Statistical mechanics

Scientists
Galileo Kepler Newton Euler Laplace d'Alembert Lagrange Hamilton Cauchy

See also: Portal:Physics

Formulas

Usually the concept of centrifugal force is used within the framework of classical (Newtonian) mechanics, which is the subject of the main part of this article (although a generalization of this concept can be quite easily obtained for relativistic mechanics in some cases).

A-priory, centrifugal force is called the force of inertia (that is, in the general case - part of the total inertial force) in a non-inertial frame of reference, which does not depend on the speed of the material point in this frame of reference, and also does not depend on the accelerations (linear or angular) of this frame of reference relative to the inertial frame reference.

For a material point, the centrifugal force is expressed by the formula:

$\backslash vec(F)=-m\; \backslash left[\; \backslash vec\; \backslash omega\; \backslash times\; \backslash left[\; \backslash vec\; \backslash omega\; \backslash times\; \backslash vec\; R\; \backslash right]\; \backslash right]\; =\; m\; \backslash left(\backslash omega^2\; \backslash vec\; R\; -\; \backslash left\; (\backslash vec\; \backslash omega\; \backslash cdot\; \backslash vec\; R\; \backslash right)\; \backslash vec\; \backslash omega\; \backslash right)\; ,$

$\backslash vec(F)$- centrifugal force applied to the body, $\backslash m$- body mass, $\backslash vec(\backslash omega)$- angular velocity of rotation of the non-inertial reference frame relative to the inertial one (the direction of the angular velocity vector is determined by the gimlet rule), $\backslash vec(R)$- radius-vector of the body in the rotating coordinate system.

The equivalent expression for the centrifugal force can be written as

$\backslash vec(F)=\; m\; \backslash omega^2\; \backslash vec(R\_0)$

if we use the notation $\backslash vec(R\_0)$ for a vector perpendicular to the axis of rotation and drawn from it to a given material point.

The centrifugal force for bodies of finite dimensions can be calculated (as is usually done for any other force) by summing the centrifugal forces acting on material points, which are elements into which we mentally divide the final body.

Conclusion

There is also a completely different understanding of the term "centrifugal force" in the literature. This is sometimes called a real force applied not to a rotating body, but acting from the side of the body on the constraints limiting its movement. In the example discussed above, this would be the name of the force acting from the side of the ball on the spring. (See, for example, the link to TSB below.)

Centrifugal force as a real force

Applied not to connections, but, on the contrary, to a body being rotated as an object of its influence, the term "centrifugal force" (literally, the force applied to a turning or rotating material body, causing it to run away from the instantaneous center of rotation), is a euphemism based on a false interpretation of the first law (Newton's principle) in the form:

Every body resists change in its state of rest or uniform rectilinear motion under the action of external force

Every body seeks maintain a state of rest or uniform rectilinear motion until an external force acts.

An echo of this tradition is the idea of ​​a certain strength, as a material factor that realizes this resistance or aspiration. It would be appropriate to talk about the existence of such a force if, for example, contrary to the acting forces, the moving body would maintain its speed, but this is not so.

The use of the term "centrifugal force" is valid when the point of its application is not a body that experiences rotation, but a connection that limits its movement. In this sense, the centrifugal force is one of the terms in the formulation of Newton's third law, the antagonist to the centripetal force that causes the rotation of the body in question and is applied to it. Both these forces are equal in magnitude and opposite in direction, but are applied to different bodies and therefore do not compensate each other, but cause a really tangible effect - a change in the direction of movement of the body (material point).

Remaining in an inertial frame of reference, consider two celestial bodies, for example, the component of a binary star with masses of the same order of magnitude $(M\_1)$ And $(M\_2)$ located at a distance $R$ from each other. In the adopted model, these stars are considered as material points and $R$ is the distance between their centers of mass. The force of universal gravitation acts as a connection between these bodies. $(F\_G):\; (G\; M\_1\; M\_2\; /R^2)$, Where $G$ is the gravitational constant. This is the only acting force here, it causes the accelerated movement of bodies towards each other.

