THE LEARNING OF OSCILLATORY SYSTEMS BASED ON PHYSICAL ANALOGS IS SYNERGY

Sefarov Nakhchıvan Yusub

UDC 37.534 DOI: 10.26140/bgz3-2021-1002-0030

THE LEARNING OF OSCILLATORY SYSTEMS BASED ON PHYSICAL ANALOGS IS SYNERGY

SEFEROV Nakhchivan Yusub, Doctor of Science, professor of Department of Engineering physics and electronics Azerbaijan Technical University (AZ1073, Azerbaijan, Baku, Javid st., 25, AzTU, Department of Physics, e-mail: nax53@mail.ru) Abstract. In the work methodology of the learning of simple harmonic oscillations by unified approach has been analyzed and shown that it is an example of synergism in learning. It is shown that, in case of each oscillatory system of a different origin the following parameters are set: the form of motion performed, extensive and intensive quantities that characterize this form of motion; subsystems of capacitive, resistive and inductive nature and relations that characterize them; differential equations on the basis of relevant extensive quantities and their derivatives. As a result, it appears that the analogue of the m mass in a spring pendulum is not the L inductance in an electric oscillating circuit, but the C capacity; the analogue of inductance is the elasticity coefficient 1/k.

Keywords: synergism in learning, elements of capacitive, resistive and inductive nature; momentum capacity; momentum current; mass current; inductance of the system.

ИЗУЧЕНИЕ КОЛЕБАТЕЛЬНЫХ СИСТЕМ НА ОСНОВЕ ФИЗИЧЕСКИХ АНАЛОГОВ КАК СИНЕРГИИ

САФАРОВ Нахчыван Юсуб, доктор технических наук, профессор кафедры инженерной физики и электроники Азербайджанский технический университет (AZ1073, Баку, Азербайджан, ул. Джавида, 25, АзТУ, e-mail: nax53@mail.ru) Аннотация. В работе проанализирована методика обучения простым гармоническим колебаниям единым подходом и показано, что это пример синергизма в обучении. Показано, что для каждой колебательной системы разного происхождения задаются следующие параметры: форма совершаемого движения, экстенсивные и интенсивные величины, характеризующие эту форму движения; подсистемы емкостной, резистивной и индуктивной природы и характеризующие их отношения; дифференциальные уравнения на основе соответствующих обширных величин и их производных. В результате оказывается, что аналогом массы m в пружинном маятнике является не индуктивность L в электрическом колебательном контуре, а емкость C; аналог индуктивности - коэффициент упругости 1 / k.

Ключевые слова: синергизм в обучении, элементы емкостной, резистивной и индуктивной природы; импульсная способность; текущий импульс; массовый ток; индуктивность системы.

Introduction

The identification of possibilities and ways to apply a unified approach in the study of simple oscillations, included in the physics program of technical universities, is of great importance for learning of this subject.

Speaking of the unified approach we mean definition of identical physical properties and relations, elements performing the same function in oscillations of various origins and creation of physical and mathematical models on the basis of these analogs [Safarov N.Y., 2015]. In our opinion, this approach to learning students results in giving them quality knowledge.

For the implementation of a unified approach the considered oscillatory systems let us allocate appropriate extensive and intensive properties (potentials) related to them [Cengel Y. A. and Boles M. A., 2006]. Oscillatory systems perform certain forms of motion and their energy is determined by an extensive quantity. Therefore, the extensive quantity in the appropriate form of motion is also called an energy carrier. For example, during the rotation of a body around a fixed axis an angular momentum is the energy carrier, etc. [Etkin V.,2008; Falk G., Herrmann F., G. G. Bruno Schmid, 1983]. Table 1 shows some characteristics of oscillatory processes that are commonly used in equipment and are included in the physics course of technical universities.

As can be seen, each of these oscillatory systems is characterized by only one form of motion. In this sense, they are simple harmonic motions (SHM).

It is known that the process of oscillation is a repetition of changes in properties (physical quantities) of a system depending on the time. In a physical system consisting of two elements of capacitive nature and connecting its elements of resistive nature (RC system), in case of a difference in potentials of these elements, there is a flow of energy carrier in the direction from the large potential to the small potential. In an RC system, if the potentials of the capacitive elements

are equal, the flow of energy carrier is stopped and does not repeat [Safarov N., Valiev D., 2013].

