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Kinetic and potential energies. Kinetic energy of rest How to find the kinetic energy of the body formula

Basic theoretical information

Mechanical work

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

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

If the force changes over time, then to find work, they build a graph of the dependence of the force on displacement and find the area of ​​the figure under the graph - this is 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 of force performed per unit of time is called power... Power P(sometimes denoted by the letter N) Is a physical quantity equal to the ratio of work A by the time interval t during which this work was completed:

This formula is used to calculate average power, i.e. power characterizing the process in general. So, work can also be expressed in terms of power: A = Pt(unless, of course, the power and time of the work are known). The unit of power is called a watt (W) or 1 joule per second. If the movement 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 the instantaneous speed into the formula. How do you know what power to count? If the problem is asked for power at a moment in time or at some point in space, then it is considered instantaneous. If you are asked about the power for a certain period of time or a section of the path, then look for the average power.

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

What kind of work is useful and what is spent is determined from the conditions of a specific problem by logical reasoning. For example, if a crane performs work on lifting 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 expended is the work done by the crane's electric motor.

So, the 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 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 the mass of a body by 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 = 100 kJ and is capable of performing work of 100 kJ. This energy can be converted into heat (when braking the car, the tires of the wheels, the road and brake discs heats up) or can be spent on deformation of 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 going, since energy, like work, is a scalar quantity.

The 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 performing work on deformation of bodies or imparting 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 with speed v it is necessary to perform work equal to the obtained value of kinetic energy. If the body mass m moves with speed v, then to stop it, it is necessary to perform work equal to its initial kinetic energy. During deceleration, kinetic energy is mainly (except for the cases of collision, when energy goes to deformation) "taken" by the friction force.

The 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 displacement. 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 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, the concept of potential energy can be introduced.

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

The physical meaning of the body's potential energy: potential energy is equal to the work performed by the force of gravity when the body is lowered to the zero level ( h Is the distance from the center of gravity of the body to the zero level). If the body has potential energy, then it is capable of doing work when this body falls from a height. h 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 energy tasks, one has to find work to raise (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 task, the zero level is chosen for reasons of convenience. The physical meaning is not the potential energy itself, but its change when the body moves from one position to another. This change is independent of the selection of the zero level.

Potential energy of a stretched spring calculated by the formula:

where: k- the stiffness of the spring. A stretched (or compressed) spring is able to set in motion a body attached to it, that is, to impart kinetic energy to this body. Consequently, such a spring has a reserve of energy. Stretching or squeezing NS one must count on 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 lengthening x 2, the elastic force will perform 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

Coefficient of performance (COP)- characteristic of the efficiency of the system (device, machine) in relation to the transformation or transmission of energy. It is determined by the ratio of the 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 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 the electromagnetic energy received in the secondary winding to the energy consumed in the primary winding.

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

Due to the inevitable loss of energy due to friction, heating of surrounding bodies, etc. The efficiency is always less than one. Accordingly, the efficiency is expressed as a fraction of the energy expended, that is, in the form of a correct 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 pressurization and pre-cooling - 40-50%, dynamos and high-power generators - 95%, transformers - 98%.

The 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 spent.

Mechanical energy conservation law

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

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 the 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 (that is, one in which no external forces act, and their work, respectively, is equal to zero) and the forces of gravity and elastic forces interacting with each other, remains unchanged:

This statement expresses energy conservation law (EEC) in mechanical processes... It is a consequence of Newton's laws. The law of conservation of mechanical energy is fulfilled only when bodies in a closed system interact with each other by the 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 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 starting and ending position of the body.
  2. Write down what or what energies the body has at these points.
  3. Equalize the initial and final energy of the body.
  4. Add other required 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 a body at two different points of the trajectory without analyzing the law of motion of the body at all intermediate points. Application of the law of conservation of mechanical energy can greatly simplify the solution of many problems.

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

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 the internal energy of bodies (heating). Thus, the energy as a whole (i.e., not only mechanical) is conserved in any case.

In any physical interactions, energy does not arise or disappear. It only transforms from one form to another. This experimentally established fact expresses the fundamental law of nature - energy conservation and transformation law.

One of the consequences of the law of conservation and transformation of energy is the statement about the impossibility of creating a "perpetuum mobile" - a machine that could perform work indefinitely without spending energy.