However, in the event that each of these bodies rotates around a common center of mass with linear velocities $(v\_1)$ = $(\backslash omega)\_1$ $(R\_1)$ And $(v\_2)$ = $(\backslash omega\_2)$ $(R\_2)$, then such a dynamic system will retain its configuration indefinitely if the angular velocities of rotation of these bodies are equal: $(\backslash omega\_1)$ = $(\backslash omega\_2)$ = $\backslash omega$, and the distances from the center of rotation (center of mass) will be related as: $(\; M\_1/M\_2\; )$ = $(R\_2/R\_1)$, moreover $(R\_2)\; +\; (R\_1)\; =\; R$, which directly follows from the equality active forces: $(F\_1)\; =\; (M\_1)(a\_1)$ And $(F\_2)\; =\; (M\_2)(\; a\_2\; )$, where the accelerations are respectively: $(a\_1\; )$= $(\backslash omega^2)(R\_1)$ And $(a\_2)\; =\; (\backslash omega^2)(\; R\_2)$ .

The centripetal forces causing the movement of bodies along circular trajectories are equal (modulo): $(F\_1)$ =$(F\_2)$ $=\; (F\_G)$. Moreover, the first of them is centripetal, and the second is centrifugal, and vice versa: each of the forces, in accordance with the Third Law, is both one and the other.

Therefore, strictly speaking, the use of each of the discussed terms is redundant, since they do not denote any new forces, being synonymous with the only force - the force of gravity. The same is true for the operation of any of the links mentioned above.

However, as the ratio between the considered masses changes, that is, as the discrepancy in the motion of the bodies possessing these masses becomes more and more significant, the difference in the results of the action of each of the considered bodies for the observer becomes more and more significant.

In a number of cases, the observer identifies himself with one of the participating bodies, and therefore it becomes motionless for him. In this case, with such a large violation of symmetry in relation to the observed picture, one of these forces turns out to be uninteresting, since it practically does not cause movement.

see also

Write a review on the article "Centrifugal force"

Notes

1. Outside the context of physics/mechanics/mathematics, for example, in philosophy, journalism or fiction, and also sometimes in colloquial speech, words centrifugal force can often be used simply as a designation of some influence directed away from some "center"; in this usage, it may not only have nothing to do with any rotation, but also with the concept of force, as it is used in physics.
2. S. E. Khaikin. Forces of inertia and weightlessness. M., 1967. Publishing house "Science". The main edition of physical and mathematical literature.
3. Let's use the centripetal acceleration formula.
4. Physical Encyclopedia, v.4 - M.: Great Russian Encyclopedia and
5. Newton I. Mathematical principles of natural philosophy. Per. and approx. A. N. Krylova. Moscow: Nauka, 1989
6. The key to this formulation is the assertion that the objects of the material world have certain volitional qualities, which was at the beginning of the formation of scientific ideas about the world around us a very common way of generalizing the results of observing natural phenomena and clarifying the general patterns inherent in it. An example of such an animalistic conception of nature was the principle that existed in natural philosophy: “Nature is afraid of emptiness”, which had to be abandoned after the Torricelli experiment (Torricelli's emptiness)
7. In this regard, Maxwell observed that one could just as well say that coffee resists becoming sweet, referring to the fact that it does not become sweet on its own, but only after sugar is put into it. .
8. S. E. Khaikin. Forces of inertia and weightlessness. M.: "Science", 1967
9. In this case, at each small moment of time, each of the bodies will approach the center at such a distance, which is equal to the difference in the distances between its trajectory and the tangent at the point of observation. In other words, the bodies fall on each other, but always miss.

Links

• Matveev A. N. Mechanics and Theory of Relativity: Textbook for university students. - 3rd edition. - M .: OOO " Publishing House"ONIX 21st century": LLC "Publishing House" World and Education ", 2003. - p. 405-406

