How to calculate the kinetic energy of the translational motion of the load. Energy

Energy is the most important concept in mechanics. What is energy. There are many definitions, and here is one of them.

What is energy?

Energy is the body's ability to do work.

Consider a body that was moving under the action of some forces and changed its speed from v 1 → to v 2 →. In this case, the forces acting on the body did a certain job A.

The work of all the forces acting on the body is equal to the work of the resultant force.

F p → = F 1 → + F 2 →

A = F 1 s cos α 1 + F 2 s cos α 2 = F p cos α.

Let us establish a connection between the change in the speed of the body and the work performed by the forces acting on the body. For simplicity, we will assume that the body is acted upon by one force F → directed along a straight line. Under the action of this force, the body moves uniformly and rectilinearly. In this case, the vectors F →, v →, a →, s → coincide in direction and they can be considered as algebraic quantities.

The work of force F → is equal to A = F s. The movement of the body is expressed by the formula s = v 2 2 - v 1 2 2 a. Hence:

A = F s = F v 2 2 - v 1 2 2 a = m a v 2 2 - v 1 2 2 a

A = m v 2 2 - m v 1 2 2 = m v 2 2 2 - m v 1 2 2.

As you can see, the work done by force is proportional to the change in the square of the body's speed.

Definition. Kinetic energy

The kinetic energy of a body is equal to half the product of the body's mass by the square of its velocity.

Kinetic energy is the energy of body movement. At zero speed, it is zero.

Kinetic energy theorem

Let us turn again to the considered example and formulate a theorem on the kinetic energy of a body.

Kinetic energy theorem

The work of the force applied to the body is equal to the change in the kinetic energy of the body. This statement is also true when the body moves under the action of a force varying in magnitude and direction.

A = E K 2 - E K 1.

Thus, the kinetic energy of a body of mass m moving with a speed v → is equal to the work that the force must perform to accelerate the body to this speed.

A = m v 2 2 = E K.

There is work to be done to stop the body

A = - m v 2 2 = - E K

Kinetic energy is the energy of motion. Along with kinetic energy, there is also potential energy, that is, the energy of interaction of bodies, which depends on their position.

For example, the body is raised above the surface of the earth. The higher it is raised, the more potential energy will be. When a body falls down under the force of gravity, that force does work. Moreover, the work of gravity is determined only by the vertical movement of the body and does not depend on the trajectory.

Important!

In general, one can speak about potential energy only in the context of those forces whose work does not depend on the shape of the body's trajectory. Such forces are called conservative.

Examples of conservative forces: gravity, elastic force.

When the body moves vertically upward, gravity does negative work.

Consider an example when the ball has moved from a point with a height h 1 to a point with a height h 2.

In this case, the force of gravity performed work equal to

A = - m g (h 2 - h 1) = - (m g h 2 - m g h 1).

This work is equal to the change in the value of m g h, taken with the opposite sign.

The value of E P = m g h is the potential energy in the gravity field. At zero level (on the ground), the potential energy of the body is zero.

Definition. Potential energy

Potential energy is a part of the total mechanical energy of the system, which is in the field of conservative forces. The potential energy depends on the position of the points that make up the system.

We can talk about potential energy in the field of gravity, potential energy of a compressed spring, etc.

The work of gravity is equal to the change in potential energy, taken with the opposite sign.

A = - (E P 2 - E P 1).

It is clear that the potential energy depends on the choice of the zero level (the origin of the OY axis). We emphasize that the physical meaning has the change potential energy when moving bodies relative to each other. With any choice of the zero level, the change in potential energy will be the same.

When calculating the motion of bodies in the gravitational field of the Earth, but at significant distances from it, one must take into account the law of universal gravitation (the dependence of the force of gravity on the distance to the center of the Earth). Let us give a formula expressing the dependence of the potential energy of the body.

E П = - G m M r.

Here G is the gravitational constant, M is the mass of the Earth.

Spring potential energy

Imagine that in the first case we took a spring and lengthened it by the amount x. In the second case, we first lengthened the spring by 2 x and then reduced it by x. In both cases, the spring was stretched by x, but this was done in different ways.

In this case, the work of the elastic force with a change in the length of the spring by x in both cases was the same and equal to

A y p p = - A = - k x 2 2.

The quantity E y p p = k x 2 2 is called potential energy compressed spring. It is equal to the work of the elastic force during the transition from a given state of the body to a state with zero deformation.

If you notice an error in the text, please select it and press Ctrl + Enter

Issues under consideration:

General theorems of the dynamics of a mechanical system. Kinetic energy: a material point, a system of material points, an absolutely rigid body (with translational, rotational and plane motion). Koenig's theorem. Force work: the elementary work of forces applied to a solid; on final displacement, gravity, sliding friction force, elastic force. Elementary work of the moment of force. The power of the force and the pair of forces. Theorem about the change in the kinetic energy of a material point. The theorem on the change in the kinetic energy of variable and unchanged mechanical systems (differential and integral form). Potential force field and its properties. Equipotential surfaces. Potential function. Potential energy. The law of conservation of total mechanical energy.

5.1 Kinetic energy

a) material point:

Definition: the kinetic energy of a material point is half the product of the mass of this point by the square of its speed:

Kinetic energy is a scalar positive quantity.

In SI, the unit of measure for energy is joule:

1 j = 1 N? M.

b) systems of material points:

The kinetic energy of a system of material points is the sum of the kinetic energies of all points of the system:

c) absolutely solid body:

1) with translational motion.

The velocities of all points are the same and equal to the velocity of the center of mass, i.e. , then:

where M- body mass.

The kinetic energy of a rigid body moving translationally is equal to half the product of the body's mass M by the square of its speed.

2) with rotational movement.

The velocities of the points are determined by the Euler formula:

Speed ​​module:

Kinetic energy of the body during rotational motion:

where: z- axis of rotation.

The kinetic energy of a rigid body rotating around a fixed axis is equal to half the product of the moment of inertia of this body relative to the axis of rotation by the square of the angular velocity of the body.

3) with flat motion.

The speed of any point is determined through the pole:

Plane motion consists of translational motion with the speed of the pole and rotational motion around this pole, then the kinetic energy is the sum of the energy of translational motion and the energy of rotational motion.

Kinetic energy through pole "A" in plane motion:

It is best to take the center of mass for the pole, then:

This is convenient because the moments of inertia about the center of mass are always known.

The kinetic energy of a rigid body during plane-parallel motion is the sum of the kinetic energy of translational motion together with the center of mass and kinetic energy from rotation around a fixed axis passing through the center of mass and perpendicular to the plane of motion.

It is often convenient to take the instantaneous center of velocities as the pole. Then:

Considering that, by definition of the instantaneous center of velocities, its speed is equal to zero, then.

Kinetic energy relative to the instantaneous center of velocities:

It must be remembered that to determine the moment of inertia relative to the instantaneous center of velocities, it is necessary to apply the Huygens - Steiner formula:

This formula is preferable in cases where the instantaneous center of velocities is at the end of the rod.

4) Koenig's theorem.

Let us assume that the mechanical system, together with the coordinate system passing through the center of mass of the system, moves translationally relative to the stationary coordinate system. Then, on the basis of the theorem on the addition of velocities during complex motion of a point, the absolute velocity of an arbitrary point of the system will be written as the vector sum of the portable and relative velocities:

where: - the speed of the origin of the moving coordinate system (portable speed, i.e. the speed of the center of mass of the system);

The speed of the point relative to the moving coordinate system (relative speed). Omitting intermediate calculations, we get:

This equality defines Koenig's theorem.

Formulation: The kinetic energy of the system is equal to the sum of the kinetic energy that would have a material point located in the center of mass of the system and having a mass equal to the mass of the system, and the kinetic energy of motion of the system relative to the center of mass.

5.2Work of strength.

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).

V general case 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 various 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 (i.e., one in which external forces do not 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 internal energy bodies (heating). Thus, the energy as a whole (i.e., not only mechanical) is conserved in any case.

In any physical interaction, 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 in a task mechanical work, then first choose a way to find 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 An absolutely elastic collision can be a 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. Center shot is very rarely implemented in practice, especially if it comes about 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.

• Back
• Forward

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 without failures solve a large number of problems 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.

A quantity in physics and mechanics that characterizes the state of a body or a whole system of bodies in interaction and motion is called energy.

Types of mechanical energy

In mechanics, there are two types of energy:

• Kinetic. This term refers to the mechanical energy of any body that moves. It is measured by the work that the body could carry out when braking to a complete stop.
• Potential. This is the combined mechanical energy of a whole system of bodies, which is determined by their location and the nature of the forces of interaction.

Accordingly, the answer to the question of how to find mechanical energy is theoretically very simple. It is necessary: ​​first to calculate the kinetic energy, then to summarize the potential and obtained results. Mechanical energy, which characterizes the interaction of bodies with each other, is a function of relative position and velocities.

Kinetic energy

Since kinetic energy is possessed by a mechanical system, which depends on the speeds at which its various points move, it can be of the translational and rotational type. The unit Joule (J) in the SI system is used to measure energy.

Let's take a look at how to find energy. Kinetic energy formula:

• Ex = mv² / 2,
  • Ek is kinetic energy measured in Joules;
  • m - body weight (kilograms);
  • v - speed (meter / second).

To determine how to find the kinetic energy for a rigid body, the sum of the kinetic energy of translational and rotational motion is derived.

The kinetic energy of a body, which is moving at a certain speed, calculated in this way, demonstrates the work that the force acting on the body at rest must perform in order to give it speed.

Potential energy

To find out how to find potential energy, you should apply the formula:

• Ep = mgh,
  • Ep is the potential energy measured in Joules;
  • g - acceleration of gravity (square meters);
  • m - body weight (kilograms);
  • h - height of the center of mass of the body above an arbitrary level (meters).

Since potential energy is characterized by the mutual influence of two or more bodies on each other, as well as a body and any field, any physical system seeks to find a position in which the potential energy will be the least, and ideally zero. potential energy. It should be remembered that velocity is characteristic of kinetic energy, and the mutual arrangement of bodies is characteristic of potential energy.

Now you know everything about how to find energy and its value according to the formulas of physics.

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 original source, a distinction is made between different types energies: 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 record low temperature, obtained in laboratory conditions above absolute zero by only 5.9 * 10 -12 K. 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.

Test by topic

Assessment of the report

Average rating: 4.2. Total ratings received: 39.