A mechanical system, which consists of a material point (body) hanging on an inextensible weightless thread (its mass is negligible compared to the weight of the body) in a uniform gravity field, is called a mathematical pendulum (another name is an oscillator). There are other types of this device. Instead of a thread, a weightless rod can be used. The mathematical pendulum can clearly reveal the essence of many interesting phenomena. With a small amplitude of oscillation, its movement is called harmonic.
General information about the mechanical system
The formula for the period of oscillation of this pendulum was derived by the Dutch scientist Huygens(1629-1695). This contemporary of I. Newton was very fond of this mechanical system. In 1656 he created the first pendulum clock. They measured time with exceptional accuracy for those times. This invention became milestone in development physical experiments and practical activities.
If the pendulum is in the equilibrium position (hanging vertically), then it will be balanced by the force of the thread tension. A flat pendulum on an inextensible thread is a system with two degrees of freedom with a connection. When you change just one component, the characteristics of all its parts change. So, if the thread is replaced by a rod, then this mechanical system will have only 1 degree of freedom. What are the properties of a mathematical pendulum? In this the simplest system under the influence of a periodic perturbation, chaos arises. In the case when the suspension point does not move, but oscillates, the pendulum has a new equilibrium position. With rapid up and down oscillations, this mechanical system acquires a stable upside down position. She also has her own name. It is called the pendulum of Kapitza.
pendulum properties
The mathematical pendulum has very interesting properties. All of them are confirmed by known physical laws. The period of oscillation of any other pendulum depends on various circumstances, such as the size and shape of the body, the distance between the point of suspension and the center of gravity, the distribution of mass relative to this point. That is why the determination of the period of a hanging body is quite challenging task. Much easier to calculate the period mathematical pendulum, the formula of which will be given below. As a result of observations of similar mechanical systems, the following regularities can be established:
If, while maintaining the same length of the pendulum, different weights are suspended, then the period of their oscillations will turn out to be the same, although their masses will differ greatly. Therefore, the period of such a pendulum does not depend on the mass of the load.
If, when starting the system, the pendulum is deflected by not too large, but different angles, then it will oscillate with the same period, but at different amplitudes. As long as the deviations from the center of equilibrium are not too large, the oscillations in their form will be quite close to harmonic ones. The period of such a pendulum does not depend on the oscillation amplitude in any way. This property of this mechanical system is called isochronism (translated from the Greek "chronos" - time, "isos" - equal).
The period of the mathematical pendulum
This indicator represents the period Despite the complex wording, the process itself is very simple. If the length of the thread of a mathematical pendulum is L, and the free fall acceleration is g, then this value is equal to:
The period of small natural oscillations in no way depends on the mass of the pendulum and the amplitude of oscillations. In this case, the pendulum moves like a mathematical pendulum with a reduced length.
Oscillations of a mathematical pendulum
A mathematical pendulum oscillates, which can be described by a simple differential equation:
x + ω2 sin x = 0,
where x (t) is an unknown function (this is the angle of deviation from the lower equilibrium position at time t, expressed in radians); ω is a positive constant that is determined from the parameters of the pendulum (ω = √g/L, where g is the gravitational acceleration and L is the length of the mathematical pendulum (suspension).
The equation of small oscillations near the equilibrium position ( harmonic equation) looks like that:
x + ω2 sin x = 0
Oscillatory movements of the pendulum
A mathematical pendulum that makes small oscillations moves along a sinusoid. Differential equation of the second order meets all the requirements and parameters of such a movement. To determine the trajectory, you must specify the speed and coordinate, from which independent constants are then determined:
x \u003d A sin (θ 0 + ωt),
where θ 0 is the initial phase, A is the oscillation amplitude, ω is the cyclic frequency determined from the equation of motion.