An excerpt characterizing centrifugal force

– Do you know how? Natasha asked. Uncle smiled without answering.
- Look, Anisyushka, that the strings are intact, or something, on the guitar? I haven’t taken it in my hands for a long time - it’s a pure march! abandoned.
Anisya Fyodorovna willingly went with her light step to carry out her master's order and brought the guitar.
Uncle, without looking at anyone, blew off the dust, tapped the lid of the guitar with his bony fingers, tuned it, and straightened himself in his chair. He took (with a somewhat theatrical gesture, leaving the elbow of his left hand) the guitar above the neck and, winking at Anisya Fyodorovna, began not the Lady, but took one sonorous, clear chord, and measured, calmly, but firmly began to finish the well-known song at a very quiet pace: and ice pavement. At the same time, in time with that sedate joy (the same that Anisya Fyodorovna's whole being breathed), the motive of the song sang in the soul of Nikolai and Natasha. Anisya Fyodorovna blushed and, covering herself with a handkerchief, laughingly left the room. Uncle continued to cleanly, diligently and energetically firmly finish the song, looking with a changed inspired look at the place from which Anisya Fyodorovna had left. A little bit something laughed in his face on one side under a gray mustache, especially laughed when the song dispersed further, the beat accelerated and something came off in places of busts.
- Charm, charm, uncle; more, more, ”Natasha shouted as soon as he finished. She jumped up from her seat, hugged her uncle and kissed him. - Nikolenka, Nikolenka! she said, looking round at her brother and as if asking him: what is this?
Nikolai also really liked the uncle's game. Uncle played the song a second time. The smiling face of Anisya Fyodorovna appeared again at the door, and from behind her there were still other faces ... "Behind the cold key, she shouts: wait a girl!" my uncle played, again made a deft enumeration, tore it off and moved his shoulders.
“Well, well, my dear, uncle,” Natasha groaned in such an imploring voice, as if her life depended on it. Uncle stood up and as if there were two people in him - one of them smiled seriously at the merry fellow, and the merry fellow made a naive and neat trick before the dance.
- Well, niece! - shouted the uncle, waving his hand to Natasha, tearing off the chord.
Natasha threw off the handkerchief that was thrown over her, ran ahead of her uncle and, propping her hands on her hips, made a movement with her shoulders and stood.
Where, how, when she sucked into herself from that Russian air that she breathed - this countess, brought up by a French emigrant, this spirit, where did she get these techniques that pas de chale should long ago have been forced out? But these spirits and methods were the same, inimitable, not studied, Russian, which her uncle expected from her. As soon as she stood up, she smiled solemnly, proudly and cunningly cheerfully, the first fear that gripped Nikolai and all those present, the fear that she would do something wrong, passed and they were already admiring her.
She did the same thing and did it so exactly, so quite exactly, that Anisya Fyodorovna, who immediately handed her the handkerchief necessary for her work, burst into tears through laughter, looking at this thin, graceful, so alien to her, educated countess in silk and velvet. who knew how to understand everything that was in Anisya, and in Anisya's father, and in her aunt, and in her mother, and in every Russian person.
“Well, the countess is a pure march,” said the uncle, laughing joyfully, having finished the dance. - Oh yes, niece! If only you could choose a good fellow for you, - march is a clean business!
“Already chosen,” said Nikolai smiling.
- ABOUT? said the uncle in surprise, looking inquiringly at Natasha. Natasha nodded her head in the affirmative with a happy smile.
