Formulate the law of conservation of the total mechanical energy of the system. The law of conservation and the transformation of energy. Formulation and definition of the law of conservation and transformation of energy
In all phenomena occurring in nature, energy does not arise and does not disappear. It only changes from one species to another, while its value is preserved.
Law of energy conservation- the fundamental law of nature, which consists in the fact that for an isolated physical system a scalar physical quantity can be introduced, which is a function of the parameters of the system and is called energy, which is conserved over time. Since the law of conservation of energy does not refer to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it can be called not a law, but the principle of conservation of energy.
Law of conservation of mechanical energy
In mechanics, the law of conservation of energy states that in a closed system of particles, the total energy, which is the sum of kinetic and potential energy and does not depend on time, that is, is the integral of motion. The law of conservation of energy is valid only for closed systems, that is, in the absence of external fields or interactions.
The forces of interaction between bodies for which the law of conservation of mechanical energy is fulfilled are called conservative forces. The law of conservation of mechanical energy is not satisfied for friction forces, since in the presence of friction forces, mechanical energy is converted into thermal energy.
Mathematical formulation
The evolution of a mechanical system of material points with masses \(m_i\) according to Newton's second law satisfies the system of equations
\[ m_i\dot(\mathbf(v)_i) = \mathbf(F)_i \]
Where
\(\mathbf(v)_i \) are the velocities of material points, and \(\mathbf(F)_i \) are the forces acting on these points.
If forces are given as the sum of potential forces \(\mathbf(F)_i^p \) and nonpotential forces \(\mathbf(F)_i^d \) , and the potential forces are written as
\[ \mathbf(F)_i^p = - \nabla_i U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]
then, multiplying all the equations by \(\mathbf(v)_i \) we can get
\[ \frac(d)(dt) \sum_i \frac(mv_i^2)(2) = - \sum_i \frac(d\mathbf(r)_i)(dt)\cdot \nabla_i U(\mathbf(r )_1, \mathbf(r)_2, \ldots \mathbf(r)_N) + \sum_i \frac(d\mathbf(r)_i)(dt) \cdot \mathbf(F)_i^d \]
The first sum on the right side of the equation is nothing more than the time derivative of a complex function, and therefore, if we introduce the notation
\[ E = \sum_i \frac(mv_i^2)(2) + U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]
and call this value mechanical energy, then, integrating the equations from time t=0 to time t, we can obtain
\[ E(t) - E(0) = \int_L \mathbf(F)_i^d \cdot d\mathbf(r)_i \]
where the integration is carried out along the trajectories of motion of material points.
Thus, the change in the mechanical energy of a system of material points over time is equal to the work of nonpotential forces.
The law of conservation of energy in mechanics is valid only for systems in which all forces are potential.
The law of conservation of energy for an electromagnetic field
In electrodynamics, the law of conservation of energy is historically formulated in the form of Poynting's theorem.
The change in the electromagnetic energy contained in a certain volume over a certain time interval is equal to the flow of electromagnetic energy through the surface that limits this volume, and the amount of thermal energy released in this volume, taken with the opposite sign.
$ \frac(d)(dt)\int_(V)\omega_(em)dV=-\oint_(\partial V)\vec(S)d\vec(\sigma)-\int_V \vec(j)\ cdot \vec(E)dV $
The electromagnetic field has energy that is distributed in the space occupied by the field. When the field characteristics change, the energy distribution also changes. It flows from one region of space to another, perhaps changing into other forms. Law of energy conservation for an electromagnetic field is a consequence of the field equations.
Inside some closed surface S, limiting the amount of space V occupied by the field contains energy W is the energy of the electromagnetic field:
W=Σ(εε 0 E i 2 / 2 +μμ 0 H i 2 / 2)ΔV i .
If there are currents in this volume, then the electric field produces work on moving charges, per unit time equal to
N=Σ ij̅ i ×E̅ i . ΔV i .