Mathematical pendulum (formulas for large amplitudes)
This mechanical system, which makes its oscillations with a significant amplitude, is subject to more complex laws of motion. For such a pendulum, they are calculated by the formula:
sin x/2 = u * sn(ωt/u),
where sn is the Jacobian sine, which for u< 1 является периодической функцией, а при малых u он совпадает с простым trigonometric sine. The value of u is determined by the following expression:
u = (ε + ω2)/2ω2,
where ε = E/mL2 (mL2 is the energy of the pendulum).
The oscillation period of a non-linear pendulum is determined by the formula:
where Ω = π/2 * ω/2K(u), K is the elliptic integral, π - 3,14.
The movement of the pendulum along the separatrix
A separatrix is a trajectory of a dynamical system that has a two-dimensional phase space. The mathematical pendulum moves along it non-periodically. At an infinitely distant moment of time, it falls from the extreme upper position to the side with zero velocity, then gradually picks it up. It eventually stops, returning to its original position.
If the amplitude of the pendulum's oscillation approaches the number π , this indicates that the motion on the phase plane approaches the separatrix. In this case, under the action of a small driving periodic force, the mechanical system exhibits chaotic behavior.
When the mathematical pendulum deviates from the equilibrium position with a certain angle φ, a tangential force of gravity Fτ = -mg sin φ arises. The minus sign means that this tangential component is directed in the opposite direction from the pendulum deflection. When the displacement of the pendulum along the arc of a circle with radius L is denoted by x, its angular displacement is equal to φ = x/L. The second law, which is for projections and force, will give the desired value:
mg τ = Fτ = -mg sinx/L
Based on this relationship, it can be seen that this pendulum is a non-linear system, since the force that tends to return it to its equilibrium position is always proportional not to the displacement x, but to sin x/L.
Only when the mathematical pendulum makes small oscillations is it a harmonic oscillator. In other words, it becomes a mechanical system capable of performing harmonic vibrations. This approximation is practically valid for angles of 15-20°. Pendulum oscillations with large amplitudes are not harmonic.
Newton's law for small oscillations of a pendulum
If a given mechanical system performs small vibrations, Newton's 2nd law will look like this:
mg τ = Fτ = -m* g/L* x.
Based on this, we can conclude that the mathematical pendulum is proportional to its displacement with a minus sign. This is the condition due to which the system becomes a harmonic oscillator. The modulus of the proportionality factor between displacement and acceleration is equal to the square of the circular frequency:
ω02 = g/L; ω0 = √g/L.
This formula reflects the natural frequency of small oscillations of this type of pendulum. Based on this,
T = 2π/ ω0 = 2π√ g/L.
Calculations based on the law of conservation of energy
The properties of a pendulum can also be described using the law of conservation of energy. In this case, it should be taken into account that the pendulum in the field of gravity is equal to:
E = mg∆h = mgL(1 - cos α) = mgL2sin2 α/2
Total equals kinetic or maximum potential: Epmax = Ekmsx = E
After the law of conservation of energy is written, the derivative of the right and left sides of the equation is taken:
Since the derivative of constants is 0, then (Ep + Ek)" = 0. The derivative of the sum is equal to the sum of the derivatives:
Ep" = (mg/L*x2/2)" = mg/2L*2x*x" = mg/L*v + Ek" = (mv2/2) = m/2(v2)" = m/2* 2v*v" = mv*α,
hence:
Mg/L*xv + mva = v (mg/L*x + mα) = 0.
Based on the last formula, we find: α = - g/L*x.
Practical application of the mathematical pendulum
Acceleration varies with latitude as the density earth's crust across the planet is not the same. Where rocks with a higher density occur, it will be somewhat higher. The acceleration of a mathematical pendulum is often used for geological exploration. It is used to search for various minerals. Just by counting the number of swings of the pendulum, you can find in the bowels of the Earth coal or ore. This is due to the fact that such fossils have a density and mass greater than the loose rocks underlying them.
The mathematical pendulum was used by such prominent scientists as Socrates, Aristotle, Plato, Plutarch, Archimedes. Many of them believed that this mechanical system could influence the fate and life of a person. Archimedes used a mathematical pendulum in his calculations. Nowadays, many occultists and psychics use this mechanical system to fulfill their prophecies or search for missing people.
The famous French astronomer and naturalist C. Flammarion also used a mathematical pendulum for his research. He claimed that with his help he was able to predict the discovery new planet, appearance Tunguska meteorite and others important events. During the Second World War in Germany (Berlin) a specialized pendulum institute worked. Today, the Munich Institute of Parapsychology is engaged in similar research. The employees of this institution call their work with the pendulum “radiesthesia”.
Mathematical pendulum- this is a material point suspended on a weightless and inextensible thread located in the Earth's gravity field. A mathematical pendulum is an idealized model that correctly describes a real pendulum only under certain conditions. A real pendulum can be considered mathematical if the length of the thread is much greater than the dimensions of the body suspended on it, the mass of the thread is negligible compared to the mass of the body, and the deformations of the thread are so small that they can be neglected altogether.
The oscillating system in this case is formed by a thread, a body attached to it, and the Earth, without which this system could not serve as a pendulum.
Where A X – acceleration, g - acceleration of gravity, X- offset, l is the length of the pendulum string.
This equation is called the equation of free oscillations of a mathematical pendulum. It correctly describes the oscillations under consideration only when the following assumptions are fulfilled:
2) only small oscillations of a pendulum with a small swing angle are considered.
Free vibrations of any systems in all cases are described by similar equations.
The reasons for the free oscillations of a mathematical pendulum are:
1. The action on the pendulum of the force of tension and the force of gravity, preventing its displacement from the equilibrium position and forcing it to fall again.
2. The inertia of the pendulum, due to which, while maintaining its speed, it does not stop in the equilibrium position, but passes through it further.
The period of free oscillations of a mathematical pendulum
The period of free oscillations of a mathematical pendulum does not depend on its mass, but is determined only by the length of the thread and the free fall acceleration in the place where the pendulum is located.
Energy conversion during harmonic vibrations
With harmonic oscillations of a spring pendulum, the potential energy of an elastically deformed body is converted into its kinetic energy, where k – elasticity coefficient, X - pendulum displacement module from the equilibrium position, m- the mass of the pendulum, v- his speed. In accordance with the equation of harmonic oscillations:
,
.
The total energy of the spring pendulum:
.
Total energy for a mathematical pendulum:
In the case of a mathematical pendulum
Energy transformations during oscillations of a spring pendulum occur in accordance with the law of conservation of mechanical energy ( 
). When the pendulum moves up or down from the equilibrium position, its potential energy increases, and its kinetic energy decreases. When the pendulum passes the equilibrium position ( X= 0), its potential energy is equal to zero and the kinetic energy of the pendulum has the largest value, equal to its total energy.
Thus, in the process of free oscillations of the pendulum, its potential energy is converted into kinetic, kinetic into potential, potential then again into kinetic, etc. But the total mechanical energy remains unchanged.
Forced vibrations. Resonance.
Oscillations that occur under the action of an external periodic force are called forced vibrations. An external periodic force, called a driving force, imparts additional energy to the oscillatory system, which is used to replenish energy losses due to friction. If the driving force changes in time according to the sine or cosine law, then the forced oscillations will be harmonic and undamped.
Unlike free oscillations, when the system receives energy only once (when the system is taken out of equilibrium), in the case of forced oscillations, the system continuously absorbs this energy from a source of external periodic force. This energy compensates for the losses spent on overcoming friction, and therefore the total energy of the oscillatory system no remains unchanged.
