Modern physics knows many types of energy associated with the movement or various relative positions of a wide variety of material bodies or particles, for example, every moving body has kinetic energy proportional to the square of its speed. This energy can change if the speed of the body increases or decreases. A body raised above the ground has gravitational potential energy that varies with changes in the height of the body.
Fixed electric charges, located at some distance from each other, have potential electrostatic energy in accordance with the fact that, according to Coulomb’s law, charges either attract (if they different sign), or repel with a force inversely proportional to the square of the distance between them.
Molecules, atoms, and particles, their components - electrons, protons, neutrons, etc., have kinetic and potential energy. Depending on the nature of the movement and the nature of the forces acting between these particles, a change in energy in systems of such particles can manifest itself in the form of mechanical work, in the flow of electric current, in the transfer of heat, in changes in the internal state of bodies, in the propagation of electromagnetic oscillations, etc.
More than 100 years ago, a fundamental law was established in physics, according to which energy cannot disappear or appear from nothing. It can only change from one type to another. This law is called law of conservation of energy.
In the works of A. Einstein, this law received significant development. Einstein established the interconvertibility of energy and mass and thereby expanded the interpretation of the law of conservation of energy, which is now generally formulated as law of conservation of energy and mass.
In accordance with Einstein's theory, any change in the energy of a body d E is associated with a change in its mass d m by the formula d E = d mс 2, where c is the speed of light in vacuum, equal to 3 x 10 8 m/s.
From this formula, in particular, it follows that if, as a result of any process, the mass of all bodies participating in the process decreases by 1 g, then energy equal to 9x10 13 J will be released, which is equivalent to 3000 tons of standard fuel.
These relationships are of paramount importance in the analysis of nuclear transformations. In most macroscopic processes, the change in mass can be neglected and we can only talk about the law of conservation of energy.
Let's follow the energy transformations using some particular example. Let's consider the entire chain of energy transformations necessary to manufacture any part on a lathe (Fig. 1). Let the initial energy 1, the amount of which we take as 100%, is obtained through the complete combustion of a certain amount of natural fuel. Consequently, for our example, 100% of the initial energy is contained in the products of fuel combustion, which are at a high (about 2000 K) temperature.
The combustion products in a power plant boiler, when cooled, give up their internal energy in the form of heat to water and water vapor. However, for technical and economic reasons, combustion products cannot be cooled to a temperature environment. They are ejected through the pipe into the atmosphere at a temperature of about 400 K, taking with them part of the original energy. Therefore, only 95% of the initial energy will be converted into the internal energy of water vapor.
The resulting water vapor will enter the steam turbine, where its internal energy will first be partially converted into the kinetic energy of the steam strings, which will then be given in the form of mechanical energy to the turbine rotor.
Only part of the steam energy can be converted into mechanical energy. The rest is given to cooling water when steam condenses in the condenser. In our example, we assumed that the energy transferred to the turbine rotor would be about 38%, which roughly corresponds to the situation in modern power plants.
When converting mechanical energy into electrical energy, due to the so-called Joule losses in the rotor and stator windings of the electric generator, about 2% more energy will be lost. As a result, about 36% of the initial energy will flow into the electrical network.
The electric motor will convert only part of the electricity supplied to it into mechanical rotational energy of the lathe. In our example, about 9% of the energy in the form of Joule heat in the motor windings and frictional heat in its bearings will be released into the surrounding atmosphere.
Thus, only 27% of the initial energy will be supplied to the working parts of the machine. But the misadventures of energy do not end there. It turns out that the overwhelming majority of energy during machining of a part is spent on friction and is removed in the form of heat with the liquid that cools the part. Theoretically, in order to obtain the required part from the initial workpiece, only a very small fraction (in our example, 2%) of the initial energy would be enough.
Rice. 1. Scheme of energy transformations when processing a part on a lathe: 1 - energy loss with exhaust gases, 2 - internal energy of combustion products, 3 - internal energy of the working fluid - water steam, 4 - heat given off to the cooling water in the turbine condenser, 5 - mechanical energy of the turbogenerator rotor, 6 - losses in the electric generator, 7 - losses in the electric drive of the machine, 8 - mechanical energy of rotation of the machine, 9 - friction work, converted into heat given off to the liquid cooling the part, 10 - increase in the internal energy of the part and chips after processing .
