Forces acting on a mathematical pendulum. Secrets of the pendulum

Mathematical pendulum call a 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 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 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 performs harmonic oscillations. 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 suspension point of the pendulum must be at rest or move uniformly in a straight line relative to the inertial reference frame 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

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 at each moment of time the angular acceleration of the pendulum, 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 its 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 trough (we again neglect the friction of the ball against the walls and air).

A mechanical system that 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 gravitational 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. A mathematical pendulum can clearly reveal the essence of many interesting phenomena. When the vibration amplitude is small, its motion is called harmonic.

Mechanical System Overview

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 interested in this mechanical system. In 1656 he created the first clock with a pendulum mechanism. They measured time with exceptional precision for those times. This invention became a major stage in the development of physical experiments and practical activities.

If the pendulum is in the equilibrium position (hanging vertically), it will be balanced by the tension force of the thread. A flat pendulum on an inextensible thread is a system with two degrees of freedom with coupling. 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 properties does a mathematical pendulum have? In this simplest system, chaos arises under the influence of periodic disturbances. In the case when the point of suspension does not move, but oscillates, the pendulum has a new equilibrium position. With rapid oscillations up and down, this mechanical system acquires a stable “upside down” position. It also has its own name. It is called the Kapitsa pendulum.

Properties of a pendulum

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, and the distribution of mass relative to this point. That is why determining the hanging period of a body is quite a difficult task. It is much easier to calculate the period of a mathematical pendulum, the formula of which will be given below. As a result of observations of similar mechanical systems, the following patterns can be established:

If, while maintaining the same length of the pendulum, we suspend different weights, then the period of their oscillations will be the same, although their masses will vary greatly. Consequently, the period of such a pendulum does not depend on the mass of the load.

If, when starting the system, the pendulum is deflected at not too large, but different angles, then it will begin to oscillate with the same period, but with different amplitudes. As long as the deviations from the center of equilibrium are not too large, the vibrations in their form will be quite close to harmonic ones. The period of such a pendulum does not depend in any way on the oscillatory amplitude. This property of a given mechanical system is called isochronism (translated from Greek “chronos” - time, “isos” - equal).

Period of a mathematical pendulum

This indicator represents the period Despite the complex formulation, the process itself is very simple. If the length of the thread of a mathematical pendulum is L, and the acceleration of free fall is g, then this value is equal to:

The period of small natural oscillations does not depend in any way on the mass of the pendulum and the amplitude of oscillations. In this case, the pendulum moves as a mathematical one 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 moment t, expressed in radians); ω is a positive constant, which is determined from the parameters of the pendulum (ω = √g/L, where g is the acceleration of free fall, and L is the length of the mathematical pendulum (suspension).

The equation for small vibrations near the equilibrium position (harmonic equation) looks like this:

x + ω2 sin x = 0

Oscillatory movements of a pendulum

A mathematical pendulum, which makes small oscillations, moves along a sinusoid. The second order differential equation meets all the requirements and parameters of such a movement. To determine the trajectory, it is necessary to set the speed and coordinate, from which independent constants are then determined:

x = 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 oscillates with a significant amplitude, is subject to more complex laws of motion. For such a pendulum they are calculated according to the formula:

sin x/2 = u * sn(ωt/u),

where sn is the Jacobi sine, which for u< 1 является периодической функцией, а при малых u он совпадает с простым тригонометрическим синусом. Значение u определяют следующим выражением:

u = (ε + ω2)/2ω2,

where ε = E/mL2 (mL2 is the energy of the pendulum).

The period of oscillation of a nonlinear pendulum is determined using the formula:

where Ω = π/2 * ω/2K(u), K is the elliptic integral, π - 3,14.

Movement of a pendulum along a separatrix

A separatrix is ​​the trajectory of a dynamical system that has a two-dimensional phase space. A mathematical pendulum moves along it non-periodically. At an infinitely distant moment in time, it falls from its highest position to the side with zero speed, then gradually gains it. It eventually stops, returning to its original position.

