Pendulum Foucault- a pendulum, which is used to experimentally demonstrate the daily rotation of the Earth.
The Foucault pendulum is a massive weight suspended on a wire or thread, the upper end of which is reinforced (for example, with a cardan joint) so that it allows the pendulum to swing in any vertical plane. If the Foucault pendulum is deflected from the vertical and released without initial speed, then the forces of gravity and tension of the thread acting on the weight of the pendulum will lie all the time in the plane of the swings of the pendulum and will not be able to cause its rotation with respect to the stars (to the inertial frame of reference associated with the stars). An observer who is on the Earth and rotates with it (i.e., located in a non-inertial frame of reference) will see that the swing plane of the Foucault pendulum slowly rotates relative to earth's surface in the direction opposite to the direction of the Earth's rotation. This confirms the fact of the daily rotation of the Earth.
At the North or South Pole, the swing plane of the Foucault pendulum will rotate 360° per sidereal day (15 o per sidereal day). finest hour). At a point on the earth's surface, geographical latitude which is equal to φ, the horizon plane rotates around the vertical with an angular velocity of ω 1 = ω sinφ (ω is the Earth's angular velocity modulus) and the pendulum swing plane rotates with the same angular velocity. Therefore, the apparent angular velocity of rotation of the plane of swing of the Foucault pendulum at latitude φ, expressed in degrees per sidereal hour, has the value rotates). AT southern hemisphere the rotation of the rocking plane will be observed in the direction opposite to that observed in the Northern Hemisphere. The refined calculation gives the value
ω m = 15 o sinφ
where a- the amplitude of oscillations of the pendulum weight, l- thread length. The additional term, which reduces the angular velocity, the less, the more l. Therefore, to demonstrate the experience, it is advisable to use the Foucault pendulum with the largest possible length of the thread (several tens of meters).
Story
For the first time this device was designed by the French scientist Jean Bernard Leon Foucault.
This device was a five-kilogram brass ball suspended from the ceiling on a two-meter steel wire.
Foucault's first experience was in the basement of his own house. January 8, 1851. This was recorded in the scientist's scientific diary.
February 3, 1851 Jean Foucault demonstrated his pendulum at the Paris Observatory to academicians who received letters like this: "I invite you to follow the rotation of the Earth."
The first public demonstration of the experience took place at the initiative of Louis Bonaparte in the Paris Panthéon in April of that year. A metal ball was suspended under the dome of the Pantheon. weighing 28 kg with a point fixed on it on a steel wire 1.4 mm in diameter and 67 m long. pendulum allowed him to freely oscillate in all directions. Under the attachment point was made a circular fence with a diameter of 6 meters, along the edge of the fence a sand path was poured in such a way that the pendulum in its movement could draw marks on the sand when crossing it. To avoid a lateral push when starting the pendulum, he was taken aside and tied with a rope, after which the rope burned out. The oscillation period was 16 seconds.
The experiment had big success and caused a wide resonance in the scientific and public circles of France and other countries of the world. Only in 1851 were other pendulums created on the model of the first, and Foucault's experiments were carried out at the Paris Observatory, in the Cathedral of Reims, in the church of St. Ignatius in Rome, in Liverpool, in Oxford, Dublin, in Rio de Janeiro, in city of Colombo in Ceylon, New York.
In all these experiments, the dimensions of the ball and the length of the pendulum were different, but they all confirmed the conclusionsJean Bernard Leon Foucault.
Elements of the pendulum, which was demonstrated in the Pantheon, are now kept in the Paris Museum of Arts and Crafts. And Foucault's pendulums are now in many parts of the world: in polytechnic and natural history museums, scientific observatories, planetariums, university laboratories and libraries.
There are three Foucault pendulums in Ukraine. One is kept in the National technical university Ukraine "KPI im. Igor Sikorsky", the second - in Kharkov national university them. V.N. Karazin, the third - at the Kharkiv Planetarium.
The pendulums shown in fig. 2, are extended bodies various shapes and dimensions that oscillate around the 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 the 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., determine the nature of its movement (oscillation). We will now consider the dynamics of oscillations in more detail using the simplest example of the 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 us now look under the influence of what forces our pendulum oscillates after it is brought out of equilibrium in some way (by push, deflection).
When the pendulum is at rest in the equilibrium position, the force of gravity acting on its weight and directed vertically downwards is balanced by the tension in the thread. In the deflected position (Fig. 15), gravity acts at an angle to the tension force directed along the thread. We decompose 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 value of the centripetal force, which causes the load to move in an arc. The component is always directed towards the equilibrium position; she seems to be striving to restore this position. Therefore, it is often called the restoring force. The modulus is greater, the more the pendulum is deflected.
Rice. 15. The 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 that slows down its movement the more, the farther it is deflected. Ultimately, this force will stop him and drag 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 turn to 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 an increase in the deviation, but now directed to the right. The movement to the left will slow down again, 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 swings?
Twice during the period - at the largest deviations to the left and to the right - the pendulum stops, that is, at these moments the speed is zero, which means that the kinetic energy is also zero. But it is precisely at these moments that the center of gravity of the pendulum is raised to the greatest height and, consequently, the potential energy is greatest. On the contrary, at the moments of passage through the equilibrium position, the potential energy is the smallest, and the speed and kinetic energy reach the maximum value.
We assume that the forces of friction of the pendulum on the air and the friction at the point of suspension can be neglected. Then, according to the law of conservation of energy, this maximum kinetic energy is just equal to the excess potential energy in the position of the greatest deviation above potential energy in a position of balance.
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 potential and kinetic energies) is constant all the time. It is equal to the energy that was imparted to the pendulum at the start, whether it is in the form of potential energy (initial deflection) or kinetic energy (initial push).
This is the case for all vibrations in the absence of friction or any other processes that take energy from the oscillating system or impart energy to it. That is why the amplitude remains unchanged and is determined by the initial deviation or the force of the push.
We get the same changes in the restoring force and the same transition 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 trough curved around the circumference. In this case, the role of the thread tension will be assumed by the pressure of the walls of the cup or trough (again, we neglect the friction of the ball against the walls and air).
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 brain do not have a mysterious power in all people, so the pendulum 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 points out to 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 bursting pains in the head in the crown area. 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 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 which channel the pendulum’s actions are directed to. this 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.
The pendulum was used and is still used by almost all famous people in Mesopotamia, Assyria, Urartu, India, China, Japan, in ancient rome, Egypt, Greece, Asia, Africa, America, Europe, the East and all over the world many countries.
Because many prominent international institutions, prominent figures different areas sciences 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. Still 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 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 effects.
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 engage in educational institutions.
And now at the level of the highest echelon modern science, they look behind 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, anyone 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.
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 acting on it and the force of elasticity F?ynp of the thread are mutually compensated.
We bring the pendulum out of the equilibrium position (deflecting it, for example, to position A) and let it go without initial velocity (Fig. 1). In this case, the forces and do not balance each other. The tangential component of gravity, acting on the pendulum, gives it a tangential acceleration a?? (component full 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 the restoring force. The normal component of gravity is directed along the thread against the elastic force. The resultant force and tells the pendulum normal acceleration, which changes the direction of the velocity vector, and the pendulum moves along the arc ABCD.
The closer the pendulum approaches 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 upward along the arc. In this case, the component is directed against the speed. With an increase in the deflection angle a, the modulus of force increases, and the modulus of speed decreases, and at point D the speed of the pendulum becomes zero. The pendulum stops for a moment and then begins to move in reverse 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|). Let us denote the length of the suspension thread as l, and the mass of the pendulum as m.
Figure 1 shows that , where . At small angles () pendulum deflection, therefore
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. We 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 equation of harmonic oscillations 
, we can conclude that the mathematical pendulum makes harmonic oscillations. And since the considered oscillations of the pendulum occurred under the action of only internal forces, then these were free oscillations of the pendulum. Consequently, free oscillations of a mathematical pendulum with small deviations are harmonic.
Denote
Cyclic frequency of pendulum oscillations.
The period of oscillation of the pendulum. Hence,
This expression is called the 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 the mathematical pendulum:
- does not depend on its mass and amplitude of oscillations;
- proportional to the square root of the length of the pendulum and inversely proportional to the square root of the acceleration free fall.
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 when two conditions are met simultaneously:
- pendulum oscillations should be small;
- the suspension point of the pendulum must be at rest or move uniformly rectilinearly 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 oscillation. As calculations show, the period of oscillation of the pendulum in this case can be calculated by the formula
where is the "effective" acceleration of the pendulum in a non-inertial frame of reference. It equals geometric sum gravitational acceleration and the vector opposite to the vector , i.e. it can be calculated using the formula
Mathematical pendulum called a material point suspended on a weightless and inextensible thread attached to a 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.