From the example considered, if it is considered fairly typical, at least three very useful conclusions can be drawn.
Firstly, at each step of energy conversion, some part of it is lost. This statement should not be understood as a violation of the law of conservation of energy. It is lost for the beneficial effect for which the corresponding transformation is carried out. The total amount of energy after conversion remains unchanged.
If a process of transformation and transmission of energy is carried out in a certain machine or apparatus, then the efficiency of this device is usually characterized by efficiency factor (efficiency). The diagram of such a device is shown in Fig. 2.
Rice. 2. Scheme for determining the efficiency of a device that converts energy.
Using the notation shown in the figure, efficiency can be determined as efficiency = Epol / Epod
It is clear that in this case, based on the law of conservation of energy, there should be Epod = Epol + Epot
Therefore, the efficiency can also be written as follows: efficiency = 1 - (Epot/Epol)
Returning to the example shown in Fig. 1, we can say that the efficiency of the boiler is 95%, the efficiency of converting the internal energy of steam into mechanical work is 40%, the efficiency of the electric generator is 95%, the efficiency of The electric drive of the machine is 75% and the efficiency of the actual part processing process is about 7%.
In the past, when the laws of energy transformation were not yet known, people's dream was to create the so-called perpetual motion machine- a device that would perform useful work without expending any energy. Such a hypothetical engine, the existence of which would violate the law of conservation of energy, is today called a perpetual motion machine of the first kind, in contrast to a perpetual motion machine of the second kind. Today, of course, no one takes seriously the possibility of creating a perpetual motion machine of the first kind.
Secondly, all energy losses ultimately turn into heat, which is given off either atmospheric air, or water from natural reservoirs.
Third, Ultimately, people usefully use only a small part of the primary energy that was expended to obtain the corresponding beneficial effect.
This is especially obvious when considering energy costs for transport. In idealized mechanics, which does not take into account friction forces, moving loads in a horizontal plane does not require energy expenditure.
In real conditions, all the energy consumed by a vehicle is spent on overcoming the forces of friction and air resistance, i.e., ultimately, all the energy consumed in transport turns into heat. In this regard, the following figures are interesting, characterizing the work of moving 1 ton of cargo over a distance of 1 km various types transport: airplane - 7.6 kWh/(t-km), car - 0.51 kWh/(t-km), train - 0.12 kWh/(t-km).
Thus, the same beneficial effect can be achieved with air transport due to 60 times greater energy consumption than with railway transport. Of course, a large expenditure of energy results in significant savings in time, but even at the same speed (car and train), the energy expenditure differs by a factor of 4.
This example suggests that people often sacrifice energy efficiency in order to achieve other goals, for example, comfort, speed, etc. As a rule, the energy efficiency of a particular process itself is of little interest to us - summary technical and economic assessments of the efficiency of processes are important . But as primary energy sources become more expensive, the energy component in technical and economic assessments becomes increasingly important.
Universal law of nature. Therefore, it also applies to electrical phenomena. Let's consider two cases of energy transformation in an electric field:
- The conductors are insulated ($q=const$).
- The conductors are connected to current sources and their potentials do not change ($U=const$).
Law of conservation of energy in circuits with constant potentials
Let us assume that there is a system of bodies that can include both conductors and dielectrics. The bodies of the system can perform small quasi-static movements. The temperature of the system is maintained constant ($\to \varepsilon =const$), that is, heat is supplied to the system or removed from it if necessary. The dielectrics included in the system will be considered isotropic, and their density will be assumed to be constant. In this case, the proportion of internal energy of bodies that is not associated with the electric field will not change. Let's consider options for energy transformations in such a system.