Table 1 - Characteristics of oscillatory processes that are commonly used in equipment

Oscillation process Form о f motion Extensive quantity Intensive quantity'

a) oscillation of a rigid body (spring pendulum) suspended on a spring attached to the support linear motion (translation) linear momentum linear velocity

b) oscillation of the piston in a vertically disposed cylindrical vessel filled with gas (translation) linear momentum linear velocity

c) oscillations of a solid body of a specific shape (physicalpendulum) suspended on a point that does not coincide with the center of gravity ro a on angular momentum angular speed

d) oscillation of the disk on a horizontalplane, suspended on a thread(torsion pendulum)that's elasticto the torsion angular momentum

e) oscillation of the liquid column in the vessels, the bottom of which is connected to the valve with a tube (U-shaped tube) laminar motion liquid volume pressure

g) oscillation of electrical charges in a circuit containing a capacitor and a coil (electric oscillating circuit) electncity electnc charge electnc potential

Therefore, in order for oscillations to occur in a physical system, aside from elements of resistive and capacitive nature, we also need an element of an inductive character. The following conditions must be met for the occurrence of oscillations in a physical system:

- the oscillatory system should include two elements (of capacitive nature), where the physical property (quantity), able to flow back and forth, can be accumulated;

- the oscillatory system must include an element of an inductive nature, in order to create a cause for counteracting with the system, in order for it to remain in the existing state; as a result, the energy carrier flow (an extensive quantity) [Herrmann F. and Bruno Schmid G., 1985] with equal potentials of capacitive elements, does not stop and keeps going;

- in an oscillatory system resistance to the back and forth flow of physical quantities must be small.

Elements of oscillatory systems

To meet all these conditions, each of the simple oscillatory systems should consist of three elements: I) that of the capacitive nature, II) that of the resistive nature; III)

that of the inductive nature. A block diagram of a simple oscillatory system can be represented as follows:

Fig.1 - System block diagram

Now let us define the elements for different oscillatory systems.

In an oscillatory system, the elements of capacitive nature provide the accumulation of energy of a corresponding form of motion and they are as follows: a) in a spring pendulum a body suspended on a spring and the spring attachment bearing on Earth (mass - the momentum capacity is infinitely large) (fig. 2a); b) during the piston oscillation the bearing of cylindrical vessel on Earth and the piston of a certain mass (fig. 2b); c) body and the Earth in a physical pendulum (fig. 2c); d) in the torsion pendulum a disk oscillating on a horizontal plane and the Earth (fig. 2d); e) left and right vessels during oscillation of the liquid column (fig. 2e); f) capacitor plates in the electric oscillating circuit (fig. 2f).

In an oscillatory system, resistive elements ensure energy carrier flow of an appropriate form of back and forth motion between the two elements of capacitive nature. The cause of the flow is the difference of potentials between the elements of capacitive nature. Elements of resistive nature are the following: a) in a spring pendulum the spring and its attachment bearing create resistance to the flow of linear momentum determined by the law, similar to Ohm's law [Herrmann F. and Bruno Schmid G., 1985], b) during piston oscillation the air column under the piston and bearing of the cylindrical vessel create resistance to the flow of linear momentum determined by the law, similar to Ohm's law, c) in a physical pendulum the body itself and the bearing create resistance to the flow of angular momentum determined by the law, similar to Ohm's law, d) in the torsion pendulum a torsion elastic thread and bearing create resistance to the flow of angular momentum determined by the law, similar to Ohm's law, e) during the oscillation of liquid column the tube connecting the vessels, creates resistance to the volume (quantity) flow of liquid determined by the law of Poiseuille [Hans U. Fuchs., 2010; Plappert D., 1979], f) in an electric oscillating circuit wires and winding connecting the capacitor plates, create resistance to electric current determined by Ohm's law.

In an oscillatory system, the element of inductive nature creates a potential difference proportional to the speed of change of energy carrier flow and equal in magnitude of the capacitive potential difference (negative), to the initial moment of oscillation. Elements of inductive nature are the following: a) in a spring pendulum the spring (as well as the spring material) create resistance to the momentum flow, b) during the oscillation of the piston the air column under the piston; c) a body suspended on a physical pendulum; d) in the torsion pendulum a thread elastic to torsion; e) during the oscillation of liquid column the liquid in the tube connecting vessels, f) the conductor coil in the electric oscillating circuit.

Table 2 shows the capacitive and resistive characteristics of elements of the considered oscillatory systems.

Table 2 - The capacitive and resistive characteristics of elements of the considered oscillatory systems

Currents of Capacity of С ар a citive p ot ential Resistance to the Resistive potential

extensive quantities extensive quantities (properties) difference cunent of quantities (properties) difference

a) and b) linear body mass [S] Vc= — <1*) R Vr = R-F (2*)

momentum current Ibis ratio takes place in

(force) the work of a damper

c) and d) angular moment of inertia J w=7 ( 1**) R m = R M (2**)

momentum current

(Л/- torque)

e) flux ofvolume V/P = K¥ [1 = II ?■ h Pi=R-iv (2***)

Iv p34] PoiseuiHe's law

f) electrical cunent I Electrical capacity С là = R '/{2****)

In tables 1 and 2, the contents of each column are physical analogs. A unified approach can be implemented on the basis of these analogs.

It should be kept in mind that each of the oscillation processes in question occurs in a closed circuit of the corresponding flow [Herrmann F., 2000]. During oscillation electric current runs through and the elements of the subsystem create a difference of potentials (drop of voltage), resisting the flow of current.

Physical and mathematical description of the elements of the oscillatory system

And now let us try to give physical and mathematical description to the elements of an inductive nature. Let us look at different oscillatory systems:

a) in a spring pendulum to start the oscillation process compress the spring and remove the body suspended onto its end from the state of equilibrium and release the spring while releasing the body. Upon compression of the spring work is done against the spring's elastic force and the spring gains potential energy. By converting the potential energy of the compressed spring into the body's kinetic energy the potential difference (velocity difference) is applied between the body and the Earth. The moment of full conversion of the potential energy of a compressed spring into the kinetic energy of the body corresponds to the time t = 0. At this point, the body (element 1) has a maximum speed relative to the bearing - Earth. This potential difference is of capacitive nature and shall be denoted via VC. Starting from time t = 0, the momentum current starts running through the spring (the springs starts to stretch thus causing the emergence of elastic force). The speed difference VC can be associated with several effects. VC can be reduced in absolute value by VR - the resistive velocity difference, resulting from the resistance to momentum current.

Capacitive velocity difference VC also may increase or decrease by VL - inductive velocity difference, arising from changes in momentum current with time. In this case, two extreme cases are possible: I) the current does not change and the capacitive velocity difference, created in the beginning, is only connected to resistance, i.e, V =-VR; II) the current is zero and the capacitive velocity difference is only connected to the rate of current change, i.e., VC =-VL;

In a spring pendulum elastic force emerges and is bound by the elasticity of spring (momentum current) and can be calculated using Hooke's law. Performing some transformations in the mathematical expression of the law, it is possible to obtain a formula of velocity difference (potential difference), responsible for the change in the elastic force depending on time. To do this, perform the following transformations:

i n*i

F= - fa ; p = ki; p = - trVl; Vl = -7P '

As can be seen, the obtained velocity difference is proportional to the change rate of momentum current. In the electric current such ratio is bound by the inductive nature of the current. As an analog, the momentum current also has an inductive nature.

In general, the capacitive velocity difference, created at the beginning, is bound by two events at the same time, in other words VC is divided between the two events, i. e. Vc = - VR -Vl (4 *)

Keep in mind that VR - resistive velocity difference, can only be negative in the direction of flow, and VL - inductive velocity difference, can be negative or positiveL. A positive value VL means that the momentum current decreases depending on time.

For the momentum current a spring plays a role of a resistor and an inductive element. The velocity difference (capacitive potential difference) applied to the body at the time t=0, by being completely converted into inductive velocity difference, changes the momentum current (force). Therefore, initially the inductive velocity difference is maximum, the resistive velocity difference amounts to zero and according to

ratios (2), we get F(0) = p'C°) = When moving away from the initial moment, the momentum current running through a spring from the subsystem body to the subsystem Earth increases, and so does VR respectively to ratios (2). At the time t=T/4, VL=0, VC=-VR, i.e. the momentum current (force) becomes maximum, and its change equals zero. The influence of the spring inductance property is reduced to zero.

