The pendulums shown in Fig. 2, are extended bodies of various shapes and sizes that oscillate around a point of suspension or support. Such systems are called physical pendulums. In a state of equilibrium, when the center of gravity is on the vertical below the point of suspension (or support), the force of gravity is balanced (through the elastic forces of a deformed pendulum) by the reaction of the support. When deviating from the equilibrium position, gravity and elastic forces determine the angular acceleration of the pendulum at each moment of time, i.e., they determine the nature of its movement (oscillation). We will now look at the dynamics of oscillations in more detail using the simplest example of a so-called mathematical pendulum, which is a small weight suspended on a long thin thread.

In a mathematical pendulum, we can neglect the mass of the thread and the deformation of the weight, i.e. we can assume that the mass of the pendulum is concentrated in the weight, and the elastic forces are concentrated in the thread, which is considered inextensible. Let's now see under what forces our pendulum oscillates after it is removed from its equilibrium position in some way (push, deflection).

When the pendulum is at rest in the equilibrium position, the force of gravity acting on its weight and directed vertically downward is balanced by the tension force of the thread. In the deflected position (Fig. 15), the force of gravity acts at an angle to the tension force directed along the thread. Let's break down the force of gravity into two components: in the direction of the thread () and perpendicular to it (). When the pendulum oscillates, the tension force of the thread slightly exceeds the component - by the amount of the centripetal force, which forces the load to move in an arc. The component is always directed towards the equilibrium position; she seems to be striving to restore this situation. Therefore, it is often called the restoring force. The more the pendulum is deflected, the greater the absolute value.

Rice. 15. Restoring force when the pendulum deviates from the equilibrium position

So, as soon as the pendulum, during its oscillations, begins to deviate from the equilibrium position, say, to the right, a force appears, slowing down its movement the more, the further it is deviated. Ultimately, this force will stop him and pull him back to the equilibrium position. However, as we approach this position, the force will become less and less and in the equilibrium position itself will become zero. Thus, the pendulum passes through the equilibrium position by inertia. As soon as it begins to deviate to the left, a force will again appear, growing with increasing deviation, but now directed to the right. The movement to the left will again slow down, then the pendulum will stop for a moment, after which the accelerated movement to the right will begin, etc.

What happens to the energy of a pendulum as it oscillates?

Twice during the period - at the greatest deviations to the left and to the right - the pendulum stops, i.e. at these moments the speed is zero, which means the kinetic energy is zero. But it is precisely at these moments that the center of gravity of the pendulum is raised to greatest height and therefore the potential energy is greatest. On the contrary, at the moments of passing through the equilibrium position, the potential energy is the lowest, and the speed and kinetic energy reach their greatest values.

We will assume that the friction forces of the pendulum against the air and the friction at the suspension point can be neglected. Then, according to the law of conservation of energy, this maximum kinetic energy is exactly equal to the excess of potential energy at the position of greatest deviation over the potential energy at the equilibrium position.

So, when the pendulum oscillates, a periodic transition of kinetic energy into potential energy and vice versa occurs, and the period of this process is half as long as the period of oscillation of the pendulum itself. However, the total energy of the pendulum (the sum of the potential and kinetic energies) is constant all the time. It is equal to the energy that was imparted to the pendulum at launch, no matter whether it is in the form of potential energy (initial deflection) or in the form of kinetic energy (initial push).

This is the case with any oscillations in the absence of friction or any other processes that take energy away from the oscillating system or impart energy to it. That is why the amplitude remains unchanged and is determined by the initial deflection or force of the push.

We will get the same changes in the restoring force and the same transfer of energy if, instead of hanging the ball on a thread, we make it roll in a vertical plane in a spherical cup or in a groove curved along the circumference. In this case, the role of thread tension will be taken over by the pressure of the walls of the cup or gutter (we again neglect the friction of the ball against the walls and air).

A mathematical pendulum is a model of an ordinary pendulum. A mathematical pendulum is a material point suspended on a long weightless and inextensible thread.

Let's move the ball out of its equilibrium position and release it. Two forces will act on the ball: gravity and the tension of the thread. When the pendulum moves, the force of air friction will still act on it. But we will consider it very small.

Let us decompose the force of gravity into two components: a force directed along the thread, and a force directed perpendicular to the tangent to the trajectory of the ball.

These two forces add up to the force of gravity. The elastic forces of the thread and the gravity component Fn impart to the ball centripetal acceleration. The work done by these forces will be zero, and therefore they will only change the direction of the velocity vector. At any moment in time, it will be directed tangentially to the arc of the circle.

Under the influence of the gravity component Fτ, the ball will move along a circular arc with a speed increasing in magnitude. The value of this force always changes in magnitude; when passing through the equilibrium position, it is equal to zero.

Dynamics of oscillatory motion

Equation of motion of a body oscillating under the action of an elastic force.

General equation of motion:

Oscillations in the system occur under the influence of elastic force, which, according to Hooke's law, is directly proportional to the displacement of the load

Then the equation of motion of the ball will take the following form:

Divide this equation by m, we get the following formula:

And since the mass and elasticity coefficient are constant quantities, the ratio (-k/m) will also be constant. We have obtained an equation that describes the vibrations of a body under the action of elastic force.

The projection of the acceleration of the body will be directly proportional to its coordinate, taken with the opposite sign.

Equation of motion of a mathematical pendulum

The equation of motion of a mathematical pendulum is described by the following formula:

This equation has the same form as the equation of motion of a mass on a spring. Consequently, the oscillations of the pendulum and the movements of the ball on the spring occur in the same way.

The displacement of the ball on the spring and the displacement of the pendulum body from the equilibrium position change over time according to the same laws.

Math pendulum is a material point suspended on a weightless and inextensible thread located in the Earth’s gravitational 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 size 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 oscillatory 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- displacement, l– length of the pendulum thread.

This equation is called equation of free oscillations of a mathematical pendulum. It correctly describes the vibrations in question only when the following assumptions are met:

2) only small oscillations of the pendulum with a small swing angle are considered.

Free vibrations of any systems are described in all cases by similar equations.

The causes of free oscillations of a mathematical pendulum are:

1. The action of tension and gravity on the pendulum, preventing it from moving from the equilibrium position and forcing it to fall again.

2. The inertia of the pendulum, due to which it, maintaining its speed, does not stop in the equilibrium position, but passes through it further.

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 acceleration free fall in the place where the pendulum is located.

Energy conversion during harmonic oscillations

During 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 - modulus of displacement of the pendulum from the equilibrium position, m- mass of the pendulum, v- its speed. According to the harmonic vibration equation:

, .

Total energy of a 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 a pendulum moves down or up from its equilibrium position, its potential energy increases, and its kinetic energy decreases. When the pendulum passes the equilibrium position ( X= 0), its potential energy is zero and the kinetic energy of the pendulum has the greatest value, equal to its total energy.

Thus, in the process of free oscillations of the pendulum, its potential energy turns into kinetic, kinetic into potential, potential then back into kinetic, etc. But the total mechanical energy remains unchanged.

Forced vibrations. Resonance.

Oscillations occurring under the influence of an external periodic force are called forced oscillations. An external periodic force, called a driving force, imparts additional energy to the oscillatory system, which goes to replenish the energy losses occurring due to friction. If the driving force changes over time according to the law of sine or cosine, then the forced oscillations will be harmonic and undamped.

Unlike free oscillations, when the system receives energy only once (when the system is brought out of equilibrium), in the case of forced oscillations the system absorbs this energy from a source of external periodic force continuously. This energy makes up for the losses spent on overcoming friction, and therefore the total energy oscillatory system no still remains unchanged.

The frequency of forced oscillations is equal to the frequency of the driving force. In the case where the driving force frequency 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 due to the fact that when υ = υ 0 the external force, acting in time with free vibrations, is always aligned 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 amplitude of forced oscillations A T on driving force frequency 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 may 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).

Let us 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 (ideal mathematical pendulum). Initially, the pendulum is at rest in the equilibrium position C. In this case, the force of gravity and the elastic force F?ynp of the thread acting on it are mutually compensated.

Let's remove the pendulum from the equilibrium position (by deflecting it, for example, to position A) and release it without an initial speed (Fig. 1). In this case, the forces do not balance each other. The tangential component of gravity, acting on the pendulum, gives it tangential acceleration a?? (component of the total acceleration directed along the tangent to the trajectory of the mathematical pendulum), and the pendulum begins to move towards the equilibrium position with a speed increasing in absolute value. The tangential component of gravity is thus a restoring force. The normal component of gravity is directed along the thread against the elastic force. The resultant of the forces gives the pendulum normal acceleration, which changes the direction of the velocity vector, and the pendulum moves along the arc ABCD.