Any body that is in an electric field is affected by pondemotive forces (forces acting on charges inside bodies). With an infinitesimal displacement, the pondemotive forces will do the work $\delta A.\ $Since the bodies move, the change in energy is dW. Also, when the conductors move, their mutual capacitance changes, therefore, in order to keep the potential of the conductors unchanged, it is necessary to change the charge on them. This means that each of the torus sources does work equal to $\mathcal E dq=\mathcal E Idt$, where $\mathcal E$ is the emf of the current source, $I$ is the current strength, $dt$ is the movement time. In our system there will be electric currents, and heat will be released in each part of it:
According to the law of conservation of charge, the work of all current sources is equal to the mechanical work of the electric field forces plus the change in electric field energy and Joule-Lenz heat (1):
If the conductors and dielectrics in the system are stationary, then $\delta A=dW=0.$ From (2) it follows that all the work of the current sources turns into heat.
Law of conservation of energy in circuits with constant charges
In the case of $q=const$, the current sources will not enter the system under consideration, then the left side of expression (2) will become equal to zero. In addition, the Joule-Lenz heat arising due to the redistribution of charges in bodies during their movement is usually considered insignificant. In this case, the law of conservation of energy will have the form:
Formula (3) shows that mechanical work electric field strength is equal to the decrease in electric field energy.
Application of the law of conservation of energy
Using the law of conservation of energy in large quantities In cases, it is possible to calculate the mechanical forces that act in an electric field, and this is sometimes much easier to do than if we consider the direct action of the field on individual parts of the bodies of the system. In this case, they act according to the following scheme. Let's say we need to find the force $\overrightarrow(F)$ that acts on a body in a field. It is assumed that the body is moving (small movement of the body $\overrightarrow(dr)$). The work done by the required force is equal to:
Example 1
Task: Calculate the force of attraction that acts between the plates of a flat capacitor, which is placed in a homogeneous isotropic liquid dielectric with a dielectric constant of $\varepsilon$. Area of the plates S. Field strength in the capacitor E. The plates are disconnected from the source. Compare the forces that act on the plates in the presence of a dielectric and in a vacuum.
Since the force can only be perpendicular to the plates, we choose the displacement along the normal to the surface of the plates. Let us denote by dx the movement of the plates, then the mechanical work will be equal to:
\[\delta A=Fdx\ \left(1.1\right).\]
The change in field energy will be:
Following the equation:
\[\delta A+dW=0\left(1.4\right)\]
If there is a vacuum between the plates, then the force is equal to:
When a capacitor, which is disconnected from the source, is filled with a dielectric, the field strength inside the dielectric decreases by $\varepsilon $ times, therefore, the force of attraction of the plates decreases by the same factor. The decrease in interaction forces between the plates is explained by the presence of electrostriction forces in liquid and gaseous dielectrics, which push the capacitor plates apart.
Answer: $F=\frac(\varepsilon (\varepsilon )_0E^2)(2)S,\ F"=\frac(\varepsilon_0E^2)(2)S.$
Example 2
Task: A flat capacitor is partially immersed in a liquid dielectric (Fig. 1). As the capacitor charges, liquid is drawn into the capacitor. Calculate the force f with which the field acts on a unit horizontal surface of the liquid. Assume that the plates are connected to a voltage source (U=const).