If the amplitude of the pendulum's oscillations approaches the number π , this indicates that the motion on the phase plane is approaching the separatrix. In this case, under the influence of a small driving periodic force, the mechanical system exhibits chaotic behavior.

When a 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 direction opposite to the deflection of the pendulum. When denoting by x the displacement of the pendulum along a circular arc with radius L, its angular displacement is equal to φ = x/L. The second law, intended for projections and force, will give the desired value:

mg τ = Fτ = -mg sin x/L

Based on this relationship, it is clear that this pendulum is a nonlinear system, since the force that tends to return it to the equilibrium position is always proportional not to the displacement x, but to sin x/L.

Only when a mathematical pendulum performs small oscillations is it a harmonic oscillator. In other words, it becomes a mechanical system capable of performing harmonic oscillations. This approximation is practically valid for angles of 15-20°. Oscillations of a pendulum with large amplitudes are not harmonic.

Newton's law for small oscillations of a pendulum

If a given mechanical system performs small oscillations, Newton's 2nd law will look like this:

mg τ = Fτ = -m* g/L* x.

Based on this, we can conclude that a 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 coefficient 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. It should be taken into account that the pendulum in the gravitational field 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, take the derivative of the right and left sides of the equation:

Since the derivative of constant quantities equals 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 + m α) = 0.

Based on the last formula, we find: α = - g/L*x.

Practical application of a mathematical pendulum

Acceleration varies with latitude because the density of the Earth's crust is not the same throughout the planet. Where rocks with higher density occur, it will be slightly higher. The acceleration of a mathematical pendulum is often used for geological exploration. It is used to search for various minerals. Simply by counting the number of oscillations of a pendulum, one can detect coal or ore in the bowels of the Earth. This is due to the fact that such fossils have a density and mass greater than the underlying loose rocks.

The mathematical pendulum was used by such outstanding 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 K. Flammarion also used a mathematical pendulum for his research. He claimed that with its help he was able to predict the discovery of a new planet, the appearance of the Tunguska meteorite and other important events. During the Second World War, a specialized Pendulum Institute operated in Germany (Berlin). Nowadays, the Munich Institute of Parapsychology is engaged in similar research. The employees of this establishment call their work with the pendulum “radiesthesia.”

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

Just as the hand and brain of not all people have a mysterious power, the pendulum, too, in the hands of not all people can become mysterious. 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, by chance, they encounter an expert, and he shows him his destined path in life.
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.
The pendulum was and is still used by almost all famous people in Mesopotamia, Assyria, Urartu, India, China, Japan, 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 in various fields of science have not yet sufficiently appreciated the action and purpose of the pendulum in favor of the coexistence of humanity with the surrounding nature symbiotically and harmoniously. Humanity has not yet completely abandoned pseudoscientific views on the universe of the Universal Normal at the level of 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 take its rightful 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 it not only to be studied, but even to be studied in educational institutions.
So now, at the level of the highest echelon of 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.

Mathematical pendulum.

A mathematical pendulum is a material point suspended on an inextensible weightless thread, performing oscillatory motion in one vertical plane under the influence 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 greater 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 force F, one of the components of weight P, it will oscillate. The other component, directed along the thread, is not taken into account, because is balanced by the tension of the thread. At small displacement angles and, then the x coordinate can be measured in the horizontal direction. From Fig. 7.3 it is clear that the weight component perpendicular to the thread is equal to

Moment of force relative to point O: , and moment of inertia:
M=FL .
Moment of inertia J in this case
Angular acceleration:

Taking these values ​​into account, we have:

(7.8)

His decision
,

As we 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 which oscillates about this axis under the influence of gravity. Unlike a mathematical pendulum, the mass of such a body cannot be considered pointlike.

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 - 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 of 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 oscillations of a physical pendulum has the form: