Kinetic energy: the formula, the definition. How to find the kinetic energy of a molecule, translational motion, a spring, a body, a gas molecule?

Everyday experience shows that immovable bodies can be set in motion, and movable ones can be stopped. We are constantly doing something, the world around is fussing, the sun is shining ... But where does a person, animals, and nature in general have the strength to carry out this work? Does the mechanical movement disappear without a trace? Will one body move without changing the movement of the other? We will discuss all this in our article.

The concept of energy

For the work of engines that give traffic to cars, tractors, diesel locomotives, aircraft, you need fuel, which is the source of energy. Electric motors give motion to machine tools by means of the electric power. Due to the energy of water falling from a height, hydroturbines are turned on, connected to electric machines that produce electric current. Human beings also need energy in order to exist and work. They say that in order to perform some work, energy is needed. What is energy?

• Observation 1. Raise the ball above the ground. While he is in a state of tranquility, mechanical work is not carried out. Let him go. Under the influence of gravity, the ball falls to the ground from a certain height. During the fall of the ball, mechanical work is performed.
• Observation 2. We will cut the spring, fix it with a thread and put the weights on the spring. Let's set the thread, the spring straightens and lifts the weights to a certain height. The spring performed mechanical work.
• Observation 3. On the trolley, fix the rod with the block at the end. Through the block we throw a thread, one end of which is wound on the axis of the trolley, and on the other hangs a weigher. Let's release the weights. Under the action of gravity, it will sink to the bottom and give the trolley traffic. Gruzik performed mechanical work.

After analyzing all the above observations, it can be concluded that if a body or several bodies perform mechanical work during an interaction, they are said to have mechanical energy or energy.

The concept of energy

Energy (from the Greek word energy is activity) is a physical quantity that characterizes the ability of bodies to perform work. The unit of energy, as well as the work in the SI system, is one Joule (1 Joule). On the letter, energy is denoted by the letter E. From the above experiments it is clear that the body performs the work when it passes from one state to another. The energy of the body changes (decreases), and the mechanical work performed by the body is equal to the result of the change in its mechanical energy.

Types of mechanical energy. The concept of potential energy

There are 2 types of mechanical energy: potential and kinetic. Now we will take a closer look at the potential energy.

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

Potential energy of a spring

The potential energy of an elastically deformed body is the physical quantity E _n, which, when the speed of translational motion changes under the action of elastic forces, decreases exactly as much as the kinetic energy increases. Springs (like other elastically deformed bodies) have a PE that is equal to half the product of their rigidity k by the square of the deformation: x = kx ² : 2.

Kinetic energy: the formula and definition

Sometimes the importance of mechanical work can be considered without using the concepts of force and displacement, emphasizing that the work characterizes the change in the energy of the body. All that we may need is the mass of a certain body and its initial and final velocities, which will lead us to kinetic energy. The kinetic energy (KE) is the energy that belongs to the body as a result of its own motion.

The wind energy is used for kinetic energy, it is used to impart movement to windmills. The movable masses of air pressurize the inclined planes of the wings of the windmills and cause them to turn around. Rotary motion by means of transmission systems is transferred to mechanisms that perform a certain work. Movable water, wrapping the turbines of a power plant, loses part of its CE, performing work. A plane flying high in the sky, in addition to PE, has a CE. If the body is at rest, that is, its velocity 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 the body and the speed with which it moves, the greater its FE. The formula for the kinetic energy of translational motion in mathematical terms is as follows:

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

Change in kinetic energy

Since the speed of motion of the body is a quantity that depends on the choice of the frame of reference, the value of the CE of the body also depends on its choice. The change in kinetic energy (IKE) of a body occurs due to the action of an external force F on the body. The physical quantity A , which is equal to the IEEEE _{to the} body due to the action of the force F on it , is called the work: A = ΔE _k . If the force that moves with velocity v ₁ acts on the body coinciding with the direction, then the velocity of the body will increase over a time interval t to a certain value of v ₂ . In this case, the ICE equals:

Where m is the mass of the body; D - traversed body path; V _f1 = (V2 - V1); V _f2 = (V2 + V1); A = F: m . It is by this formula that 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 the velocity V.

Average kinetic energy

Kinetic energy is an energy determined by the speed of motion of different points that belong to this system. However, it should be remembered that it is necessary to distinguish between two energies characterizing different types of motion: translational and rotational. The average kinetic energy (SCE) is then the average difference between the total energy of the entire system and its energy of tranquility, that is, in fact, its magnitude is the average value of the 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.