This is the amount of field energy that goes into other forms. It follows from Maxwell's equations that
∆W + N∆t = -∆t∮ SS̅ × n̅ . da,
Where ∆W is the change in the energy of the electromagnetic field in the volume under consideration over time Δt, a vector S̅ = E̅ × H̅ called the Poynting vector.
This law of conservation of energy in electrodynamics.
Through a small area ΔA with unit normal vector n̅ per unit of time in the direction of the vector n̅ flowing energy S̅ × n̅ .ΔA, Where S̅- meaning Pointing vectors within the site. The sum of these quantities over all elements of a closed surface (denoted by the integral sign), which is on the right side of the equality , is the energy flowing out of the volume bounded by the surface per unit time (if this value is negative, then energy flows into the volume). Pointing vector determines the energy flow of the electromagnetic field through the area, it is non-zero everywhere where the vector product of the electric and magnetic field strength vectors is non-zero.
There are three main directions practical application electricity: transmission and transformation of information (radio, television, computers), transmission of momentum and momentum (electric motors), transformation and transmission of energy (electric generators and power lines). Both momentum and energy are transferred by the field through empty space, the presence of a medium leads only to losses. Energy is not transmitted through wires! Wires with current are needed to form electric and magnetic fields of such a configuration that the energy flow, determined by the Poynting vectors at all points in space, is directed from the energy source to the consumer. Energy can be transmitted without wires, then it is carried by electromagnetic waves. (The internal energy of the Sun decreases, is carried away by electromagnetic waves, mainly light. Thanks to part of this energy, life on Earth is supported.)
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The total mechanical energy of a system of bodies is the sum of the kinetic and potential energies:
The change in the kinetic energy of the system is equal to the total work of all forces acting on the bodies of this system:
∆Ek = Apot + Anepot + Aext (1)
The change in the potential energy of the system is equal to the work of potential forces with the opposite sign:
∆Ep = - Apot (2)
Obviously, the change in total mechanical energy is equal to:
∆E = ∆Ep + ∆Ek (3)
From equations (1-3) we obtain that the change in total mechanical energy is equal to the total work of all external forces and internal non-potential forces.
∆Ek = Avext + Anepot (4)
Formula (4) is law of change of total mechanical energy phone systems
What does it consist of law of conservation of mechanical energy? The law of conservation of mechanical energy is that the total mechanical energy of a closed system remains unchanged.
4) Rotational movement. moment of impulse. Tensor of inertia. Kinetic energy and angular momentum of a rigid body. Theorems of König and Steiner-Huygens.
Rotational movement.
rotational movement- a type of mechanical movement. During the rotational motion of an absolutely rigid body, its points describe circles located in parallel planes. The centers of all circles lie in this case on one straight line, perpendicular to the planes of the circles and called the axis of rotation. The axis of rotation can be located inside the body and outside it. The axis of rotation in a given reference system can be either movable or fixed.
With uniform rotation (T revolutions per second),
§ Rotation frequency is the number of revolutions of the body per unit time.
,
§ Rotation period- the time of one complete revolution. Rotation period T and its frequency are related by .
§ Line speed a point located at a distance R from the axis of rotation
§ Angular velocity body rotation
.
§ Kinetic energy of rotational motion
Where Iz- the moment of inertia of the body about the axis of rotation. - angular velocity
moment of impulse.
angular momentum characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed about the axis of rotation, and how fast the rotation occurs.
It should be noted that rotation here is understood in a broad sense, not only as a regular rotation around an axis. For example, even with a rectilinear motion of a body past an arbitrary imaginary point that does not lie on the line of motion, it also has an angular momentum. Perhaps the greatest role is played by the angular momentum in describing the actual rotational motion.
The angular momentum of a closed system is conserved.
The angular momentum of a particle with respect to some origin is determined by the vector product of its radius vector and momentum:
where is the radius-vector of the particle relative to the selected reference point, which is motionless in the given reference frame, is the momentum of the particle.