Different tasks for work

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

  1. The job can be found by the formula: A = FS∙ cos α ... Find the force performing the work and the amount of movement of the body under the action of this force in the selected frame of reference. 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 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 of lifting a body at a constant speed can be found by the formula: A = mgh, where h- the 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. Work can be found as the area of ​​the figure under the force versus displacement or power versus time graph.

Energy conservation law and dynamics of rotational motion

The tasks of this topic are quite complex mathematically, but if you know 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 boil down 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 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 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, one must remember that:

  • The condition for passing the top point when rotating on the thread with a minimum speed is the reaction force of the support N at the top point is 0. The same condition is fulfilled when passing the top point of the dead loop.
  • When rotating on a rod, the condition for passing the entire circle: 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 point of separation is equal to 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 kind of problem is the impact interaction of bodies.

By blow (or collision) it is customary to call a short-term interaction of bodies, as a result of which their speeds undergo significant changes. During the collision of bodies between them, short-term impact forces act, 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 from consideration the collision process itself and to obtain a relationship between the velocities of bodies before and after the collision, bypassing all intermediate values ​​of these quantities.

The impact interaction of bodies often has to be dealt with in everyday life, in technology and in physics (especially in the physics of the atom and elementary particles). Two models of impact interaction are often used in mechanics - absolutely elastic and absolutely inelastic impacts.

With a completely inelastic blow is called such an impact interaction in which the bodies are connected (stick together) with each other and move on as one body.

With a completely inelastic impact, mechanical energy is not conserved. It partially or completely passes into the internal energy of bodies (heating). To describe any shocks, 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 make a drawing beforehand).

Absolutely resilient impact

Absolutely resilient impact a collision is called, 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 is the central impact of two billiard balls, one of which was at rest before the collision.

Center blow balls called collision, in which the speed of the balls before and after 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 collision, if their velocities before collision are known. Central impact is very rarely realized in practice, especially when it comes to collisions of atoms or molecules. In the case of off-center elastic collision, the velocities of the particles (balls) before and after the collision are not directed along one straight line.

A particular case of off-center elastic impact can be the collision of two billiard balls of the same mass, one of which was motionless before the collision, and the velocity of the second was directed not 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. Challenging tasks

Multiple bodies

In some problems on the law of conservation of energy, the cables with the help of which some objects are moved may have mass (i.e. not be weightless, as you might already get 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 weights is located and make a drawing;
  2. write down the law of conservation of mechanical energy, in which the sum of the kinetic and potential energy of both bodies in the initial situation is recorded on the left side, and the sum of the kinetic and potential energy of both bodies in the final situation is recorded 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.

Shell burst

In the event of a projectile bursting, 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.

Heavy slab collisions

Let towards a heavy plate that moves at a speed v, a light ball with a mass of m with speed u n. Since the momentum of the ball is much less than the momentum of the plate, then 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 off 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 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 moved in the same direction before the impact leads to the result according to which the speed of the ball decreases by twice the speed of the wall:

Problems on the maximum and minimum values ​​of the energy of colliding balls

In problems of this type, the main thing is to understand that the potential energy of elastic deformation of balls is maximum, if the kinetic energy of their motion is minimum - this follows from the law of conservation of mechanical energy. The sum of the kinetic energies of the balls is minimal at the moment when the velocities of the balls are the same in magnitude and directed in the same direction. At this moment, the relative velocity of the balls is zero, and the deformation and the associated potential energy are maximum.

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How to successfully prepare for a CT in physics and mathematics?

In order to successfully prepare for the CT in physics and mathematics, among other things, three important conditions must be met:

  1. Explore all topics and complete all tests and tasks given in the training 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 CT is an exam, where it is not enough just to know physics or mathematics, you still need to be able to quickly and smoothly 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 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 the basic level of complexity, which are also quite possible to learn, and thus, completely automatically and without difficulty, at the right time, solve most of the CG. After that, you will only have to think about the most difficult tasks.
  3. Attend all three physics and mathematics rehearsal tests. Each RT can be visited twice to solve both options. Again, on the CT, 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 correctly plan the time, distribute forces, and most importantly, fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own surname. Also, during RT, it is important to get used to the style of posing questions in tasks, which on the CT may seem very unusual to an unprepared person.