Let us denote by h the height of the liquid column, dh the change (increase) in the liquid column. The work done by the required force will be equal to:
where S is the horizontal cross-sectional area of the capacitor. The change in electric field is:
An additional charge dq will be transferred to the plates, equal to:
where $a$ is the width of the plates, take into account that $E=\frac(U)(d)$ then the work of the current source is equal to:
\[\mathcal E dq=Udq=U\left(\varepsilon (\varepsilon )_0E-(\varepsilon )_0E\right)adh=E\left(\varepsilon (\varepsilon )_0E-(\varepsilon )_0E\right )d\cdot a\cdot dh=\left(\varepsilon (\varepsilon )_0E^2-(\varepsilon )_0E^2\right)Sdh\left(2.4\right).\]
If we assume that the resistance of the wires is small, then $\mathcal E $=U. We use the law of conservation of energy for systems with direct current, provided that the potential difference is constant:
\[\sum(\mathcal E Idt=\delta A+dW+\sum(RI^2dt\ \left(2.5\right).))\]
\[\left(\varepsilon (\varepsilon )_0E^2-(\varepsilon )_0E^2\right)Sdh=Sfdh+\left(\frac(\varepsilon (\varepsilon )_0E^2)(2)-\frac ((\varepsilon )_0E^2)(2)\right)Sdh\to f=\frac(\varepsilon (\varepsilon )_0E^2)(2)-\frac((\varepsilon )_0E^2)(2 )\ .\]
Answer: $f=\frac(\varepsilon (\varepsilon )_0E^2)(2)-\frac((\varepsilon )_0E^2)(2).$
Law of conservation of energy in capacitor circuits Problem 1 A Q 0 W A kmech ist Option 1 With switch K2 open, switch K1 is closed and opened after the end of transient processes. After this, key K2 is closed. Solution. According to the law of conservation of energy, the change in energy in a capacitor is determined by the relation mechA the work of mechanical forces is equal to zero, since there are no movements inside the capacitors. istA the work of the current source is zero, since when key K2 is closed, key K1 is open, the current source is turned off. Q is the amount of heat that is released when charges move. W W kn The initial and final energies of the capacitors correspond, respectively, to the open and closed switch K2. For the initial state (the capacitors are charged from the current source): Q Q W W kk 0 kN kk For the final state (only capacitor C2 and capacitor C3 parallel to it remain in the circuit). The charges of the capacitors are conserved, since the circuit is open. q 23 2 Ec W кк 2 q 23 2 C 23 2 2 E c 4 2 (c 2) c 2 3 2 E c Substitute the energies of the capacitors into the relation for Q and get the answer. 2 Q E c Option 2. 2 3 2 E c 1 3 2 E c 2 c C o q o W kn 2) c 2 c Ec 2 1 () C C C 6 (c c 3 c C C C c 6 3 2 1 q q q 2 E C 1 3 2 C U 2 s E o 2 2 cE 2 2 o 2 o kn ist Q A kk ist kkkn When open key K2, key K1 is closed and after the end of transient processes, key K2 is closed. Solution. In this case, key K2 is closed under voltage, the current source remains connected constantly, participates in recharging the capacitors, and therefore does work. The law of conservation of energy in this case takes the form: - W W Q W W A The initial state of the circuit is the same as in option 1, therefore the initial charges and energy of the capacitors correspond to the calculated ones. In the final state, after closing the switch K2, the remaining parallel-connected capacitors C2 and C3 will be charged (recharged) from the current source. C q ok c C C 3 2 ok 3 Ec E C ok 2 2 C E 3 E c ok 2 2 Work of the current source: E q E q A (source ok We substitute the energies of the capacitors into the relation for Q and get the answer. E (3 Ec 2 Ec) q on) 2 E c 2 c 3 c W kk 2 Q E c 2 2 E c E c 2 E c 3 2 1 3 The same answer in the first and second options is not a pattern, but an accident. Problem 2 In the initial state for the circuit in Fig. 2 C1=2C, C2=3C, emf. current source is equal to E. In a flat air capacitor C1, with the help of an external force, the plates were very quickly moved apart, increasing the distance between the plates by 2 times. How much heat will be released in the circuit during the subsequent transient process? Solution. When the plate moves rapidly against the Coulomb force, the charge of the plates is maintained, the Coulomb force does negative work, and the external force does positive work. The second plate moves in the field of the first plate, the electrical capacity of the first capacitor decreases by 2 times. A fur F k