As seen, from time t=0 to t=T/4, the momentum current running through the spring into Earth (the force onto the bearing - Earth system) does not increase from zero immediately, but gradually. In other words, in order to achieve the value of momentum current from zero to the maximum a finite amount of time is required.

Since the moment t= T/4 the momentum current starts to decrease. The current starts changing and VL and VR are different from zero. At the moment t=T/2, VL=0, the capacitive potential difference is -VC, the resistive vLelocity difference is VR=VC. At this point, the momentum applied to the body throuRgh Cexternal energy, is fully returned to the subsystem Earth.

The second half-period starts with the flow of momentum current from the Earth to the body (the body begins to acquire a positive momentum), i.e., the process begins and is repeated in the opposite direction. At the moment of time 3T/4, the spring reaches its maximum compression, the potential energy of the spring becomes maximum. At the moment of time T, this energy is completely converted into the kinetic energy of the body and it obtains the maximum momentum (mVC).

It is clear fCrom the above that the capacitive potential difference (VC), transmitted to the element of capacitive nature initially from the external energy sources is used for the resistive (V ) and inductive (VL) potential differences. In fact, we are talRking about a closedL circuit of momentum current. According to Kirchhoffs law derived from the law of conservation and transformation of energy the following condition is met in a closed circuit:

VC+ V + VL = 0 (5 *)

b) to obtain piston oscillations it is necessary to compress the gas by moving the piston's center of gravity down and release the piston. In this case, the compressed gas itself acts as an elastic spring. The compression of gas also produces force proportional to the displacement (x) of the piston's center of gravity. Making the corresponding transformations in the mathematical expression of this force, we can obtain the formula of velocity difference (potential difference), responsible for the change in force depending on time, which is an analog of formula (3*). Here, however, k is replaced by a coefficient, determined by parameters of gas (Ricardsons metod) [Stephen et al, 2006] p. S 2 ^ (7)

k T"

Y - ratio of gas heat capacity, P - gas pressure and V gas volume, S - the piston's cross-section area. The law of conservation and transformation of energy in a closed circuit of momentum current also remains in the form of (5*).

c) for obtaining oscillations of a body in a physical pendulum, it should be moved to the right from its equilibrium position and released. The potential energy of the body will then be transformed into its rotational kinetic energy. Due to the rotational motion of the body an angular momentum emerges as an energy carrier. Continue the arguments as in a) and b). At the moment when t=0, angular velocity difference m is at its maximum value and has a capacitive nature, while the angular momentum current equals zero. In later moments the current gradually increases. The impression is made that the body is a subsystem, which accumulates angular momentum. The angular velocity difference mC, applied to the rotating body, may be connected to several effects. This value can be changed by the amount of m - the resistive angular velocity difference arising due to the current's resistance to the angular momentum. The velocity difference

may increase or decrease by the amount of mL - the inductive angular velocity difference arising due to changes in the angular momentum current depending on time. Two extreme cases are possible here: I) the current does not change and the capacitive velocity difference initially is caused only by the power of resistance, i.e., mC = - m„; Il) the current amounts to zero and the capacitive velocity difference is due to only the rate of current change, i.e., mQ = - mL;

In a physical pendulum arises a moment of force proportional to the body's angle of inclination (0) from the equilibrium position. Making the corresponding transformations in the mathematical expression of this force, we can obtain the formula of the difference of angular velocities (potential difference) responsible for the change in the moment of force depending on time, which is an analog of formula (3*).

M= -m%bm

M= —mob • Ô: £ =-mgb-Wz; --L

trzgo

(3**)

It can be seen that the angular velocity difference is proportional to the rate of change of current angular momentum (torque). In the electric current this ratio is bound by the inductive nature of the current. By analogy, we can come to a conclusion about the inductive nature of the angular momentum current.

In general, the capacitive angular velocity difference, created at the beginning, is simultaneously bound by both events, in other words, mC is divided between the two events,

i.e. C

mC = - mR- mL (4**)

It should be taken into account that mR - the angular velocity difference, arising due to the resistive nature, can only be negative in the direction of flow, while mL _ the angular velocity difference, arising due to the inductive nature, can be negative or positive. mL is positive, which means that the angular momentum current decreases with time.

At the time t = 0 an extreme condition II) occurs while at T/4 an extreme condition I) occurs. The following arguments can be continued like in case of a spring pendulum.