Successful, diligent and responsible fulfillment of these three points, as well as responsible elaboration of the final training tests, will allow you to show excellent results on the CT, the maximum of what you are capable of.

Found a bug?

If you, as it seems to you, found an error in the training materials, please write about it by e-mail (). In the letter, indicate the subject (physics or mathematics), the title or number of the topic or test, the number of the problem, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not an error.

A4. What changes does a person note in sound with an increase in the frequency of oscillations in a sound wave?
1) Raising the pitch
2) Lowering the pitch
3) Increase the volume
4) Decrease volume

A5. Distances from two coherent sources of waves to point M are equal to a and b. The phase difference of oscillations of the sources is equal to zero, the wavelength is equal to l. If only one source of waves radiates, then the amplitude of oscillations of the particles of the medium at point M is equal to A1, if only the second, then - A2. If the difference between the wave paths a - b = 3l / 2, then at point M the amplitude of the total oscillation of the particles of the medium is
1) is equal to zero 2) is equal to | A1 - A2 | 3) is equal to | A1 + A2 |
4) changes over time periodically

A6. Choose the correct statement.
A. Based on Faraday's experiments on the study of electromagnetic induction, Maxwell theoretically predicted the existence of electromagnetic waves.
B. Based on theoretical predictions of Maxwell, Hertz discovered electromagnetic waves experimentally.
C. Based on Hertz's experiments on the study of electromagnetic waves, Maxwell created a theory of their propagation in a vacuum.
1) Only A and B 2) Only A and C 3) Only B and C 4) And A, and B, and C

A7. Which statement is correct?
In Maxwell's electromagnetic field theory
A - an alternating electric field generates a vortex magnetic field
B - an alternating magnetic field generates a vortex electric field

A8. One scientific laboratory uses a linear accelerator to accelerate charged particles, and the other uses a cyclotron, in which particles are accelerated along a spiral trajectory. Which laboratory should take into account the possibility of electromagnetic radiation hazardous to humans.
1) Only in the first 2) Only in the second 3) In both laboratories
4) None of the laboratories

A9. Which statement is correct?
Radiation of electromagnetic waves occurs when
A - the motion of an electron in a linear accelerator
B - oscillatory motion of electrons in the antenna
1) Only A 2) Only B 3) Both A and B 4) Neither A nor B

A10. A charged particle does not emit electromagnetic waves in a vacuum
1) uniform rectilinear motion
2) uniform motion around the circumference
3) oscillatory motion
4) any movement with acceleration

A11. The speed of propagation of electromagnetic waves
1) has a maximum value in vacuum
2) has a maximum value in dielectrics
3) has a maximum value in metals
4) is the same in any environment

A12. In the first experiments to study the propagation of electromagnetic waves in air, the wavelength cm and the radiation frequency MHz were measured. Based on these imprecise experiments, the value of the speed of light in air was obtained, equal to approximately
1) 100,000 km / s 2) 200,000 km / s 3) 250,000 km / s 4) 300,000 km / s

A13. Oscillations of the electric field in an electromagnetic wave are described by the equation: E = 10sin (107t). Determine the vibration frequency (in Hz).
1) 107 2) 1.6 * 106 3) (107 t) 4) 10

A14. When an electromagnetic wave propagates in a vacuum
1) only energy transfer occurs
2) only momentum transfer occurs
3) there is a transfer of both energy and momentum
4) there is no transfer of either energy or momentum

A15. When an electromagnetic wave passes through the air, vibrations occur
1) air molecules
2) air density
3) the strength of the electric and induction of magnetic fields
4) oxygen concentration

A16. The phenomenon proving that in an electromagnetic wave the vector of the electric field strength oscillates in a direction perpendicular to the direction of propagation of the electromagnetic wave is
1) interference 2) reflection 3) polarization 4) diffraction

A17. Indicate the combination of those parameters of the electromagnetic wave that change when the wave passes from air to glass
1) speed and wavelength 2) frequency and speed
3) wavelength and frequency 4) amplitude and frequency

A18. What phenomenon is characteristic of electromagnetic waves, but is not a common property of waves of any nature?
1) interference 2) refraction 3) polarization 4) diffraction

A19. What wavelength should the radio be tuned to in order to listen to the Europe + radio station, which broadcasts at 106.2 MHz?
1) 2.825 dm 2) 2.825 cm 3) 2.825 km 4) 2.825 m

A20. Amplitude modulation of high-frequency electromagnetic waves in a radio transmitter is used to
1) increasing the power of the radio station
2) changes in the amplitude of high-frequency oscillations
3) changes in the amplitude of the sound frequency oscillations
4) setting a certain frequency of radiation of a given radio station

Everyday experience shows that immovable bodies can be set in motion, and movable ones can be stopped. We are constantly doing something, the world is bustling around, the sun is shining ... But where do humans, animals, and nature as a whole get the strength to do this work? Does it disappear without a trace? Will one body begin to move without changing the movement of the other? We will talk about all this in our article.