dE q 1 2 q d q n 1 2 S 0 2 n d 2 q d 1 n 2 S 0 2 q 1 n 2 C n For the initial state ( before the start of movement) : C 0 n 1 n C C 2 C C 2 1 n q 0 n q 1 n q 2 n 2 3 c c 3 2 c c Ec 6 5 6 5 c A fur 2 2 36 E c 25 2 0.72 2 E c W kn 2 6 sE 5 2 0.6 2 E c Since the electrical capacity C1 decreased quickly, then during the subsequent transient process the voltage on it should increase , therefore, in order for the sum of the voltages on C1 and C2 to remain equal to E, the charge will go into the current source, which means that the current source will do negative work. For the final state: 3 c c 3 c c C C 2 C C 1 3 4 C 0 c 1 k 2 k k k n 0 2 2 () E q 0 W kk A source (E q 0 3 cE 2 4 C E k 2 3 4 3 8 Law of conservation of energy W W Q Q W W AA Problem 3 kkкн fur kkкн ist fur ist AA cE Ec 6 5 Ec) 9 20 2 E c 0.45 2 E c 2 0.375 cE 2 (0.375 0.6 0.72 0. 45) E c 2 0.495 E c 2 In the initial state for the circuit Fig. 3 C1= C2=C, e.m.f. current source is equal to E. In a flat air capacitor C1, with the help of an external force, the plates were very quickly moved, reducing the distance between the plates by 2 times. How much heat will be released in the circuit during the subsequent transient process? Solution. For the initial state: s 2 CC on s 2 qE C he sE W he 2 kn 2 S 1 n 2 sE sE 2 2 When the capacitor plates move quickly, all charges are conserved, and the electrical capacitance of the first capacitor increases by 2 times. At the same time, for a constant potential difference on the current source, a larger charge is required, therefore, in the subsequent transient process, the charge will flow from the current source, and the current source will do positive work. 2 c sE) qсE c ok 3 c 2 3 C ok сЭ 2 C c 1 к 2 (3 AЕ сЭ ist 2 3 сЭ 2 W кк A mech 2 q 1 n W WA kk kn Problem 4 A ist sE mech 2 nd 1 2 cE 1.5 2 sE 2 0.25 cE 2 0.25 cE 2 1 01 02 0 Solution. This problem has non-zero initial conditions and its peculiarity is that when switch K is closed, the total charge of the right plate of the capacitor C1 and the left plate of the capacitor C2 is not equal to zero: for consonant switching of the capacitors q U C U C 0 2 (polarity as in Figure 4). This charge will be conserved (according to the law of conservation of electric charge) during any subsequent transient processes. Since the circuit is connected to a current source, when switch K is closed, the charges of the capacitors (right plates) will change and will be equal after the transient process q1 and q2, and the voltages U1 and U2. These charges and voltages must correspond to the law of conservation of charge and the voltage ratio for series consonant connection. We obtain a system of two equations. If capacitor C2 were connected in opposite directions (in polarity), then the signs of both q2 and U2 would change to the opposite. 1 U U q q 2 1 2 E q 0 q q 1 2 C C 1 q q 1 2 2 E q 0 q C 1 2 (q 1 q C EC C 0 2) 1 1 Find the charges of the capacitors. q 1 q 2 EC C q C 1 0 EC C U C U C C 2 02 1 2 EC C q C 2 0 EC C U C U C C 2 01 2 1 1 2 C C 2 1 2 C C 2 1 1 1 2 01 2 1 C C 2 2 01 C C q p , that is, 0 1 1 2 1 q p or 0 From the relations it is clear that situations are possible when capacitors, as a result of a transient process, can be recharged to opposite polarities. Work of the current source (for the positive pole): isTAE q 2 1 2 q 2 q 2 q 02 It can be shown that EC C U C U C C 1 01 1 2 2 02 2 C C 1 q q 2 1 2 2 U C 2 02 EC C U C C U C C 1 01 1 2 02 2 C C 1 1 2 2 Capacitor energy for the initial state: W W W n 1 n n 2 2 01 C U 1 2 2 02 C U 2 2 For the final state: W k 2 q 2 2 C 2 2 q 1 2 C 1 2 C U about 2 about It should be noted that W k , since under non-zero initial conditions the total charge is unequal to the charges of series-connected capacitors . Let us determine the value of the released heat at the following numerical values: C1=c, C2=3c, E= 8 V, U01 =4 V, U02 =2 V. q 0 q 1 q 4 8 2 3 2 c c c 3 2 c 11 c c c 3 c 2 c 4 s 3 3 s c 4 s 14 2 s 3 s q 2 8 s 8 s 3 s 4 s c 2 3 s 15 2 s 3 2 s Wсн W k 2 s 16 2 11 (2 8 1.5 s c) 3 4 s 2 2 s 12 s A source Q W W As source Problem 5. 15 s (2 2) 2 3 s 121 s 8 75 s 8 24.5 s 14 s 24.5 s 12 s 1.5 2 1 E U U , therefore charges will not flow either from the source or to the source Solution. 