If the sum of the moments of forces acting on a body rotating around a fixed axis is zero, then the angular momentum is conserved (the law of conservation of the angular momentum):
The time derivative of the angular momentum of a rigid body is equal to the sum of the moments of all forces acting on the body:
Tensor of inertia.
Tensor of inertia- in the mechanics of an absolutely rigid body - a tensor quantity that relates the angular momentum of the body and kinetic energy its rotation with its angular velocity:
where is the inertia tensor, is the angular velocity, is the angular momentum
Kinetic energy.
Kinetic energy- the energy of the mechanical system, which depends on the speed of movement of its points. The SI unit of measure is Joule. Kinetic energy is the difference between the total energy of a system and its rest energy. Often allocate the kinetic energy of translational and rotational motion.
For an absolutely rigid body, the total kinetic energy can be written as the sum of the kinetic energy of translational and rotational motion:
where: - mass of the body, - speed of the center of mass of the body, - moment of inertia of the body, - angular velocity of the body.
König's theorem.
König's theorem allows expressing the total kinetic energy of the system in terms of the energy of motion of the center of mass and the energy of motion relative to the center of mass.
The kinetic energy of the system is the energy of motion of the center of mass plus the energy of motion relative to the center of mass:
,
where is the total kinetic energy, is the energy of the center of mass movement, is the relative kinetic energy.
In other words, the total kinetic energy of a body or a system of bodies in complex motion is equal to the sum of the system's energy in translational motion and the system's energy in rotational motion relative to the center of mass.
The Steiner-Huygens theorem.
The Huygens-Steiner theorem: the moment of inertia of a body about an arbitrary axis is equal to the sum of the moment of inertia of this body about an axis parallel to it, passing through the center of mass of the body, and the product of the body's mass by the square of the distance between the axes:
Where is the known moment of inertia about the axis passing through the center of mass of the body, is the desired moment of inertia about the parallel axis, is the mass of the body, is the distance between the indicated axes.
5) System of two particles. Reduced mass. Central field. Kepler's laws.
Reduced mass.
Reduced mass- a conditional characteristic of the mass distribution in a moving mechanical system, depending on the physical parameters of the system (masses, moments of inertia, etc.) and on its law of motion.
Usually the reduced mass is determined from the equality 
, where is the kinetic energy of the system, and is the speed of that point of the system to which the mass is reduced. In more general view the reduced mass is the coefficient of inertia in the expression for the kinetic energy of a system with stationary constraints, the position of which is determined by the generalized coordinates
where the dot means differentiation with respect to time, and are functions of generalized coordinates.
System of two particles.
The task of two bodies is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star.
The two-body problem can be represented as two independent one-body problems that involve a solution for the motion of one particle in an external potential. Since many single-body problems can be solved exactly, the corresponding two-body problem can also be solved. In contrast, the three-body problem (and, more generally, the n-body problem) cannot be solved except in special cases.
In the two-body problem, which arises, for example, in celestial mechanics or scattering theory, the reduced mass appears as a kind of effective mass when the two-body problem is reduced to two problems about one body. Consider two bodies: one with mass and the other with mass . In the equivalent one-body problem, one considers the motion of a body with a reduced mass equal to
where the force acting on this mass is given by the force acting between these two bodies. It can be seen that the reduced mass is equal to half the harmonic mean of the two masses.
Central field.
Having reduced the problem of the motion of two bodies to the problem of the motion of one body, we have come to the question of determining the motion of a particle in external field, in which its potential energy depends only on the distance to a certain fixed point; such a field is called central. Force
acting on the particle, in absolute value also depends only on and is directed at each point along the radius vector.
When moving in the central field, the moment of the system relative to the center of the field is conserved. For one particle it is
Kepler's laws.