- Another one! - she said. But as soon as she said it, another new system thoughts and feelings rose up in her. What did Nikolai's smile mean when he said: "already chosen"? Is he happy about it or not? He seems to think that my Bolkonsky would not have approved, would not have understood our joy. No, he would understand. Where is he now? thought Natasha, and her face suddenly became serious. But it only lasted for one second. “Don’t think about it, don’t dare to think about it,” she said to herself, and smiling, she sat down again with her uncle, asking him to play something else.
Uncle played another song and a waltz; then, after a pause, he cleared his throat and sang his favorite hunting song.
Like powder from the evening
Turned out good...
Uncle sang the way the people sing, with that complete and naive conviction that in a song all meaning lies only in the words, that the melody comes by itself and that there is no separate melody, but that the melody is only for the warehouse. Because of this, this unconscious tune, like the song of a bird, was unusually good with my uncle. Natasha was delighted with her uncle's singing. She decided that she would no longer study the harp, but would only play the guitar. She asked her uncle for a guitar and immediately picked up the chords for the song.
At ten o'clock a line, a droshky, and three riders arrived for Natasha and Petya, sent to look for them. The count and countess did not know where they were and were very worried, as the messenger said.
Petya was taken down and laid like a dead body in a ruler; Natasha and Nikolai got into the droshky. Uncle wrapped up Natasha and said goodbye to her with a completely new tenderness. He escorted them on foot to the bridge, which had to be bypassed into a ford, and ordered the hunters to go ahead with lanterns.
“Farewell, dear niece,” his voice shouted out of the darkness, not the one that Natasha had known before, but the one that sang: “Like powder since the evening.”
The village we passed had red lights and a cheerful smell of smoke.
- What a charm this uncle is! - said Natasha, when they drove out onto the main road.
“Yes,” said Nikolai. - Are you cold?
- No, I'm fine, fine. I feel so good, - Natasha even said with bewilderment. They were silent for a long time.
The night was dark and damp. The horses were not visible; all you could hear was their paddling through the invisible mud.
What was going on in this childish, receptive soul, which so greedily caught and assimilated all the most diverse impressions of life? How did it fit into her? But she was very happy. Already approaching the house, she suddenly sang the motive of the song: “Like powder from the evening,” a motive that she caught all the way and finally caught.
- Got it? Nikolay said.
“What are you thinking now, Nikolenka?” Natasha asked. They liked to ask each other that.
- I? - said Nikolai remembering; - you see, at first I thought that Rugai, the red male, looked like an uncle and that if he were a man, he would still keep the uncle with him, if not for the jump, then for the frets, he would keep everything. How good he is, uncle! Is not it? - Well, what about you?
- I? Hold on, hold on. Yes, at first I thought that here we are going and we think that we are going home, and God knows where we are going in this darkness and suddenly we will arrive and see that we are not in Otradnoye, but in a magical kingdom. And then I thought… No, nothing more.
“I know, I was thinking about him right,” Nikolai said smiling, as Natasha recognized by the sound of his voice.
“No,” answered Natasha, although at the same time she really thought both about Prince Andrei and about how he would like his uncle. “And I also repeat everything, I repeat all the way: how Anisyushka performed well, well ...” said Natasha. And Nikolai heard her sonorous, causeless, happy laughter.
“You know,” she said suddenly, “I know that I will never be as happy and calm as I am now.
“That’s nonsense, nonsense, lies,” said Nikolai and thought: “What a charm this Natasha of mine is! I don't have another friend like him and never will. Why should she get married, everyone would go with her!
“What a charm this Nikolai is!” thought Natasha. - A! there’s still a fire in the living room,” she said, pointing to the windows of the house, which shone beautifully in the wet, velvet darkness of the night.