1. Heat is released only when charge redistribution occurs, i.e. current flows. When the key is opened, this can only happen from a current source. The potential difference between points A and B does not change since ABU (charges can flow if the potential of the positive pole of the current source is unequal to the potential t.B, and the potential of the negative pole of the source is unequal to the potential t.A). This means that the charges of the capacitors will not change, the work of the current source is zero, so no heat will be released when the switch is opened. 2. The constancy of capacitor charges can be proven using the law of conservation of charge for the midpoint of the circuit. For the initial state: 2 q 1 n q 23 C he q he C C C) 1 3 C C C 1 3 ( 2 EC 1 C C C 3 1 2 2) (EC C C 3 1 C C C 1 3 EC C 3 C C C 3 1 1 2 2 q 23 (C C U U ) 23 23 2 3 q 3 n C U 3 23 Since when the key is opened it turns off the left plate of the capacitor C3 is from the middle point, then its negative charge q3n goes with it. Therefore, according to the law of conservation of charge for the middle point, we obtain: q 1 q 2 q 3 n 1 EC C 3 C C C 3 1 2 Solving this equation together with the equation for voltages in a series connection U U 1 2 E q q 2 1 C C 1 2 E , we can determine q1 and q2 the capacitor charges established after the transition process. We obtain q 1 ) EC C C 3 C C C 1 3 (1 2 2, the value of which is equal to q1н, which means that charge redistribution will not occur when the key is opened.
In all phenomena occurring in nature, energy neither appears nor disappears. It only transforms from one type to another, while its meaning remains the same.
Law of energy conservation- a fundamental law of nature, which consists in the fact that for an isolated physical system a scalar physical quantity, which is a function of the system parameters and is called energy, which is conserved over time. Since the law of conservation of energy does not apply 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 energy conservation
In electrodynamics, the law of conservation of energy is historically formulated in the form of Poynting's theorem.
The change in electromagnetic energy contained in a certain volume over a certain time interval is equal to the flow of electromagnetic energy through the surface limiting 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 $
An 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 area of space to another, possibly transforming into other forms. Law of energy conservation For 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— electromagnetic field energy:
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 equal to
N=Σ ij̅ i ×E̅ i . ΔV i .
This is the amount of field energy that transforms into other forms. From Maxwell's equations it follows that
ΔW + NΔt = -Δt∮ SS̅ × n̅. dA,
Where ΔW— change in the energy of the electromagnetic field in the volume under consideration over time Δt, a vector S̅ = E̅ × H̅ called Poynting vector.
This law of conservation of energy in electrodynamics.
Through a small area the size ΔA with unit normal vector n̅ per unit time in the direction of the vector n̅ energy flows S̅ × n̅.ΔA, Where S̅- meaning Poynting vector within the site. The sum of these quantities over all elements of a closed surface (indicated by the integral sign), standing on the right side of the equality, represents the energy flowing out of the volume bounded by the surface per unit time (if this quantity is negative, then the energy flows into the volume). Poynting vector determines the flow of electromagnetic field energy through the site; it is non-zero wherever the vector product of the electric and magnetic field strength vectors is non-zero.
Three main directions can be distinguished practical application electricity: transmission and transformation of information (radio, television, computers), transmission of impulse and angular 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 only leads to losses. Energy is not transmitted through wires! Current-carrying wires are needed to form electric and magnetic fields of such a configuration that the energy flow, determined by 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. ( Internal energy The sun is waning, carried away electromagnetic waves, mainly by light. Part of this energy supports life on Earth.)