The average kinetic energy of gas molecules

Numerous experiments have established that the average kinetic energy of gas molecules in translational motion for a given temperature is the same, and does not depend on the nature of the gas. In addition, it was also found that when the gas is heated by 1 ° C, the SCE increases by the same value. To say more precisely, this value is equal to: ΔЕ = 2,07 x 10 ^-23 J / ^o C. In order to calculate what the average kinetic energy of the gas molecules in the translational motion is equal to, it is necessary, in addition to this relative quantity, to know at least one more Absolute value of the energy of translational motion. In physics, these values are fairly accurately determined for a wide range of temperatures. For example, at a temperature of t = 500 ^о С the kinetic energy of translational motion of the molecule Ек = 1600 х 10 ^-23 J. Knowing the 2 values ( ΔЕ _к and Е _к ), we can both calculate the energy of translational motion of molecules at a given temperature, and solve The inverse problem is to determine the temperature from the given values of the energy.

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

The law of conservation of total mechanical energy

The study of the motion of bodies under the influence of gravity and elastic forces showed that there is a certain physical quantity, which is called the potential energy E _n ; It depends on the coordinates of the body, and its change is equated with the ICE, which is taken with the opposite sign: ΔE _{n =} - ΔE _k . So, the sum of changes in the CE and PE of the body that interact with gravitational forces and elastic forces is 0 : ΔE _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 was proved by the most accurate experiments,
Called the law of conservation of mechanical energy . If the bodies interact with forces that depend on the speed of relative motion, the mechanical energy in the system of interacting bodies is not conserved. An example of forces of this type, which are called non-conservative , are frictional forces. If friction forces act on the body, then it is necessary to expend energy to overcome them, that is, part of it is used to perform work against frictional forces. However, the violation of the law of conservation of energy is only imaginary, because it is a separate case of the general law of conservation and transformation of energy. The energy of the bodies never disappears and does not reappear: it is only transformed from one species into another. This law of nature is very important, it is carried out everywhere. It is sometimes called the general law of conservation and transformation of energy.

The relationship 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 CE 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 chaotic motion of molecules, PE interaction between them and intramolecular energy. Internal energy is a single-valued function of the state of the system, which suggests the following: if the system is in a given state, its internal energy assumes its intrinsic values, regardless of what happened before.

Relativism

When the speed of the body is close to the speed of light, the 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 on the basis of this principle:

Examples of problems in finding kinetic energy

1. Compare the kinetic energy of a ball weighing 9 g flying at a speed of 300 m / s, and a man weighing 60 kg, 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.

Decision:

• The kinetic energy (formula): E _k = mv ² : 2.
• We have all the data to calculate, and therefore we find E _for both the person and 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 the ball is less than that of the person.

2. The body with a mass of 10 kg was raised to a height of 10 m, after which it was released. Which CE will it have at a height of 5 m? The air resistance can 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 -?

Decision:

• The body of a certain mass, raised to a certain height, has a potential energy: E _n = mgh. If the body falls, then at some height h _{1 it} will have sweat. Energy E _n = mgh ₁ and kin. The energy E _k1. In order for the kinetic energy to be correctly found, the formula given above does not help, and therefore we solve the problem using the following algorithm.
• In this step we use the law of conservation of energy and write: E _n1 + E _k1 = E _n.
• Then E _k1 = E _n - E _n1 = mgh - mgh ₁ = mg (hh ₁ ).
• Substituting our values in the formula, we get: E _k1 = 10 x 9.81 (10-5) = 490.5 J.

The answer is: E _k1 = 490.5 Joules.

3. A flywheel having a mass m and a radius R, wraps around an axis passing through its center. The angular speed of turning the flywheel is ω . In order to stop the flywheel, a brake shoe, acting on it with _friction force F, _is pressed to its rim. How many turns does the flywheel go to a complete stop? Consider that the mass of the flywheel is concentrated along the rim.

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

Decision:

• In solving the problem, we will consider the flywheel turns to be similar to those of a thin uniform hoop with radius R and mass m, which turns around with angular velocity ω.
• The kinetic energy of such a body is: E _k = (J ω ² ): 2, where J = m R ² .
• The flywheel will stop, provided that all of its TEs are spent for work on overcoming the frictional force F of the _friction between the brake shoe and the rim: E _k = F of the friction * s, where S is the stopping distance, which is 2 πRN.
• Therefore, F _friction * 2 πRN = (M R ² ω ² ): 2, whence N = ( m ω ² R): (4 π F тр).

Answer: N = (mω ² R): (4πF _tr).

Finally

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