The closer the pendulum comes to the equilibrium position C, the smaller the value of the tangential component 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 in an upward arc. In this case, the component is directed against the speed. As the angle of deflection a increases, the magnitude of the force increases, and the magnitude of the velocity decreases, and at point D the speed of the pendulum becomes zero. The pendulum stops for a moment and then begins to move in the opposite direction to the equilibrium position. Having passed it again by inertia, the pendulum, slowing down its movement, will reach point A (there is no friction), i.e. will complete a complete swing. After this, the movement of the pendulum will be repeated in the sequence already described.

Let us obtain an equation describing the free oscillations of a mathematical pendulum.

Let the pendulum in this moment time is at point B. Its displacement S from the equilibrium position at this moment is equal to the length of the arc SV (i.e. S = |SV|). Let us denote the length of the suspension thread as l, and the mass of the pendulum as m.

From Figure 1 it is clear that , where . At small angles () the pendulum deflects, therefore

The minus sign is placed in this formula 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. Let us project the vector quantities of this equation onto the direction of the tangent to the trajectory of the mathematical pendulum

From these equations we get

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 as

Comparing it with the harmonic vibration equation , we can conclude that the mathematical pendulum makes harmonic vibrations. And since the considered oscillations of the pendulum occurred under the influence of only internal forces, these were free oscillations of the pendulum. Consequently, free oscillations of a mathematical pendulum with small deviations are harmonic.

Let's denote

Cyclic frequency of pendulum oscillations.

Period of oscillation of a pendulum. Hence,

This expression is called Huygens' formula. It determines the period of free oscillations of a mathematical pendulum. From the formula it follows that at small angles of deviation from the equilibrium position, the period of oscillation of a mathematical pendulum is:

1. does not depend on its mass and vibration amplitude;
2. is proportional to the square root of the length of the pendulum and inversely proportional to the square root of the acceleration of gravity.

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 pendulum's oscillations should be small;
2. the pendulum's suspension point must be at rest or move uniformly in a straight line relative to inertial system the reference in which it is located.

If the suspension point of a mathematical pendulum moves with acceleration, then the tension force of the thread changes, which leads to a change in the restoring force, and, consequently, the frequency and period of oscillations. As calculations show, the period of oscillation of the pendulum in this case can be calculated using the formula

where is the “effective” acceleration of the pendulum in a non-inertial reference frame. It is equal to the geometric sum of the acceleration of free fall and the vector opposite to the vector, i.e. it can be calculated using the formula

don't believe it though case. Read all these articles carefully. Then it will become as clear as the shining Sun.

How not all people have a hand and a brain mysterious power, the pendulum also cannot become mysterious in the hands of all people. This strength is not acquired, but is born with a person. In one family, one is born rich and the other poor. No one has the power to make the naturally rich poor or vice versa. Now you understand with this what I wanted to tell you. If you don’t understand, blame yourself, you were born this way.

What is a pendulum? What is it made of? A pendulum is any freely moving body attached to a string. In the hands of a master, even a simple reed sings like a nightingale. Also, in the hands of a talented biomaster, a pendulum makes incredible effects in the sphere of human existence and 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 the pendulum with me. I looked around and a wine cork caught my eye. From about the middle of the cork, I made a small cut with a knife and attached the thread. The pendulum is ready.
I asked him: “Will you work with me honestly?” He was spinning strongly in a clockwise direction, as if responding cheerfully. Mentally let him know: “Let’s find the missing ring then.” The pendulum moved again as a sign of agreement. I started walking around the yard.

Because the daughter-in-law said that she had not yet entered 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, because her fingers had become thinner and the ring had begun to fall off. Suddenly, in my hands, the pendulum moved a little, turned a little back, the pendulum went silent. I moved forward, but the pendulum moved again. He walked on, became silent again, I was amazed. To the left the pendulum is silent, forward it is silent. To the right go nowhere. There is a small ditch flowing there. Suddenly I realized and held the pendulum directly above the water. The pendulum began to spin intensively clockwise. I called my daughter-in-law and showed me the location of the ring.
With joy in her eyes, she began to rummage through the ditch and quickly found the ring. It turns out that she was washing her hands in a ditch, and at that time the ring fell, but she did not notice. Everyone present admired the work of the wine cork.