The frequency of forced oscillations is equal to the frequency of the driving force. When the frequency of the driving force υ coincides with the natural frequency of the oscillatory system υ 0 , there is a sharp increase in the amplitude of forced oscillations - resonance. Resonance occurs because υ = υ 0 the external force, acting in time with free vibrations, is always co-directed with the speed of the oscillating body and does positive work: the energy of the oscillating body increases, and the amplitude of its oscillations becomes large. Graph of the dependence of the amplitude of forced oscillations A T on the frequency of the driving force υ shown in the figure, this graph is called the resonance curve:
The phenomenon of resonance plays an important role in a number of natural, scientific and industrial processes. For example, it is necessary to take into account the phenomenon of resonance when designing bridges, buildings and other structures that experience vibration under load, otherwise, under certain conditions, these structures can be destroyed.
Mathematical pendulum called material point suspended on a weightless and inextensible thread attached to the suspension and located in the field of gravity (or other force).
We study the oscillations of a mathematical pendulum in an inertial frame of reference, relative to which the point of its suspension is at rest or moves uniformly in a straight line. We will neglect the force of air resistance (an ideal mathematical pendulum). Initially, the pendulum is at rest in the equilibrium position C. In this case, the force of gravity \(\vec F\) and the elastic force \(\vec F_(ynp)\) of the thread acting on it are mutually compensated.
Let's bring the pendulum out of the equilibrium position (deflecting it, for example, to position A) and let it go without initial speed (Fig. 13.11). In this case, the forces \(\vec F\) and \(\vec F_(ynp)\) do not balance each other. The tangential component of gravity \(\vec F_\tau\), acting on the pendulum, gives it a tangential acceleration \(\vec a_\tau\) (component of the total acceleration directed along the tangent to the trajectory of the mathematical pendulum), and the pendulum begins to move to the equilibrium position with increasing modulus of speed. The tangential component of gravity \(\vec F_\tau\) is thus the restoring force. The normal component \(\vec F_n\) of gravity is directed along the thread against the elastic force \(\vec F_(ynp)\). The resultant of the forces \(\vec F_n\) and \(\vec F_(ynp)\) gives the pendulum a normal acceleration \(~a_n\), which changes the direction of the velocity vector, and the pendulum moves along an arc ABCD.
The closer the pendulum approaches the equilibrium position C, the smaller the value of the tangential component \(~F_\tau = F \sin \alpha\) becomes. In the equilibrium position, it is equal to zero, and the speed reaches its maximum value, and the pendulum moves further by inertia, rising upward along the arc. In this case, the component \(\vec F_\tau\) is directed against the speed. With an increase in the deflection angle a, the modulus of force \(\vec F_\tau\) increases, and the modulus of velocity decreases, and at point D the pendulum's velocity becomes equal to zero. The pendulum stops for a moment and then begins to move in the opposite direction to the equilibrium position. Having again passed it by inertia, the pendulum, slowing down, will reach point A (no friction), i.e. makes a full swing. After that, the movement of the pendulum will be repeated in the sequence already described.
We obtain an equation describing the free oscillations of a mathematical pendulum.
Let the pendulum at a given moment of time be at point B. Its displacement S from the equilibrium position at this moment is equal to the length of the arc CB (i.e. S = |CB|). Denote the length of the suspension thread l, and the mass of the pendulum - m.
Figure 13.11 shows that \(~F_\tau = F \sin \alpha\), where \(\alpha =\frac(S)(l).\) At small angles \(~(\alpha<10^\circ)\) отклонения маятника \(\sin \alpha \approx \alpha,\) поэтому
\(F_\tau = -F\frac(S)(l) = -mg\frac(S)(l).\)
The minus sign in this formula is put because the tangential component of gravity is directed towards the equilibrium position, and the displacement is counted from the equilibrium position.