In this case we are talking about a closed current circuit of angular momentum and according to Kirchhoffs law derived from the law of conservation and transformation of energy in a closed circuit, the following condition is met, i.e.

mC + mR+m= 0 (5**)

d) in the torLion pendulum a form of motion is similar to that of a physical pendulum, therefore it is possible to argue similarly. In order to achieve oscillations it is necessary to displace the disk from the state of equilibrium at an angle 0, and let it go. At the time t = 0, the disk passes through the state of equilibrium and has a maximum angular velocity mC. All ratios for the time 0, t/4, T/2, 3T/4, T are identical to those of the point c).

In the torsion pendulum the moment of force occurs proportionally to the angle of torsion (0) of the body from the state of equilibrium. Making the corresponding transformations in the mathematical expression of this force, we can obtain the formula of angular velocity difference (potential difference) responsible for the change of the moment of force through time, which is analogous to the formula (3).

Af=-c0:ii = -c0 :I=-cMi: »! = - = £ (3***)

Here, M is a moment of elastic force, which occurs at torsioning of an elastic thread, c-coefficient, which characterizes the elastic properties of the thread. The ratio between (4**) and (5**) remains the same.

e) in order to study the oscillations of liquid in the connected vessels, it is necessary to understand the nature of pressure difference existing in the various cross sections of a closed tube with the liquid. Imagine that pressure difference arises in the sections connecting the first and second vessels with the tube. This pressure difference may be related to a number of events. The pressure difference may change in the direction of flow and as a result of friction of the liquid on the tube walls. Resulting from the rate of change of the flux of volume, the pressure difference may increase or decrease in the direction from the first to the second section.

Two extreme cases are possible in this situation: I) the flux of volume does not change and the capacitive pressure difference arising initially is caused only by the friction force of the liquid, i.e., PC = - PR; II) the flux of volume is equal to zero and the capacitive pre ssure difference is caused only by the rate of change of the flux of volume, i.e. PC = - PL;

In general, the capacitive pressure difference, created at the beginning, is associated with both events at the same time, in other words PC is divided between two events, i.e. P = - PR - PL (4***)

This condition is a physical analog of the electric circuit where inductors and resistors are separate elements placed one after another i.e., in series, and can be visualized more easily.

It is necessary to take into account that PR - pressure difference arising due to the resistive nature can oRnly be negative in the direction of flow, and PL - pressure difference resulting from the inductive nature caLn be both positive and negative. A positive value of PL means that the fluid flux is decreasing with time. In the literature [Hans U. Fuchs.,2010] P, is defined as

Pl=-L-U=J9

(3****)

Ui = -lJ = -Ltq

(3*****)

In fact, suppose that the left vessel has the liquid of a certain level, and the vessel on the right is empty. At the time t=0 open the valve in the tube connecting the vessels and let the liquid flow. The right vessel is empty, that is why the liquid begins to flow gradually in the direction from the left to the right vessel and the flux of volume begins to increase from zero not instantly but gradually. The flux of volume achieves a maximum from zero in a finite amount of time. The experiment shows that the time, when liquid levels in the vessels (with equal hydrostatic pressures of liquid columns in the sections) are equilibrated, the flow of liquid doesn't stop, on the contrary, its flux of volume has the largest value. At this time (T/4), the above-mentioned extreme condition I) occurs (PL=0). The flow of liquid continues (the system doesn't go out of the existing state) due to the inductive properties. Since the flow of liquid doesn't stop, the liquid level increases in the right vessel and continues to decrease in the left vessel. At some point of time (T/2) the level of liquid in the right vessel reaches maximum value and the flow of liquid stops. After this the liquid flows in the opposite direction until the liquid in the left vessel achieves its maximum level. This way the liquid oscillates moving back and forth between the connected vessels.

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It is clear from the above that the hydrostatic pressure (PC) of a column of a certain liquid volume initially filled wiCthin the left vessel (element 1 of capacitive nature) due to external sources of energy is used for resistive (PR) and inductive pressure differences (PL). In fact, we are talking about a closed circuit of the liLquid flow. According to Kirchhoffs law derived from the law of conservation and transformation of energy in a closed circuit, the following condition is met, i.e.,

PC + PR + PL = 0 (5***)

f) in order to start the electrical oscillations in the oscillating circuit it is necessary to charge the capacitor and concatenate it to the resistor and the coil. As a result of charging a potential difference UC arises between the plates of the capacitor. According to theC block diagram of the oscillation system, the capacitor plates can be considered subsystems of capacitive nature. In fact, one capacitor performs the function of two subsystems of capacitive nature. In order to understand the mechanism, the circuit can be connected to two capacitors. Except the connected vessels, in all other analyzed oscillatory systems, it is not clear that there are two subsystems of capacitive nature.