Count Ilya Andreich resigned from the leaders because this post was too expensive. But things didn't get better for him. Often Natasha and Nikolai saw the secret, restless negotiations of their parents and heard rumors about the sale of a rich, ancestral Rostov house and a suburban one. Without leadership, it was not necessary to have such a large reception, and the life of congratulations was conducted more quietly than in previous years; but the huge house and outbuilding were still full of people, more people were still sitting at the table. All of these were people who had settled down in the house, almost members of the family, or those who, it seemed, had to live in the count's house. These were Dimmler - a musician with his wife, Yogel - a dance teacher with his family, the old lady Belova, who lived in the house, and many others: Petya's teachers, the former governess of young ladies and just people who were better or more profitable to live with the count than at home. There was no such big visit as before, but the course of life was the same, without which the count and countess could not imagine life. There was the same, still increased by Nikolai, hunting, the same 50 horses and 15 coachmen at the stable, the same expensive gifts on name days, and solemn dinners for the whole county; the same count whists and bostons, behind which he, dissolving cards for everyone to see, allowed himself to be beaten every day by hundreds of neighbors who looked at the right to play the game of Count Ilya Andreich as the most profitable lease.
The count, as if in huge snares, went about his affairs, trying not to believe that he was entangled, and with each step he became more and more entangled and feeling himself unable either to break the nets that entangled him, or carefully, patiently begin to unravel them. The Countess, with a loving heart, felt that her children were going bankrupt, that the count was not to blame, that he could not be different from what he was, that he himself was suffering (although he hides it) from the consciousness of his and his children's ruin, and was looking for means to help the cause. From her feminine point of view, there was only one way - the marriage of Nicholas to a rich bride. She felt that this was the last hope, and that if Nikolai refused the party that she had found for him, she would have to say goodbye forever to the opportunity to improve things. This party was Julie Karagina, the daughter of a beautiful, virtuous mother and father, known from childhood to Rostov, and now a rich bride on the occasion of the death of the last of her brothers.
The Countess wrote directly to Karagina in Moscow, offering her the marriage of her daughter to her son, and received a favorable response from her. Karagina replied that she, for her part, agreed that everything would depend on the inclination of her daughter. Karagina invited Nikolai to come to Moscow.
Several times, with tears in her eyes, the Countess told her son that now that both her daughters were added, her only wish was to see him married. She said that she would lie down in the coffin calm, if that were the case. Then she said that she had a beautiful girl in mind and elicited his opinion about marriage.
In other conversations, she praised Julie and advised Nikolai to go to Moscow for the holidays to have fun. Nikolai guessed what his mother's conversations were leading to, and in one of these conversations he called her to complete frankness. She told him that all the hope of getting things right was now based on his marriage to Karagina.
- Well, if I loved a girl without a fortune, would you really demand, maman, that I sacrifice feeling and honor for a fortune? he asked his mother, not understanding the cruelty of his question and wishing only to show his nobility.
“No, you didn’t understand me,” said the mother, not knowing how to justify herself. “You didn’t understand me, Nikolinka. I wish you happiness,” she added, and felt that she was telling a lie, that she was confused. She started crying.
“Mamma, don’t cry, but just tell me that you want it, and you know that I will give my whole life, I will give everything so that you are calm,” said Nikolai. I will sacrifice everything for you, even my feelings.
But the countess did not want to put the question that way: she did not want a sacrifice from her son, she herself would like to sacrifice to him.
“No, you didn’t understand me, let’s not talk,” she said, wiping her tears.
“Yes, maybe I love the poor girl,” Nikolai said to himself, well, should I sacrifice feeling and honor for the state? I wonder how my mother could tell me this. Because Sonya is poor, I can’t love her, he thought, I can’t respond to her true, devoted love. And I'll probably be happier with her than with some sort of Julie doll. I can always sacrifice my feelings for the good of my relatives, he said to himself, but I cannot command my feelings. If I love Sonya, then my feeling is stronger and higher than anything for me.
Nikolai did not go to Moscow, the countess did not resume the conversation with him about marriage, and with sadness, and sometimes with anger, she saw signs of an ever greater rapprochement between her son and the dowry Sonya. She reproached herself for that, but she could not help but grumble, find fault with Sonya, often stopping her for no reason, calling her "you" and "my dear." Most of all, the kind countess was angry with Sonya because this poor, black-eyed niece was so meek, so kind, so devotedly grateful to her benefactors, and so faithfully, unfailingly, selflessly in love with Nikolai, that it was impossible to reproach her for anything. .
Nikolai spent his vacation with his relatives. The 4th letter was received from the fiancé Prince Andrei, from Rome, in which he wrote that he would have been on his way to Russia long ago if his wound had not suddenly opened in a warm climate, which makes him postpone his departure until the beginning of next year . Natasha was just as in love with her fiancé, just as reassured by this love, and just as receptive to all the joys of life; but at the end of the fourth month of separation from him, moments of sadness began to come over her, against which she could not fight. She felt sorry for herself, it was a pity that she had been lost for nothing, for no one, all this time, during which she felt herself so capable of loving and being loved.
It was sad in the Rostovs' house.