Energy concept

For the operation of engines that give motion to cars, tractors, diesel locomotives, airplanes, you need fuel, which is a source of energy. Electric motors move machines using electricity. Due to the energy of water falling from a height, hydraulic turbines are wrapped, connected to electric machines that produce electric current. A person also needs energy in order to exist and work. They say that in order to do any work, energy is needed. What is energy?

  • Observation 1. Lift the ball off the ground. As long as he is calm, no mechanical work is done. Let's let him go. Gravity causes the ball to fall to the ground from a certain height. When the ball falls, mechanical work is performed.
  • Observation 2. Let's close the spring, fix it with a thread and put a weight on the spring. Let's set fire to the thread, the spring will straighten and raise the weight to a certain height. The spring has done mechanical work.
  • Observation 3. On the trolley we fix the rod with the block at the end. We will throw a thread through the block, one end of which is wound on the axis of the trolley, and a weight hangs on the other. Let's release the weight. Under the action, it will drop downward and give the cart movement. The weight has done mechanical work.

After analyzing all the above observations, we can conclude that if a body or several bodies perform mechanical work during interaction, then they say that they have mechanical energy, or energy.

Energy concept

Energy (from the Greek word energy- activity) is a physical quantity that characterizes the ability of bodies to do work. The unit of energy, as well as work in the SI system, is one Joule (1 J). In writing, energy is denoted by the letter E... From the above experiments, it can be seen that the body does work when it passes from one state to another. At the same time, the energy of the body changes (decreases), and the mechanical work performed by the body is equal to the result of a change in its mechanical energy.

Types of mechanical energy. Potential energy concept

There are 2 types of mechanical energy: potential and kinetic. Now let's take a closer look at potential energy.

Potential energy (PE) - determined by the mutual position of the bodies that interact, or by parts of the same body. Since any body and the earth attract each other, that is, they interact, the PE of the body raised above the ground will depend on the height of the rise h... The higher the body is lifted, the greater its PE. It has been experimentally established that PE depends not only on the height to which it is raised, but also on body weight. If the bodies were raised to the same height, then a body with a large mass will also have a large PE. The formula for this energy is as follows: E p = mgh, where E p is potential energy, m- body weight, g = 9.81 N / kg, h - height.

Spring potential energy

Bodies are called physical quantities E p, which, when the speed of translational motion changes under the action, decreases by exactly as much as the kinetic energy increases. Springs (like other elastically deformed bodies) have such a PE, which is equal to half the product of their stiffness k per strain square: x = kx 2: 2.

Kinetic energy: formula and definition

Sometimes the meaning of mechanical work can be considered without using the concepts of force and movement, focusing on the fact that work characterizes a change in the energy of the body. All we may need is the mass of a body and its initial and final velocities, which will lead us to kinetic energy. Kinetic energy (KE) is the energy that belongs to the body due to its own motion.

Wind has kinetic energy, it is used to give motion to wind turbines. The propelled ones put pressure on the inclined planes of the wings of wind turbines and force them to turn around. Rotational motion is transmitted by transmission systems to mechanisms that perform a specific job. The propelled water that turns the turbines of a power plant loses some of its EC while doing work. An airplane flying high in the sky, in addition to a PE, has an EE. If the body is at rest, that is, its speed relative to the Earth is zero, then its CE relative to the Earth is zero. It has been experimentally established that the greater the mass of a body and the speed with which it moves, the greater its FE. The formula for the kinetic energy of translational motion in mathematical expression is as follows:

Where TO- kinetic energy, m- body mass, v- speed.