As noted above, in these oscillatory systems the Earth is considered to be the second subsystem. Experiments show that at t=0, the electric current in the circuit is equal to zero. At this moment the conditions UC=-UL and UR=0 should be met. According to the law of electromagnetic induction UL is calculated by the formula

At the time T/4 the condition UC =-UR and UL=0 are met. The resistive potential difference is maximum, that's why the electric current is also maximum.

At this moment the capacitive voltage completely converts into the resistive voltage and the electric field energy entirely transforms into magnetic field energy. In all other analyzed oscillations such splitting of motion forms is absent. This is the major difference between electrical and other oscillations. This difference has no influence on a common approach. It's the opposite, two oscillating systems are studied at the same time, in which the electric and magnetic forms of motion take place under the same laws.

At the time T/2 the ratios corresponding to the time t=0 are formed and the energy is fully accumulated in the capacitor. In the next moments of time these events are repeated.

According to Kirchhoffs law derived from the law of conservation and transformation of energy in a closed circuit, the following condition is met, i.e.,

UC + U + UL = 0 (5****)

InCoscilRlatoryL systems of different nature there are elements of the same physical properties (capacitive, inductive and resistive properties) and taking into account the oscillations of the values, characterizing the same physical properties (extensive property or an energy carrier), it is possible to apply a unified approach to all oscillatory processes of a similar type.

Table 3 shows differential equations of oscillations for the analyzed oscillatory systems.

Table 3 - Differential equations of oscillations for the analyzed oscillatory systems

Vc+Vr+Vl=C: - + jtp--p = oip + kRp —-p= 0: 0 =-

equations and transformations are the &

' V - 0 m ntif " 1

>i = 0: - + £t + rt=0; L+-L = 0; ft = n

equations and ttansformations are the same a sine). Instead ofthei coefficient the-c coefficient is applied:

; Kv = , ; T 20 1' l..V ' = U

Ji■ jj- ] = ~—: = Taidng into account the ratio n — — Y it is possible to define the law of liquid quantity

transformation m the vessels.

Uc+ I/r + 14 = 0; f+ leQ = 0; Q + CRQ+ CLeQ = 0; <? + - (J - — Q = 0; £ = ^ = Q+ 23Q - m O= 0.

Traditionally, in oscillations with a linear form of motion (for example, in a spring pendulum), the differential equation is written in accordance with the spatial coordinate and its derivatives [Pippard A.B.,2006; Walter Fox Smith., 2010]. In table 3, in point a) the equation is written in accordance with the linear momentum and its derivatives.

As an example let us analyze the laws of resistive and inductive potential difference and the momentum current in the spring pendulum.

As described above at the moment time t=0, the capacitive potential difference is completely consumed by the inductive potential difference and the resistive potential difference remains zero. Therefore, the momentum current

also becomes zero. At the moment of time when t=T/4, it's vice versa, the capacitive potential differential is completely consumed by resistive potential difference, and the inductive potential difference remains zero. Therefore, at this moment of time, the change in the momentum current becomes zero, and the momentum current becomes maximum.

Synergy in learning

The term synergy comes from the Attic Greek word GDvepyia synergia from synergos, GDvepyo^, meaning "working togethef'. There are different definitions of synergy. Determination close to the goal of this paper is the following: Synergy or synergism is the phenomenon of two or more discrete influences or agents acting in common to create an effect, which is greater than the sum of the effects each is able to create independently (wikipedia).