Kepler's laws- three empirical ratios. Describe the idealized heliocentric orbit of the planet. Within the framework of classical mechanics, they are derived from the solution of the two-body problem by passing to the limit / → 0, where , are the masses of the planet and the Sun.
1. Every planet solar system revolves around an ellipse with the Sun at one of its foci.
2. Each planet moves in a plane passing through the center of the Sun, and for equal periods of time, the radius vector connecting the Sun and the planet describes equal areas.
3. The squares of the periods of revolution of the planets around the Sun are related as the cubes of the semi-major axes of the orbits of the planets. It is true not only for the planets, but also for their satellites.
6) Lagrange function. Lagrange's equations. Generalized impulses, energy. Cyclic coordinates. Hamilton function and Hamilton equations.
Lagrange function.
7) Harmonic vibrations. Amplitude. Frequency. Spring pendulum, mathematical pendulum, physical pendulum.
Harmonic vibrations.
Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:
Where X- the value of the changing quantity, t- time, other parameters - constant: A- oscillation amplitude, ω - cyclic frequency of oscillations, - full phase of oscillations, - initial phase of oscillations.
Generalized harmonic oscillation in differential form
(Any non-trivial solution of this differential equation is a harmonic oscillation with a cyclic frequency)
§ Free vibrations are made under the action of the internal forces of the system after the system has been brought out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there should be no energy dissipation in it (the latter would cause damping).
§ Forced vibrations performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).
Amplitude.
Amplitude - the maximum value of the displacement or change of a variable from the average value during oscillatory or wave motion. A non-negative scalar quantity, the dimension of which coincides with the dimension of the physical quantity being defined.
Otherwise: Amplitude - the module of the maximum deviation of the body from the equilibrium position. For example:
§ amplitude for mechanical vibration of a body (vibration), for waves on a string or spring - this is the distance and is written in units of length.
Frequency.
Frequency- a physical quantity, a characteristic of a periodic process, equal to the number of complete cycles of the process completed per unit of time. The standard notation in formulas is , , or . The SI unit of frequency is general case is Hz. The reciprocal of the frequency is called the period.
Periodic processes are known in nature with frequencies ranging from ~10 −16 Hz (the frequency of revolution of the Sun around the center of the Galaxy) to ~1035 Hz (the frequency of field oscillations characteristic of the most high-energy cosmic rays).
Spring pendulum.
A spring pendulum is a mechanical system consisting of a spring with a coefficient of elasticity (stiffness) k (Hooke's law), one end of which is rigidly fixed, and at the other there is a load of mass m.
When an elastic force acts on a massive body, returning it to the equilibrium position, it oscillates around this position. Such a body is called a spring pendulum. The vibrations are caused by an external force. Oscillations that continue after the external force has ceased to act are called free oscillations. Oscillations caused by the action of an external force are called forced. In this case, the force itself is called compelling.
In the simplest case, a spring pendulum is a rigid body moving along a horizontal plane, attached to a wall by a spring.
Mathematical pendulum.
Mathematical pendulum- an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small natural oscillations of a mathematical pendulum of length L motionless suspended in a uniform gravitational field with free fall acceleration g equals
and does not depend on the amplitude and mass of the pendulum.
A flat mathematical pendulum with a rod is a system with one degree of freedom. If the rod is replaced by an tensile thread, then this is a system with two degrees of freedom with a connection. An example of a school problem in which the transition from one to two degrees of freedom is important.
For small oscillations, a physical pendulum oscillates in the same way as a mathematical pendulum with a reduced length.
physical pendulum.
Physical pendulum - an oscillator, which is solid, which oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.
8) Vibrations with friction. dissipative function.
In real systems, energy dissipation always occurs. If energy losses are not compensated by external devices, then the oscillations will dampen over time and eventually stop altogether. Consider the oscillations of a spring pendulum in a viscous medium.
For a body moving in a homogeneous viscous medium, the friction force depends only on the speed. At low speeds, we can assume that the friction force
, where beta is a positive constant coefficient.