According to Newton's second law \(m \vec a = m \vec g + F_(ynp).\) We project the vector quantities of this equation onto the direction of the tangent to the trajectory of the mathematical pendulum
\(~F_\tau = ma_\tau .\)
From these equations we get
\(a_\tau = -\frac(g)(l)S\) - dynamic equation of motion of a mathematical pendulum. The tangential acceleration of a mathematical pendulum is proportional to its displacement and is directed towards the equilibrium position. This equation can be written in the form \. Comparing it with the equation of harmonic oscillations \(~a_x + \omega^2x = 0\) (see § 13.3), we can conclude that the mathematical pendulum performs harmonic oscillations. And since the considered oscillations of the pendulum occurred under the action of only internal forces, these were free oscillations of the pendulum. Hence, free oscillations of a mathematical pendulum with small deviations are harmonic.
Denote \(\frac(g)(l) = \omega^2.\) Whence \(\omega = \sqrt \frac(g)(l)\) is the cyclic frequency of the pendulum.
The period of oscillation of the pendulum \(T = \frac(2 \pi)(\omega).\) Therefore,
\(T = 2 \pi \sqrt( \frac(l)(g) )\)
This expression is called Huygens formula. It determines the period of free oscillations of the mathematical pendulum. It follows from the formula that at small angles of deviation from the equilibrium position, the oscillation period of a mathematical pendulum: 1) does not depend on its mass and oscillation amplitude; 2) is proportional to the square root of the length of the pendulum and inversely proportional to the square root of the gravitational acceleration. This is consistent with the experimental laws of small oscillations of a mathematical pendulum, which were discovered by G. Galileo.
We emphasize that this formula can be used to calculate the period if two conditions are met simultaneously: 1) the oscillations of the pendulum must be small; 2) the suspension point of the pendulum must be at rest or move uniformly rectilinearly relative to the inertial frame of reference in which it is located.
If the suspension point of a mathematical pendulum moves with acceleration \(\vec a\), then the tension force of the thread changes, which leads to a change in the restoring force, and, consequently, the frequency and period of oscillation. As calculations show, the period of oscillation of the pendulum in this case can be calculated by the formula
\(T = 2 \pi \sqrt( \frac(l)(g") )\)
where \(~g"\) is the "effective" acceleration of the pendulum in a non-inertial reference frame. It is equal to the geometric sum of the free fall acceleration \(\vec g\) and the vector opposite to the vector \(\vec a\), i.e. it can be calculated using the formula
\(\vec g" = \vec g + (- \vec a).\)
Literature
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - S. 374-376.
Mathematical pendulum.
A mathematical pendulum is a material point suspended on an inextensible weightless thread, oscillating in one vertical plane under the action of gravity.
Such a pendulum can be considered a heavy ball of mass m, suspended on a thin thread, the length l of which is much larger than the size of the ball. If it is deflected by an angle α (Fig. 7.3.) from the vertical line, then under the influence of the force F - one of the components of the weight P, it will oscillate. The other component , directed along the thread, is not taken into account, because balanced by the tension in the string. At small displacement angles and, then the x-coordinate can be counted in the horizontal direction. From Fig. 7.3 it can be seen that the weight component perpendicular to the thread is equal to
The moment of force relative to the point O: , and the moment of inertia:
M=FL .
Moment of inertia J in this case
Angular acceleration:
Taking into account these values, we have: 
 |
(7.8) |
His decision 
,
| where and | (7.9) |
As you can see, the period of oscillation of a mathematical pendulum depends on its length and the acceleration of gravity and does not depend on the amplitude of the oscillations.
physical pendulum.
A physical pendulum is a rigid body fixed on a fixed horizontal axis (suspension axis) that does not pass through the center of gravity and oscillates about this axis under the action of gravity. Unlike a mathematical pendulum, the mass of such a body cannot be considered as a point mass.
At small deflection angles α (Fig. 7.4), the physical pendulum also performs harmonic oscillations. We will assume that the weight of the physical pendulum is applied to its center of gravity at point C. The force that returns the pendulum to the equilibrium position, in this case, will be the component of gravity - the force F.
