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.

Projectile burst

In the event of a projectile burst, 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 on selected axes.

Collisions with a heavy plate

Let towards a heavy plate that moves at a 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, the plate's speed will not change after impact, and it will continue to move at the same speed and in the same direction. As a result of elastic impact, the ball will fly off the plate. Here it is important to understand that the speed of the ball relative to the plate will not change. In this case, for the final speed of the ball we get:

Thus, the speed of the ball after impact is increased by twice the speed of the wall. A similar reasoning for the case when the ball and the plate were moving in the same direction before the impact leads to the result that the speed of the ball is reduced by twice the speed of the wall:

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

1. Study all the topics and complete all the tests and tasks given in the study materials on this site. To do this, you need nothing at all, namely: to 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 various topics and varying complexity. The latter can only be learned by solving thousands of problems.
2. Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, 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, solve most of the digital transformation at the right time. After that, 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 solve both options. Again, on the DT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and problems, or your own name. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

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

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To be able to characterize the energy characteristics of motion, the concept of mechanical work was introduced. And it is to her in her various manifestations that the article is devoted. To understand the topic is both easy and quite complex. The author sincerely tried to make it more understandable and understandable, and one can only hope that the goal has been achieved.

What is mechanical work?

What is it called? If some force works on the body, and as a result of the action of this force, the body moves, then this is called mechanical work. When approached from the point of view of scientific philosophy, several additional aspects can be distinguished here, but the article will cover the topic from the point of view of physics. Mechanical work is not difficult if you think carefully about the words written here. But the word "mechanical" is usually not written, and everything is reduced to the word "work". But not every job is mechanical. Here a man sits and thinks. Does it work? Mentally yes! But is it mechanical work? No. What if the person is walking? If the body moves under the influence of a force, then this is mechanical work. Everything is simple. In other words, the force acting on the body does (mechanical) work. And one more thing: it is work that can characterize the result of the action of a certain force. So if a person walks, then certain forces (friction, gravity, etc.) perform mechanical work on a person, and as a result of their action, a person changes his point of location, in other words, he moves.

Work as a physical quantity is equal to the force that acts on the body, multiplied by the path that the body made under the influence of this force and in the direction indicated by it. We can say that mechanical work was done if 2 conditions were simultaneously met: the force acted on the body, and it moved in the direction of its action. But it was not performed or is not performed if the force acted, and the body did not change its location in the coordinate system. Here are small examples where mechanical work is not done:

1. So a person can fall on a huge boulder in order to move it, but there is not enough strength. The force acts on the stone, but it does not move, and work does not occur.
2. The body moves in the coordinate system, and the force is equal to zero or they are all compensated. This can be observed during inertial motion.
3. When the direction in which the body moves is perpendicular to the force. When the train moves along a horizontal line, the force of gravity does not do its work.

Depending on certain conditions, mechanical work can be negative and positive. So, if the directions and forces, and the movements of the body are the same, then positive work occurs. An example of positive work is the effect of gravity on a falling drop of water. But if the force and direction of movement are opposite, then negative mechanical work occurs. An example of such an option is a balloon rising up and gravity, which does negative work. When a body is subjected to the influence of several forces, such work is called "resultant force work".

Features of practical application (kinetic energy)

We pass from theory to practical part. Separately, we should talk about mechanical work and its use in physics. As many probably remembered, all the energy of the body is divided into kinetic and potential. When an object is in equilibrium and not moving anywhere, its potential energy is equal to the total energy, and its kinetic energy is zero. When the movement begins, the potential energy begins to decrease, the kinetic energy to increase, but in total they are equal to the total energy of the object. For a material point, kinetic energy is defined as the work of the force that accelerated the point from zero to the value H, and in formula form, the kinetics of the body is ½ * M * H, where M is the mass. To find out the kinetic energy of an object that consists of many particles, you need to find the sum of all the kinetic energy of the particles, and this will be the kinetic energy of the body.

Features of practical application (potential energy)

In the case when all the forces acting on the body are conservative, and the potential energy is equal to the total, then no work is done. This postulate is known as the law of conservation of mechanical energy. Mechanical energy in a closed system is constant in the time interval. The conservation law is widely used to solve problems from classical mechanics.

Features of practical application (thermodynamics)

In thermodynamics, the work done by a gas during expansion is calculated by the integral of pressure multiplied by volume. This approach is applicable not only in cases where there is an exact function of volume, but also to all processes that can be displayed in the pressure/volume plane. The knowledge of mechanical work is also applied not only to gases, but to everything that can exert pressure.

Features of practical application in practice (theoretical mechanics)

In theoretical mechanics, all the properties and formulas described above are considered in more detail, in particular, these are projections. She also gives her own definition for various formulas of mechanical work (an example of the definition for the Rimmer integral): the limit to which the sum of all the forces of elementary work tends when the fineness of the partition tends to zero is called the work of the force along the curve. Probably difficult? But nothing, with theoretical mechanics everything. Yes, and all the mechanical work, physics and other difficulties are over. Further there will be only examples and a conclusion.

Mechanical work units

The SI uses joules to measure work, while the GHS uses ergs:

1. 1 J = 1 kg m²/s² = 1 Nm
2. 1 erg = 1 g cm²/s² = 1 dyne cm
3. 1 erg = 10 −7 J

Examples of mechanical work

In order to finally understand such a concept as mechanical work, you should study a few separate examples that will allow you to consider it from many, but not all, sides:

1. When a person lifts a stone with his hands, then mechanical work occurs with the help of the muscular strength of the hands;
2. When a train travels along the rails, it is pulled by the traction force of the tractor (electric locomotive, diesel locomotive, etc.);
3. If you take a gun and shoot from it, then thanks to the pressure force that the powder gases will create, work will be done: the bullet is moved along the barrel of the gun at the same time as the speed of the bullet itself increases;
4. There is also mechanical work when the friction force acts on the body, forcing it to reduce the speed of its movement;
5. The above example with balls, when they rise in the opposite direction relative to the direction of gravity, is also an example of mechanical work, but in addition to gravity, the Archimedes force also acts when everything that is lighter than air rises up.

What is power?

Finally, I want to touch on the topic of power. The work done by a force in one unit of time is called power. In fact, power is such a physical quantity that is a reflection of the ratio of work to a certain period of time during which this work was done: M = P / B, where M is power, P is work, B is time. The SI unit of power is 1 watt. A watt is equal to the power that does the work of one joule in one second: 1 W = 1J \ 1s.