To energy
Conclusions.
· The nature of natural oscillations in the presence of a friction force is determined by the ratio between and . At – aperiodic mode (3); – fluctuations are described by a periodic law with amplitude exponentially decreasing with time (4); – critical damping mode (5).
The quality factor of the oscillating system is very important parameter characterizing dissipative processes in the system.
dissipative function(scattering function) - a function introduced to take into account the transition of the energy of ordered motion into the energy of disordered motion, ultimately into heat, for example, to take into account the influence of viscous friction forces on the motion of a mechanical system. The dissipative function characterizes the degree of decrease in the mechanical energy of this system. The dissipative function divided by the absolute temperature determines the rate at which the entropy in the system increases (the so-called entropy production). The dissipative function has the dimension of power.
9) Forced vibrations without friction. beats. Resonance.
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The energy characteristics of motion are introduced on the basis of the concept of mechanical work or the work of a force.
If a force acts on a body and the body moves under the action of this force, then the force is said to do work.
Mechanical work -this is a scalar value equal to the product of the module of the force acting on the body, the module of displacement and the cosine of the angle between the force vector and the displacement (or velocity) vector.
Work is a scalar quantity. It can be both positive (0° ≤ α< 90°), так и отрицательна (90° < α ≤ 180°). При α = 90° работа, совершаемая силой, равна нулю.
In the SI system, work is measured in joules (J). A joule is equal to the work done by a force of 1 N in a displacement of 1 m in the direction of the force.
The work done by a force per unit of time is called power.
Power N – physical quantity, equal to the ratio work A to the time interval t during which this work is done:
N=A/t
In the International System (SI), the unit of power is called watt (W). A watt is equal to the power of a force that does 1 J of work in 1 second.
Off-system unit of power 1 hp=735 W
Relationship between power and speed in uniform motion:
N=A/t since A=FScosα then N=(FScosα)/t but S/t = v hence
N=Fvcosα
Units of work and power are used in engineering:
1 W s = 1 J; 1Wh \u003d 3.6 10 3 J; 1 kWh \u003d 3.6 10 6 J
If a body is capable of doing work, then it is said to have energy.
mechanical energy body -it is a scalar value equal to the maximum work that can be done under given conditions.
Denoted E SI unit of energy
Mechanical work is a measure of the change in energy in various processes.A =ΔE.
There are two types of mechanical energy - kinetic Ek And potential Ep energy.
The total mechanical energy of a body is equal to the sum of its kinetic and potential energies
E = Ek + Ep
Kinetic energy - is the energy of a body due to its motion.
A physical quantity equal to half the product of the body's mass and the square of its speed is called kinetic energybody:
Kinetic energy is the energy of motion. Kinetic energy of a body of mass m moving at a speed is equal to the work that must be done by the force applied to a body at rest to tell it this speed:
If the body is moving at a speed , then to stop it completely, work must be done
Along with kinetic energy or energy of motion in physics important role plays concept potential energy or interaction energies of bodies.
Potential energy – the energy of a body due to the mutual arrangement of interacting bodies or parts of one body.
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. Such forces are called conservative. The work of conservative forces on a closed trajectory is zero.
have the property of conservatism gravity And elastic force. For these forces, we can introduce the concept of potential energy.
Ppotential energy bodies in the field of gravity(potential energy of a body raised above the ground):
Ep = mgh
It is equal to the work done by gravity when the body is lowered to the zero level.
The concept of potential energy can also be introduced for elastic force. This force also has the property of being conservative. By stretching (or compressing) a spring, we can do this in a variety of ways.
You can simply lengthen the spring by x, or first lengthen it by 2x and then reduce the extension to x, and so on. In all these cases, the elastic force does the same work, which depends only on the length of the spring x in the final state if the spring was initially undeformed. This work is equal to the work of the external force A, taken with the opposite sign:
where k is the stiffness of the spring.