The minus sign on the right side means that the force F is directed towards decreasing the angle α. Taking into account the smallness of the angle α
To derive the law of motion of mathematical and physical pendulums, we use the basic equation for the dynamics of rotational motion
Moment of force: cannot be determined explicitly. Taking into account all the quantities included in the original differential equation of the oscillations of a physical pendulum, it has the form
don't believe your case. Read all of these articles carefully. Then it will become as clear as the shining Sun.
Just as the hand and the brain do not have a mysterious power in all people, the pendulum, too, in the hands of not all people can become mysterious. This power is not acquired, but is born together with a person. In one family, one is born rich and the other poor. No one is able to make the natural rich poor or vice versa. Now you understand with this what I wanted to tell you. If you do not understand, blame yourself, you were born that way.
What is a pendulum? What is it made from? A pendulum is any freely moving body attached to a thread. In the hands of the master, even a simple reed sings like a nightingale. Also, in the hands of a talented biomaster, the pendulum makes incredible impacts in the sphere of being and human existence.
It doesn't always happen that you carry a pendulum with you. So I had to find a lost ring from one family, but I didn’t have a pendulum with me. I looked around and a wine cork caught my eye. From about the middle of the cork, I made a little incision with a knife and attached the thread. The pendulum is ready.
I asked him: “Will you work honestly with me?” He affirmatively strongly spun clockwise, as if responding cheerfully. Mentally let him know: "Let's find the missing ring then." The pendulum swung again in agreement. I started walking around the yard.
Because the daughter-in-law said that she had not yet managed to enter the house when she noticed that she did not have a ring on her finger. She also said that she had long wanted to go to the jeweler, as her fingers had grown thin, and the ring began to fall off. Suddenly, on my hands, the pendulum moved a little, turned back a little, the pendulum fell silent. I walked forward, but the pendulum moved again. I went on, quieted down again, I was amazed. To the left the pendulum is silent, forward is silent. Right go nowhere. There is a small ditch there. Suddenly I enlightened and held the pendulum directly above the water. The pendulum began to spin clockwise intensively. I called my daughter-in-law and showed the location of the ring.
With joy in her eyes, she began to rummage along the canal and quickly found a ring. It turns out that she was washing her hands in the ditch, and at that time the ring fell, but she did not notice. All those present admired the work of the wine cork.
Not all people are born fortune tellers or fortune tellers. Not all fortunetellers or fortune-tellers work successfully. Single predictors work with smaller errors, and many cheat like gypsies. So is the pendulum. It is a good-for-nothing thing for an inept person, although it is made of gold, it has no value. In the hands of a real master, a piece of ordinary stone or a nut does wonders.
I remember like yesterday. At one meeting, I took off my jacket and went out for a while. When he returned, he felt something was wrong with his heart. Mechanically he began to rummage in his pocket. It turned out that someone took my silver pendulum. I kept silent and did not tell anyone about what had happened.
Many days passed, and one day one of those people who sat with us at the gathering where my pendulum was lost came to my house. He apologized deeply and handed me the pendulum. It turns out that he thought that all the power was on my pendulum and thought that this pendulum would work for him as well as for me.
When he realized his mistake, his conscience tormented him for a long time and finally decided to return the pendulum to its owner. I accepted his apology and even treated him to tea and even diagnosed. I found many diseases in him with a pendulum and prepared appropriate medicines for him.
Some people have a natural gift for healing and divination. This talent has not come out for years. Sometimes, on occasion, they come across a connoisseur, and he shows him his destined life path.
Recently, a middle-aged woman came for diagnostics. You can't tell by her appearance that she's sick. She complained about the high warmth in her limbs, both the palms and the soles of the feet were constantly hot, and often felt wild arching pains in her head in the region of the crown. First diagnosing it by pulse, noticing an increase in vascular tone, I began to measure blood pressure with a semi-automatic device. The values eventually went off scale both systolic and diastolic. They indicated 135 to 241, and the heart rate was below normal for such hypertension: 62 beats per minute. In front of me, a woman with such high blood pressure sat calmly. As if not feeling discomfort, from his condition of the vessels. Essential (incomprehensible) hypertension did not oppress her.