Change in kinetic energy

Since the speed of movement of a body is a quantity that depends on the choice of the frame of reference, the value of the FE of the body also depends on its choice. A change in the kinetic energy (IKE) of the body occurs due to the action of an external force on the body F... Physical quantity A, which is equal to IQE ΔE to body due to the action of force on it F, called work: A = ΔE c. If on a body that moves with speed v 1 , the force is acting F coinciding with the direction, then the speed of movement of the body will increase over a period of time t to some value v 2 ... In this case, the IQE is equal to:

Where m- body mass; d- the traversed path of the body; V f1 = (V 2 - V 1); V f2 = (V 2 + V 1); a = F: m... It is this formula that calculates how much the kinetic energy changes. The formula can also have the following interpretation: ΔЕ к = Flcos ά , where cosά is the angle between the force vectors F and speed V.

Average kinetic energy

Kinetic energy is energy determined by the speed of movement of different points that belong to this system. However, it should be remembered that it is necessary to distinguish between 2 energies that characterize different translational and rotational. (SKE) in this case is the average difference between the totality of the energies of the entire system and its energy of tranquility, that is, in fact, its value is the average value of potential energy. The formula for the average kinetic energy is as follows:

where k is the Boltzmann constant; T is the temperature. It is this equation that is the basis of the molecular kinetic theory.

Average kinetic energy of gas molecules

Numerous experiments have established that the average kinetic energy of gas molecules in translational motion at a given temperature is the same and does not depend on the type of gas. In addition, it was also found that when the gas is heated by 1 ° C, the SEE increases by the same value. To be more precise, this value is equal to: ΔE k = 2.07 x 10 -23 J / o C. In order to calculate what the average kinetic energy of gas molecules in translational motion is equal to, it is necessary, in addition to this relative value, to know at least one more absolute value of the energy of translational motion. In physics, these values ​​are quite accurately determined for a wide range of temperatures. For example, at a temperature t = 500 о С kinetic energy of the translational motion of the molecule Ek = 1600 x 10 -23 J. Knowing 2 quantities ( ΔE to and E k), we can both calculate the energy of the translational motion of molecules at a given temperature, and solve the inverse problem - to determine the temperature from the given energy values.

Finally, we can conclude that the average kinetic energy of molecules, the formula of which is given above, depends only on the absolute temperature (and for any state of aggregation of substances).

Total mechanical energy conservation law

The study of the motion of bodies under the influence of gravity and elastic forces has shown that there is a certain physical quantity, which is called potential energy E n; it depends on the coordinates of the body, and its change is equated to the IQE, which is taken with the opposite sign: Δ E n =-ΔE c. So, the sum of changes in the FE and PE of the body, which interact with gravitational forces and elastic forces, is equal to 0 : Δ E n +ΔE k = 0. Forces that depend only on the coordinates of the body are called conservative. The forces of attraction and elasticity are conservative forces. The sum of the kinetic and potential energies of the body is the total mechanical energy: E n +E k = E.

This fact, which has been proven by the most accurate experiments,
are called mechanical energy conservation law... If bodies interact with forces that depend on the speed of relative motion, mechanical energy is not conserved in the system of interacting bodies. An example of this type of force called non-conservative, are the friction forces. If friction forces act on the body, then to overcome them it is necessary to expend energy, that is, part of it is used to perform work against friction forces. However, violation of the law of conservation of energy is only imaginary here, because it is a separate case of the general law of conservation and transformation of energy. The energy of bodies never disappears or reappears: it only transforms from one type to another. This law of nature is very important, it is carried out everywhere. It is also sometimes called the general law of conservation and transformation of energy.

The connection between the internal energy of the body, kinetic and potential energies

The internal energy (U) of a body is its total energy of the body minus the FE of the body as a whole and its PE in the external field of forces. From this we can conclude that the internal energy consists of the CE of the chaotic movement of molecules, the PE interaction between them, and intramolecular energy. Internal energy is an unambiguous function of the state of the system, which suggests the following: if the system is in a given state, its internal energy takes on its inherent values, regardless of what happened earlier.

Relativism

When the speed of a body is close to the speed of light, kinetic energy is found by the following formula:

The kinetic energy of the body, the formula of which was written above, can also be calculated according to the following principle:

Examples of tasks for finding kinetic energy

1. Compare the kinetic energy of a 9 g ball flying at a speed of 300 m / s and a 60 kg man running at a speed of 18 km / h.