A stretched (or compressed) spring is capable of setting in motion a body attached to it, that is, imparting kinetic energy to this body. Therefore, such a spring has a reserve of energy. The potential energy of a spring (or any elastically deformed body) is the quantity
Potential energy of an elastically deformed body is equal to the work of the elastic force during the transition from given state to a zero strain state.
If in the initial state the spring was already deformed, and its elongation was equal to x1, then upon transition to a new state with elongation x2, the elastic force will do work equal to the change in potential energy, taken with the opposite sign:
Potential energy at elastic deformation is the interaction energy separate parts bodies between themselves by elastic forces.
If the bodies that make up closed mechanical system, interact with each other only by the forces of gravity and elasticity, then the work of these forces is equal to the change in the potential energy of the bodies, taken with the opposite sign:
A = -(Ep2 - Ep1).
According to the kinetic energy theorem, this work is equal to the change in the kinetic energy of bodies:
Hence Ek2 – Ek1 = –(Ep2 – Ep1) or Ek1 + Ep1 = Ek2 + Ep2.
The sum of the kinetic and potential energy of the bodies that make up a closed system and interact with each other by gravitational and elastic forces remains unchanged.
This statement expresses law of energy conservation in mechanical processes. It is a consequence of Newton's laws.
The sum E = Ek + Ep is called full mechanical energy.
The total mechanical energy of a closed system of bodies interacting with each other only by conservative forces does not change with any movements of these bodies. There are only mutual transformations of the potential energy of bodies into their kinetic energy, and vice versa, or the transfer of energy from one body to another.
E = Ek + Ep = const
The law of conservation of mechanical energy is fulfilled only when bodies in a closed system interact with each other by conservative forces, that is, forces for which the concept of potential energy can be introduced.
In real conditions, almost always moving bodies, along with gravitational forces, elastic forces and other conservative forces, are affected by friction forces or resistance forces of the medium.
The friction force is not conservative. The work of the friction force depends on the length of the path.
If friction forces act between the bodies that make up a closed system, then mechanical energy is not conserved. Part of the mechanical energy is converted into internal energy bodies (heating).
This video tutorial is intended for self-acquaintance with the topic "The Law of Conservation of Mechanical Energy". Let us first define the total energy and the closed system. Then we formulate the Law of conservation of mechanical energy and consider in which areas of physics it can be applied. We will also define work and learn how to define it by looking at the formulas associated with it.
Topic: Mechanical oscillations and waves. Sound
Lesson 32
Yeryutkin Evgeny Sergeevich
The topic of the lesson is one of the fundamental laws of nature -.
We talked earlier about potential and kinetic energy, and also about the fact that a body can have both potential and kinetic energy together. Before talking about the law of conservation of mechanical energy, let's remember what total energy is. Full of energy called the sum of the potential and kinetic energies of the body. Let's remember what is called a closed system. This is a system in which there is a strictly defined number of bodies interacting with each other, but no other bodies from the outside act on this system.
When we have decided on the concept of total energy and a closed system, we can talk about the law of conservation of mechanical energy. So, the total mechanical energy in a closed system of bodies interacting with each other through gravitational forces or elastic forces remains unchanged during any movement of these bodies.
It is convenient to consider the conservation of energy using the free fall of a body from a certain height as an example. If a body is at rest at a certain height relative to the Earth, then this body has potential energy. As soon as the body begins its movement, the height of the body decreases, and the potential energy also decreases. At the same time, the speed begins to increase, kinetic energy appears. When the body approached the Earth, the height of the body is 0, the potential energy is also 0, and the maximum will be the kinetic energy of the body. This is where the transformation of potential energy into kinetic energy is seen. The same can be said about the movement of the body in reverse, from the bottom up, when the body is thrown vertically upwards.
Of course, it should be noted that given example we considered taking into account the absence of friction forces, which in reality act in any system. Let's turn to the formulas and see how the law of conservation of mechanical energy is written:.