Not all people are born fortune tellers or fortune tellers. Not all fortune tellers or fortune tellers are successful. A few predictors work with smaller errors, but many cheat like gypsies. So is the pendulum. An incompetent person has it as a useless thing, even though it is made of gold, it has no meaning. In the hands of a true master, a piece of ordinary stone or nut does wonders.
I remember it like yesterday. At one gathering, I took off my jacket and went out for a while. When I returned, I felt something was wrong in my heart. Mechanically he began to rummage in his pocket. It turned out that someone took my silver pendulum. I fell silent and didn’t tell anyone about what happened.
Many days passed, and one day one of those people who sat with us at that 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 also work for him just like mine.
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 also treated him to tea and even diagnosed him. I found many illnesses in him with a pendulum and prepared proper medicines for him.
Some people have a natural gift for healing and divination. This talent does not come out for years. Sometimes on occasion they encounter an expert, and he shows him his intended life path.
Recently a middle-aged woman came for a diagnosis. You can't tell by her appearance that she is sick. She complained of high warmth in her extremities, heat was constantly coming out of both her palms and soles, and she often felt wild bursting pains in her head in the crown area. Having first diagnosed it by pulse, noticing an increase in vascular tone, I began measuring 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 turned out to be below the norm for such hypertension: 62 beats per minute. A woman with such high blood pressure sat calmly in front of me. As if without feeling any discomfort from my vascular condition. Essential (unexplained) hypertension did not depress her.

I didn’t notice anything wrong with her pulse and during the pulse diagnostics either. I diagnosed her with a less common essential (unexplained cause) hypertension. If a regular doctor would have measured her blood pressure, he would have immediately called an ambulance and put her on a stretcher. He wouldn't even allow her to move. The fact is that a person with such an increase in blood pressure is considered to have a hypertensive crisis. It may be followed by a cerebral stroke or heart attack.
According to her, regular antihypertensive medications make her feel so much worse that they even make her feel nauseous. At the insistence of her son, she learned to use a 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 the doctor Muhiddin recommended to her four years ago. If her head hurts badly, then she drinks aspirin; in extremely severe cases, she takes pentalgin. Doctors and neighbors of a hypertensive patient laugh at her self-medication.
I used my pendulum to check all the medications she takes for headaches and high blood pressure. All of them turned out 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 treatment for herself, in order to get rid of essential hypertension, she must treat the diseases of other people, laying hands or feet on them. Now I often refer patients to her, and she successfully treats them psychic passes. He directs the warmth of his hand to diseases up to the waist, to diseases below the waist, in a lying position over 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 either aspirin or pentalgin, and the pendulum sometimes allows her to drink a decoction of willow or quince leaves for minor headaches.
Who needs her help, write to me, she will help you for a meager fee. I even taught her how to treat people at great distances in a non-contact way.
A person who truly works with a pendulum during the operation of the pendulum must be in synchronous communication with it and must know and feel in advance in which direction the actions of the pendulum are directed at the moment. With the energetic potency of his brain, the person holding the thread of the pendulum should help him subconsciously, and not speculatively, in further actions on this object, and not look indifferently at the action of the pendulum as a spectator.
Almost everyone has used and still uses the pendulum. famous people in Mesopotamia, Assyria, Urartu, India, China, Japan, in ancient Rome, Egypt, Greece, Asia, Africa, America, Europe, the East and many countries around the world.
Due to the fact that many prominent international institutions, prominent figures different areas sciences have not yet sufficiently assessed the action and purpose of the pendulum in favor of the coexistence of humanity with surrounding nature symbiotic and harmonious. Even humanity has not completely abandoned pseudoscientific views on the universe of the Universal normal at the level modern natural science. There is a stage of blurring the line 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 occupy people’s lives along with information science decent place. 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, but even to engage in educational institutions.
So it is now at the top echelon level modern science, they look at the idea of ​​a pendulum as if it were a backward industry. It is necessary to systematize the pendulum, dowsing, and frame under a single section of computer science, and it is necessary to create a computer program module.
With the help of this module, anyone can find missing things, determine the location of objects, and finally, diagnose people, animals, birds, insects, and all of nature in general.
To do this, you need to study the ideas of L. G. Puchko about 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 have achieved amazing results with the help of a pendulum.