Law of energy conservation
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 satisfied 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 we submit the forces as the sum of potential forces \(\mathbf(F)_i^p \) and non-potential forces \(\mathbf(F)_i^d \) , and write the potential forces in the form
\[ \mathbf(F)_i^p = - \nabla_i U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]
then, multiplying all 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 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 name this value mechanical energy, then by 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 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 non-potential forces.
The law of conservation of energy in mechanics is satisfied only for systems in which all forces are potential.
Javascript is disabled in your browser.To perform calculations, you must enable ActiveX controls!
The law of conservation of energy states that the energy of a body never disappears or appears again, it can only be transformed from one type to another. This law is universal. It has its own formulation in various branches of physics. Classical mechanics considers the law of conservation of mechanical energy.
Total mechanical energy closed system physical bodies between which conservative forces act is a constant value. This is how Newton's law of conservation of energy is formulated.
A closed, or isolated, physical system is considered to be one that is not affected by external forces. There is no exchange of energy with the surrounding space, and the own energy that it possesses remains unchanged, that is, it is conserved. In such a system, only internal forces act, and the bodies interact with each other. Only the transformation of potential energy into kinetic energy and vice versa can occur in it.
The simplest example of a closed system is a sniper rifle and a bullet.
Types of mechanical forces
The forces that act inside a mechanical system are usually divided into conservative and non-conservative.
Conservative forces are considered whose work does not depend on the trajectory of the body to which they are applied, but is determined only by the initial and final position this body. Conservative forces are also called potential. The work done by such forces along a closed loop is zero. Examples of conservative forces – gravity, elastic force.
All other forces are called non-conservative. These include friction force and resistance force. They are also called dissipative forces. These forces, during any movements in a closed mechanical system, perform negative work, and under their action, the total mechanical energy of the system decreases (dissipates). It transforms into other, non-mechanical forms of energy, for example, heat. Therefore, the law of conservation of energy in a closed mechanical system can be fulfilled only if there are no non-conservative forces in it.
The total energy of a mechanical system consists of kinetic and potential energy and is their sum. These types of energies can transform into each other.
Potential energy
Potential energy is called the energy of interaction of physical bodies or their parts with each other. It is determined by their relative position, that is, the distance between them, and is equal to the work that needs to be done to move the body from the reference point to another point in the field of action of conservative forces.
Any motionless physical body raised to some height has potential energy, since it is acted upon by gravity, which is a conservative force. Such energy is possessed by water at the edge of a waterfall, and a sled on a mountain top.
Where did this energy come from? While the physical body was raised to a height, work was done and energy was expended. It is this energy that is stored in the raised body. And now this energy is ready to do work.
The amount of potential energy of a body is determined by the height at which the body is located relative to some initial level. We can take any point we choose as a reference point.
If we consider the position of the body relative to the Earth, then the potential energy of the body on the Earth’s surface is zero. And on top h it is calculated by the formula:
E p = m ɡ h ,
Where m - body mass
ɡ - acceleration of gravity
h – height of the body’s center of mass relative to the Earth
ɡ = 9.8 m/s 2
When a body falls from a height h 1 up to height h 2 gravity does work. This work is equal to the change in potential energy and has a negative value, since the amount of potential energy decreases when the body falls.
A = - ( E p2 – E p1) = - ∆ E p ,
Where E p1 – potential energy of the body at height h 1 ,
E p2 - potential energy of the body at height h 2 .
If the body is raised to a certain height, then work is done against the forces of gravity. In this case it has a positive value. And the amount of potential energy of the body increases.
An elastically deformed body (compressed or stretched spring) also has potential energy. Its value depends on the stiffness of the spring and on the length to which it was compressed or stretched, and is determined by the formula:
E p = k·(∆x) 2 /2 ,
Where k – stiffness coefficient,
∆x – lengthening or compression of the body.
The potential energy of a spring can do work.
Kinetic energy
Translated from Greek, “kinema” means “movement.” The energy that a physical body receives as a result of its movement is called kinetic. Its value depends on the speed of movement.
A soccer ball rolling across a field, a sled rolling down a mountain and continuing to move, an arrow shot from a bow - all of them have kinetic energy.