Imagine that a body in some frame of reference has kinetic energy and potential energy. If the system is closed, then with any change, a redistribution occurs, the transformation of one type of energy into another, but the total energy remains the same in its value. Imagine a situation where a car is moving along a horizontal road. The driver turns off the engine and continues driving with the engine turned off. What happens in this case? IN this case the car has kinetic energy. But you know perfectly well that over time the car will stop. Where did the energy go in this case? After all, the potential energy of the body in this case also did not change, it was some kind of constant relative to the Earth. How did the energy change happen? In this case, the energy went to overcome the forces of friction. If friction occurs in the system, then it also affects the energy of this system. Let's see how the energy change is recorded in this case.
Energy changes, and this change in energy is determined by the work against the friction force. We can determine the work using the formula, which is known from grade 7: A \u003d F. * S.
So, when we talk about energy and work, we must understand that every time we must take into account the fact that part of the energy is spent on overcoming the forces of friction. Work is being done to overcome the forces of friction.
In conclusion of the lesson, I would like to say that work and energy are inherently related quantities through acting forces.
Additional task 1 "On the fall of a body from a certain height"
Task 1
The body is at a height of 5 m from the ground and begins to fall freely. Determine the speed of the body at the moment of contact with the ground.
Given: Solution:
H \u003d 5 m 1. EP \u003d m * g *. H
V0 = 0 ; m*g*H=
_______ V2 = 2gH
VK - ? Answer:
Consider the law of conservation of energy.
Rice. 1. Body movement (task 1)
At the top point, the body has only potential energy: EP \u003d m * g * H. When the body approaches the ground, the height of the body above the ground will be equal to 0, which means that the potential energy of the body has disappeared, it has turned into kinetic energy.
According to the law of conservation of energy, we can write: m*g*H=. Body weight is reduced. Transforming the above equation, we get: V2 = 2gH.
The final answer will be: 
. Plugging in the whole value, we get: .
Additional task 2
A body falls freely from a height H. Determine at what height the kinetic energy is equal to a third of the potential.
Given: Solution:
H EP \u003d m. g. H; ; 
M.g.h = m.g.h + m.g.h
h-? Answer: h = H.
Rice. 2. To problem 2
When a body is at height H, it has potential energy, and only potential energy. This energy is determined by the formula: EP \u003d m * g * H. This will be the total energy of the body.
When the body begins to move down, the potential energy decreases, but at the same time, the kinetic energy increases. At the height to be determined, the body will already have some speed V. For the point corresponding to the height h, the kinetic energy has the form: . The potential energy at this height will be denoted as follows: .
According to the law of conservation of energy, our total energy is conserved. This energy EP \u003d m * g * H remains constant. For point h we can write the following relation: (according to Z.S.E.).
Recalling that the kinetic energy according to the condition of the problem is , we can write the following: m.g.Н = m.g.h + m.g.h.
Please note that the mass is reduced, the acceleration of gravity is reduced, after simple transformations, we get that the height at which this relationship is satisfied is h = H.
Answer: h= 0.75H
Additional task 3
Two bodies - a bar of mass m1 and a plasticine ball of mass m2 - move towards each other with the same speeds. After the collision, the plasticine ball stuck to the bar, the two bodies continue to move together. Determine how much energy has been converted into internal energy of these bodies, taking into account the fact that the mass of the bar is 3 times the mass of the plasticine ball.
Given: Solution:
m1 = 3. m2 m1.V1- m2.V2= (m1+m2).U; 3.m2V- m2.V= 4 m2.U2.V=4.U; .
This means that the speed of the bar and plasticine ball together will be 2 times less than the speed before the collision.
The next step is this.
.
In this case, the total energy is the sum of the kinetic energies of the two bodies. Bodies that haven't touched yet haven't hit. What happened after the collision? Look at the following entry: 
.