So, what is given to us: m 1 = 0.009 kg; V 1 = 300 m / s; m 2 = 60 kg, V 2 = 5 m / s.

Solution:

  • Kinetic energy (formula): E k = mv 2: 2.
  • We have all the data for the calculation, and therefore we will find E to both for the person and for the ball.
  • E k1 = (0.009 kg x (300 m / s) 2): 2 = 405 J;
  • E k2 = (60 kg x (5 m / s) 2): 2 = 750 J.
  • E k1< E k2.

Answer: the kinetic energy of a ball is less than that of a person.

2. A body with a mass of 10 kg was raised to a height of 10 m, after which it was released. What kind of FE will it have at a height of 5 m? Air resistance may be neglected.

So, what is given to us: m = 10 kg; h = 10 m; h 1 = 5 m; g = 9.81 N / kg. E k1 -?

Solution:

  • A body of a certain mass, raised to a certain height, has potential energy: E p = mgh. If the body falls, then it will have sweat at a certain height h 1. energy E p = mgh 1 and kin. energy E k1. In order to correctly find the kinetic energy, the formula given above will not help, and therefore we will solve the problem according to the following algorithm.
  • In this step, we use the law of conservation of energy and write: E n1 +E k1 = E NS.
  • Then E k1 = E NS - E n1 = mgh - mgh 1 = mg (h-h 1).
  • Substituting our values ​​into the formula, we get: E k1 = 10 x 9.81 (10-5) = 490.5 J.

Answer: E k1 = 490.5 J.

3. Flywheel having mass m and radius R, wraps around an axis passing through its center. Flywheel turning speed - ω ... In order to stop the flywheel, a brake shoe is pressed against its rim, acting on it with force F friction... How many revolutions will the flywheel make until it comes to a complete stop? Note that the mass of the flywheel is centered on the rim.

So, what is given to us: m; R; ω; F friction. N -?

Solution:

  • When solving the problem, we will consider the revolutions of the flywheel to be similar to the revolutions of a thin homogeneous hoop with a radius R and mass m, which turns at angular velocity ω.
  • The kinetic energy of such a body is equal to: E k = (J ω 2): 2, where J = m R 2 .
  • The flywheel will stop provided that all of its FE is spent on work to overcome the friction force F friction, arising between the brake pad and the rim: E k = F friction * s, where 2 πRN = (m R 2 ω 2) : 2, where N = ( m ω 2 R): (4 π F tr).

Answer: N = (mω 2 R): (4πF tr).

Finally

Energy is the most important component in all aspects of life, because without it, no body could do work, including a person. We think that the article made it clear to you what energy is, and a detailed presentation of all aspects of one of its components - kinetic energy - will help you understand many of the processes taking place on our planet. And you can learn how to find kinetic energy from the above formulas and examples of problem solving.

The ability or ability of physical bodies to do work is characterized by a concept that is basic for all branches of physics, which is called energy. Depending on the initial source, different types of energy are distinguished: mechanical, internal, electromagnetic, nuclear, gravitational, chemical. Mechanical energy is of two types: potential and kinetic. Kinetic energy is inherent only in moving bodies. Can we then talk about the kinetic energy of rest?

What is the kinetic energy

Let's remember how kinetic energy is calculated. If the body mass m force acting F, then its speed v will begin to change. When moving a body a distance s, work will be done A:

$ A = F * s $ (1)

According to Newton's second law, the force is:

$ F = m * a $ (2)

where a- acceleration.

From the well-known formulas obtained in the section of mechanics, it follows that the displacement modulus s with uniformly accelerated rectilinear motion is associated with the modules of the final v 2 , initial v 1 speeds and accelerations a by the following formula;

$ s = ((v_2 ^ 2-v_1 ^ 2) \ over (2 * a)) $ (3)

Then you can get the formula for calculating the work:

$ A = F * s = m * a * ((v_2 ^ 2 - v_1 ^ 2) \ over 2 * a) = (m * v_2 ^ 2 \ over 2) - (m * v_1 ^ 2 \ over 2) $ (4)

A quantity equal to the product of body weight m by the square of its speed, divided in half is called the kinetic energy of the body E k:

$ E_k = (m * v ^ 2 \ over 2) $ (5)

From formulas (4) and (5) it follows that the work A is equal to:

$ A = E_ (k2) - E_ (k1) $ (6)

Thus, the work done by the force applied to the body turned out to be equal to the change in the kinetic energy of the body. This means that any physical body moving with a nonzero speed has kinetic energy. Therefore, at rest, at a speed v equal to zero and the kinetic energy of rest will also be equal to zero.