If a body is at rest, its kinetic energy is zero. As soon as a force or several forces act on a body, it will begin to move. And since the body moves, the force acting on it does work. The work of force, under the influence of which a body from a state of rest goes into motion and changes its speed from zero to ν , called kinetic energy body mass m .
If at the initial moment of time the body was already in motion, and its speed mattered ν 1 , and at the final moment it was equal to ν 2 , then the work done by the force or forces acting on the body will be equal to the increase in the kinetic energy of the body.
∆ E k = E k 2 - Ek 1
If the direction of the force coincides with the direction of movement, then positive work is done and the kinetic energy of the body increases. And if the force is directed in the direction opposite to the direction of movement, then negative work is done, and the body gives off kinetic energy.
Law of conservation of mechanical energy
Ek 1 + E p1= E k 2 + E p2
Any physical body located at some height has potential energy. But when it falls, it begins to lose this energy. Where does she go? It turns out that it does not disappear anywhere, but turns into the kinetic energy of the same body.
Suppose , the load is fixedly fixed at a certain height. Its potential energy at this point is equal to its maximum value. If we let go of it, it will begin to fall at a certain speed. Consequently, it will begin to acquire kinetic energy. But at the same time its potential energy will begin to decrease. At the point of impact, the kinetic energy of the body will reach a maximum, and the potential energy will decrease to zero.
The potential energy of a ball thrown from a height decreases, but its kinetic energy increases. A sled at rest on a mountain top has potential energy. Their kinetic energy at this moment is zero. But when they begin to roll down, the kinetic energy will increase, and the potential energy will decrease by the same amount. And the sum of their values will remain unchanged. The potential energy of an apple hanging on a tree when it falls is converted into its kinetic energy.
These examples clearly confirm the law of conservation of energy, which says that the total energy of a mechanical system is a constant value . The total energy of the system does not change, but potential energy transforms into kinetic energy and vice versa.
By what amount the potential energy decreases, the kinetic energy increases by the same amount. Their amount will not change.
For a closed system of physical bodies the following equality is true:
E k1 + E p1 = E k2 + E p2,
Where E k1, E p1
- kinetic and potential energies of the system before any interaction, E k2 , E p2
- the corresponding energies after it.
The process of converting kinetic energy into potential energy and vice versa can be seen by watching a swinging pendulum.
Click on the picture
Being in the extreme right position, the pendulum seems to freeze. At this moment its height above the reference point is maximum. Therefore, the potential energy is also maximum. And the kinetic value is zero, since it is not moving. But the next moment the pendulum begins to move downwards. Its speed increases, and, therefore, its kinetic energy increases. But as the height decreases, so does the potential energy. At the lowest point it will become equal to zero, and the kinetic energy will reach its maximum value. The pendulum will fly past this point and begin to rise up to the left. Its potential energy will begin to increase, and its kinetic energy will decrease. Etc.
To demonstrate energy transformations, Isaac Newton came up with a mechanical system called Newton's cradle or Newton's balls .
Click on the picture
If you deflect to the side and then release the first ball, its energy and momentum will be transferred to the last through three intermediate balls, which will remain motionless. And the last ball will deflect at the same speed and rise to the same height as the first. Then the last ball will transfer its energy and momentum through the intermediate balls to the first, etc.
The ball moved to the side has maximum potential energy. Its kinetic energy at this moment is zero. Having started to move, it loses potential energy and gains kinetic energy, which at the moment of collision with the second ball reaches a maximum, and potential energy becomes equal to zero. Next, the kinetic energy is transferred to the second, then the third, fourth and fifth balls. The latter, having received kinetic energy, begins to move and rises to the same height at which the first ball was at the beginning of its movement. Its kinetic energy at this moment is zero, and its potential energy is equal to its maximum value. Then it begins to fall and transfers energy to the balls in the same way in the reverse order.
This continues for quite a long time and could continue indefinitely if non-conservative forces did not exist. But in reality, dissipative forces act in the system, under the influence of which the balls lose their energy. Their speed and amplitude gradually decrease. And eventually they stop. This confirms that the law of conservation of energy is satisfied only in the absence of non-conservative forces.