On the left side we leave full energy, and on the right side we must write kinetic energy bodies after interaction and take into account that part of the mechanical energy has turned into heat Q.
Thus, we have: 
. As a result, we get the answer .
Please note: as a result of this interaction, most of the energy is converted into heat, i.e. goes into internal energy.
List of additional literature:
Are you familiar with conservation laws? // Quantum. - 1987. - No. 5. - S. 32-33.
Gorodetsky E.E. Law of conservation of energy // Kvant. - 1988. - No. 5. - S. 45-47.
Soloveichik I.A. Physics. Mechanics. Handbook for entrants and high school students. - St. Petersburg: Agency IGREK, 1995. - S. 119-145.
Physics: Mechanics. Grade 10: Proc. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakishev. - M.: Bustard, 2002. - C. 309-347.
The rocket and the gases ejected by its engine interact with each other. Based on the momentum conservation law, in the absence of external forces, the sum of the momentum vectors of the interacting bodies remains constant. Before the start of the engines, the momentum of the rocket and fuel was equal to zero; therefore, even after the engines are turned on, the sum of the vectors of the rocket's momentum and the momentum of the outflowing gases is equal to zero: , (17.1) where is the mass of the rocket; - rocket speed; - mass of emitted gases; - the speed of the outflow of gases. From here we obtain , (17.2) and for the module of rocket speed we have . (17.3) This formula can be used to calculate the module of the rocket's velocity under the condition of a small change in the mass of the rocket as a result of the operation of its engines. 4) Reactive force. The movement of most modern aircraft is jet, because. occurs as a result of the expiration of gases heated in the engine with great speed. In this case, the aircraft moves in the direction opposite to the velocity of the outflow of gases. Rockets move in the same way, ejecting fuel combustion products from the nozzle. An example of jet propulsion is the recoil of a gun barrel when fired. The force acting on a body during reactive motion is called jet force. Ticket number 12- Non-inertial frames of reference In non-inertial frames, Newton's laws, generally speaking, are no longer valid. However, the laws of dynamics can also be applied to them, if, in addition to the forces due to the action of bodies on each other, we introduce into consideration forces of a special kind - the so-called inertia forces. If we take into account the forces of inertia, then Newton's second law will be valid for any frame of reference: the product of a body's mass and acceleration in the frame under consideration is equal to the sum of all forces acting on a given body (including the forces of inertia). Forces of inertia Fin at the same time, they must be such that, together with the forces F, due to the influence of bodies on each other, they imparted to the body acceleration a "as it has in non-inertial frames of reference, i.e. Since F \u003d ma (a is the acceleration of the body in an inertial frame of reference), then the forces of inertia The forces of inertia are forces due to accelerated motion of a non-inertial frame of reference (NFR) relative to an inertial frame of reference (IRF).The basic law of dynamics for non-inertial frames of reference: 
, where is the force acting on the body from other bodies; - the force of inertia acting on the body relative to the progressively moving NSO. - acceleration of NSO relative to ISO. It appears, for example, in an airplane during acceleration on the runway; - centrifugal force of inertia acting on the body relative to the rotating NSO. - the angular velocity of the NSO relative to the ISO, - the distance from the body to the center of rotation; 
- Coriolis force of inertia acting on a body moving at a speed relative to the rotating NSO. - angular velocity of the NSO relative to the IFR (the vector is directed along the axis of rotation in accordance with the rule of the right screw). The forces of inertia are directed in the direction opposite to the acceleration. The forces of inertia arise only in a reference frame moving with acceleration, i.e. these are apparent forces. Centrifugal force of inertia Let us consider a rotating disk with racks fixed on it with balls suspended on threads (Fig. 2). When the disk rotates at a constant angular velocity , the balls deviate by a certain angle, the greater, the farther it is from the axis of rotation. Relative to the inertial frame of reference (fixed), all balls move along a circle of the corresponding radius