Rice. 1. Examples of kinetic energy:

Stationary body and temperature

Any physical body consists of atoms and molecules, which are in a state of continuous chaotic motion at a temperature T not equal to zero. With the help of molecular kinetic theory, it has been proved that the average kinetic energy E to the chaotic movement of molecules depends only on temperature. So for a monatomic gas, this relationship is expressed by the formula:

$ E_k = (3 \ over 2) * k * T $ (7)

where: k = 1.38 * 10 -23 J / K - Boltzmann's constant.

Thus, when the body as a whole is at rest, each molecule and atom of which it is composed can nevertheless have nonzero kinetic energy.

Rice. 2. Chaotic movement of molecules in gas, liquid, solid :.

The temperature of absolute zero is naturally equal to 0 0 K or -273.15 0 C. Scientists working in this field strive to cool matter to this temperature in order to gain new knowledge. So far, the record low temperature obtained in laboratory conditions is only 5.9 * 10 -12 K above absolute zero. To achieve such values, lasers and magnetic cooling are used.

Rest energy

Formula (5) for kinetic energy is valid for speeds much less than the speed of light with, which is equal to 300,000 km / s. Albert Einstein (1879-1955) created a special theory of relativity in which the kinetic energy E to particles of mass m 0 moving at speed v, there is:

$ E_k = m_0 * c ^ 2 \ over \ sqrt (1 - (v ^ 2 \ over c ^ 2)) - m_0 * c ^ 2 $ (8)

At speed v much less than the speed of light with (v << c) formula (8) becomes classical, i.e. into formula (5).

At v= 0 kinetic energy will also be equal to zero. However, the total energy E 0 will be equal to:

$ E_0 = m_0 * c ^ 2 $ (9)

The expression $ m_0 * c ^ 2 $ is called the rest energy. The existence of non-zero energy in a body at rest means that the physical body has energy due to its existence.

Rice. 3. Portrait of Albert Einstein :.

According to Einstein, the sum of the rest energy (9) and kinetic energy (8) gives the total energy of the particle ENS:

$ Ep = m_0 * c ^ 2 \ over \ sqrt (1 - v ^ 2 \ over c ^ 2) = m * c ^ 2 $ (10)

Formula (10) shows the relationship between the mass of a body and its energy. It turns out that a change in body weight leads to a change in its energy.

What have we learned?

So, we learned that the kinetic energy of rest of an ordinary physical body (or particle) is equal to zero, because its speed is zero. The kinetic energy of the particles that make up a body at rest will be nonzero if its absolute temperature is not zero. There is no separate formula for the kinetic energy of rest. To determine the energy of a body at rest, it is permissible to use expressions (7) - (9), bearing in mind that this is the internal energy of the particles that make up the body.

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Kinetic energy is a scalar physical quantity equal to half of the product of the mass of a body by the square of its speed.

To understand what the kinetic energy of a body is, consider the case when a body of mass m under the action of a constant force (F = const) moves in a rectilinear uniformly accelerated manner (a = const). Let us determine the work of the force applied to the body when the modulus of the velocity of this body changes from v1 to v2.

As we know, the work of constant force is calculated by the formula. Since in the case we are considering, the direction of the force F and the displacement s coincide, then, and then we get that the work of the force is equal to A = Fs. According to Newton's second law, we find the force F = ma. For rectilinear uniformly accelerated motion, the following formula is valid:

From this formula, we express the movement of the body:

We substitute the found values ​​of F and S into the work formula, and we get:

It can be seen from the last formula that the work of the force applied to the body when the speed of this body changes is equal to the difference between two values ​​of a certain quantity. And mechanical work is a measure of energy change. Consequently, on the right side of the formula is the difference between the two values ​​of the energy of a given body. This means that the quantity is the energy due to the movement of the body. This energy is called kinetic energy. It is denoted by Wк.

If we take the formula of work we have derived, then we get

The work done by the force when the speed of a body changes is equal to the change in the kinetic energy of this body

There is also:

Potential energy:

In the formula, we used:

Kinetic energy