According to her pulse, I did not notice anything wrong during the pulse diagnostics either. I diagnosed her with rare essential (unexplained cause) hypertension. If an ordinary doctor would measure her blood pressure, he immediately called an ambulance and put her on a stretcher. Wouldn't even let her move. The fact is that a person with such an increase in pressure is considered a hypertensive crisis. It may be followed by a stroke or a heart attack.
According to her, from conventional antihypertensive drugs she feels so bad that after them she even feels sick. At the urging of her son, she learned to use the pendulum, when her head hurts badly, she asks the pendulum whether or not to drink aspirin or pentalgin. More rarely, with the consent of the pendulum, she takes a decoction of willow leaves or a decoction of quince leaves, which were recommended to her by the healer Mukhiddin four years ago. If her head hurts badly, then she drinks aspirin, in extremely severe cases, she takes pentalgin. Doctors and neighbors of hypertension laugh at her self-medication.
I checked with my pendulum all the medicines she takes for headaches and high blood pressure. All of them proved to be effective.I also asked the pendulum. “Will her health improve if she begins to heal people with her warmth?” The pendulum immediately swung strongly clockwise, in the affirmative. So I prescribed her a treatment from herself, in order to get rid of essential hypertension, she must deal with the treatment of diseases of other people, laying hands or feet on them. Now I myself often refer patients to her, and she successfully treats them. psychic passes. On diseases up to the waist, he directs the warmth of the hand, on diseases below the waist, in a lying position above the patient, he holds the right or left leg, respectively, in the problem area.
Both she and the patients are satisfied with the results. For two years now she has not taken aspirin or pentalgin, and the pendulum sometimes allows her to drink a decoction of willow or quince leaves, with minor headaches.
Who needs her help, write to me, she will help you for a meager fee. I even taught her to treat people who are at great distances in a non-contact way.
A person who truly works with the pendulum during the operation of the pendulum must be in synchronous communication with it and must know and feel in advance to which channel the pendulum’s actions are directed at the moment. With the energy potential of his brain, a person holding the thread of the pendulum should help him subconsciously, and not speculatively, in further actions on this object, but indifferently not look at the action of the pendulum as a spectator.
Almost all famous people in Mesopotamia, Assyria, Urartu, India, China, Japan, ancient Rome, Egypt, Greece, Asia, Africa, America, Europe, in the East and all over the world in many countries used the pendulum and still use it.
Due to the fact that many prominent international institutions, prominent figures in various fields of science have not yet sufficiently appreciated the action and purpose of the pendulum in favor of the coexistence of mankind with the surrounding nature in a symbiotic and harmonious way. The pseudoscientific views on the universe of the Universal Normal at the level of modern natural science have not completely left humanity yet. There is a stage of erasing the edge of knowledge between religion, esotericism and natural science. Naturally, natural science should become the basis of all fundamental sciences without any side views.
There is hope that the science of the pendulum will also take a worthy place in people's lives along with information science. After all, there was a time when the leaders of our multinational country declared cybernetics a pseudoscience and did not allow not only to study, even to study in educational institutions.
So now, at the level of the highest echelon of modern science, they are looking at the idea of a pendulum as if at a backward industry. It is necessary to systematize the pendulum, the dowsing, the frame under a single section of computer science, and it is necessary to create a computer program module.
With the help of this module, any person can find missing things, locate objects, and finally, diagnose people, animals, birds, insects, in general, all nature.
To do this, you need to study the ideas of L. G. Puchko on multidimensional medicine and the work of the psychic Geller, as well as the ideas of the Bulgarian healer Kanaliev and the work of many other people who achieved amazing results with the help of the pendulum.