In learning one can define the analogy between syner-gism and the interference. There is the following analogy between the learning parameters and the interference:

Synergism in lea riling Interference of waves Notation ofthe parameters

The quality of learning The intensity of the wave I

Max quality Max intensity Im»

Analog learning elements (the same properties, relations, processes of different nature) Sources with the same frequency (with same or different amplitudes)

Learning quality ofthe first of the analogleaming elements The intensity ofthe first source la

Learning quality ofthe second of the analogleaming elements The intensity of the second source lo

Learning atleast mo analog elements together as a unified The meeting of two waves coming from two sources with the same

Synchronized (one afterthe other at the same time) learning of two analog elements together as a unified whole The constant phase difference p=0

I = 4In cos2 — 2

the capacitance of the capacitor is an analog of the technique in learning. The simpler (less) learning techniques, the higher the quality of learning, that is, the quality of learning may be considered as inversely proportional to the content. Ultimately, for the learning process one can write the following analog formula:

In the above described learning of simple harmonic oscillation by the unified approach content is rich, technique is simple. Therefore, high-quality learning is guaranteed.

It is known from the interference phenomenon that the final intensity (I) of the interfering waves of the same intensity is related to the wave intensity (I0) and phase difference (j) as follows:

(*)

This indicates that the maximum of the final intensity is obtained for j = 0. From the analogy of the synergy and the interference it turns out that the highest quality in teaching is obtained for synchronously (in sequence, at the same time) learning analog elements (this means j = 0 in teaching). Consequently, in order to obtain a high quality, the teaching must meet the terms of synergies: in the teaching program the analog elements must be identified and their synchronous study should be prescribed.

Described above learning of the harmonic oscillations by the unified approach adequately satisfy the conditions of the synergy. Two (or more) different oscillatory process in the study of nature as whole oscillatory processes is understood deeper and easier to remember. For example, capacitive element in one oscillatory process, followed by capacitive element in other oscillatory processes; inductive element, followed by inductive element in other oscillatory processes; resistive element, followed by resistive element in other oscillatory processes, etc.

In our opinion, (*) can be called "formula of synergism".

Let us interpret the concept of the "quality of learning" using a "powerful" analogy. For this purpose, we use existing analogy between the capacitor charging process in the electricity and the learning process.

At first, we will show the existence of an analogy between the concepts of voltage (potential difference) and the quality of learning. It is known that for a capacitor U =Q/C. The charge (Q) of the capacitor is the analog of the content in learning. The richer (more) content the higher the quality of learning, that is, the quality of learning may be considered as directly proportional to the content.

The capacitance (C) of the capacitor determines the procedure of the charge distribution, which depends on the shape and size "charge capacity" (capacitor). In the learning, procedure of learning (distribution) of the content determines the learning techniques (forms and methods). Consequently,

Fig.2 - Oscillatory system diagrams

Conclusion

1. In case of linear motion of the body, the extensive property (energy carrier) is the body's momentum. Therefore, the equation of motion, as well as in the electrical form of motion, must be bound by an extensive quantity and its derivatives. In oscillations, where the linear motion is performed (e.g., a spring pendulum), the equation should be written as a function of the momentum and its derivatives, instead of the spatial coordinate x. The spatial coordinate x cannot be an extensive value in a linear form of the body motion. The spatial coordinate does not include properties inherent to the body. The division of the body into two parts has no effect on x.

2. Taking a spatial coordinate as an extensive quantity (coordinates of state) at the linear shape of body motion may lead to inaccurate results in the analogy to the constructive parameters in oscillatory systems such similarity of mass and inductance. The research performed in the article shows that the mass is an analog of electrical capacity and the elasticity is an analog to inductance. These pairs are characteristics of identical physical properties of various motion forms._

3. The correct definition of natural analogs allows to implement a unified approach; in the analyzed oscillatory systems the laws of conservation and transformation of energy were applied, expressing the transformation of energy through extensive quantities of appropriate forms of motion and their derivatives, thus receiving the same kind of mathematical equations.

4. It has been determined that the oscillatory systems are composed of elements with the same properties, and it also allows using a unified approach.

5. A unified approach saves time in the teaching of oscillation processes, allows gaining system knowledge, correctly understanding the processes, generalizing, drawing conclusions and simplifying the process of remembering.

6. We believe that the teaching of simple harmonic oscillations by the unified approach is an example of the synergy in teaching.

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Статья поступила в редакцию 27.01.2021

Статья принята к публикации 27.05.2021

THE LEARNING OF OSCILLATORY SYSTEMS BASED ON PHYSICAL ANALOGS IS SYNERGY Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Аннотация научной статьи по электротехнике, электронной технике, информационным технологиям, автор научной работы — Sefarov Nakhchıvan Yusub

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