Christmas time came, and apart from the ceremonial mass, except for the solemn and boring congratulations from neighbors and courtyards, except for all the new dresses put on, there was nothing special commemorating Christmas time, but in a windless 20 degree frost, in a bright blinding sun during the day and in starry winter light at night, the need for some kind of commemoration of this time was felt.

To calculate the acceleration of bodies through the balance of forces.

This is often convenient. For example, when the entire laboratory rotates, it may be more convenient to consider all movements relative to it, introducing only additional inertia forces, including centrifugal, acting on all material points, than to take into account the constant change in the position of each point relative to the inertial frame of reference.

Often, especially in technical literature, they implicitly pass into a non-inertial frame of reference rotating with the body, and they speak of the manifestations of the law of inertia as a centrifugal force acting from the side of the moving along a circular path bodies on the bonds causing this rotation, and consider it, by definition, to be equal in absolute value to the centripetal force and always directed in the opposite direction to it.

However, in the general case, when the instantaneous center of rotation of the body along a circular arc, which approximates the trajectory at each of its points, may not coincide with the beginning of the force vector causing the movement, it is incorrect to call the force acting on the connection a centrifugal force. After all, there is also a component of the connection force directed tangentially to the trajectory, and this component will change the speed of the body along it. Therefore, some physicists generally avoid using the term "centrifugal force" as unnecessary.

Encyclopedic YouTube

• 1 / 5

  Usually the concept of centrifugal force is used within the framework of classical (Newtonian) mechanics, to which the main part of this article concerns (although a generalization of this concept can be quite easily obtained in some cases for relativistic mechanics).

  By definition, centrifugal force is the force of inertia (that is, in the general case, part of the total inertial force) in a non-inertial frame of reference, which does not depend on the speed of a material point in this frame of reference, and also does not depend on the accelerations (linear or angular) of this frame itself. frames of reference relative to the inertial frame of reference.

  For a material point, the centrifugal force is expressed by the formula:

  F → = − m [ ω → × [ ω → × R → ] ] = m (ω 2 R → − (ω → ⋅ R →) ω →) , (\displaystyle (\vec (F))=-m\ left[(\vec (\omega ))\times \left[(\vec (\omega ))\times (\vec (R))\right]\right]=m\left(\omega ^(2)( \vec (R))-\left((\vec (\omega ))\cdot (\vec (R))\right)(\vec (\omega ))\right),) F → (\displaystyle (\vec (F)))- centrifugal force applied to the body, m(\displaystyle\m)- body mass, ω → (\displaystyle (\vec (\omega )))- angular speed of rotation of the non-inertial reference frame relative to the inertial one (the direction of the angular velocity vector is determined by the gimlet rule), R → (\displaystyle (\vec (R)))- radius-vector of the body in the rotating coordinate system.

  The equivalent expression for the centrifugal force can be written as

  F → = m ω 2 R 0 → (\displaystyle (\vec (F))=m\omega ^(2)(\vec (R_(0))))

  if we use the notation R 0 → (\displaystyle (\vec (R_(0)))) for a vector perpendicular to the axis of rotation and drawn from it to a given material point.

  The centrifugal force for bodies of finite dimensions can be calculated (as is usually done for any other forces) by summing up the centrifugal forces acting on material points, which are the elements into which we mentally divide the finite body.

  Conclusion

  There is also a completely different understanding of the term "centrifugal force" in the literature. This is sometimes called a real force applied not to a rotating body, but acting from the side of the body on the constraints limiting its movement. In the example discussed above, this would be the name of the force acting from the side of the ball on the spring. (See, for example, the link to TSB below.)

  Centrifugal force as a real force

  Applied not to connections, but, on the contrary, to a body being rotated as an object of its influence, the term "centrifugal force" (literally, the force applied to a turning or rotating material body, causing it to run away from the instantaneous center of rotation), is a euphemism based on a false interpretation of the first law (Newton's principle) in the form:

  Every body resists change in its state of rest or uniform rectilinear motion under the action of an external force

  Every body seeks maintain a state of rest or uniform rectilinear motion until an external force acts.

  An echo of this tradition is the idea of ​​a certain strength, as a material factor that realizes this resistance or aspiration. It would be appropriate to talk about the existence of such a force if, for example, contrary to the acting forces, the moving body would maintain its speed, but this is not so.

  The use of the term "centrifugal force" is valid when the point of its application is not a body that experiences rotation, but a connection that limits its movement. In this sense, the centrifugal force is one of the terms in the formulation of Newton's third law, the antagonist to the centripetal force that causes the rotation of the body in question and is applied to it. Both these forces are equal in magnitude and opposite in direction, but are applied to different bodies and therefore do not compensate each other, but cause a really tangible effect - a change in the direction of movement of the body (material point).

  Remaining in an inertial frame of reference, consider two celestial bodies, for example, a component of a binary star with masses of the same order of magnitude M 1 (\displaystyle (M_(1))) And M 2 (\displaystyle (M_(2))) located at a distance R (\displaystyle R) from each other. In the adopted model, these stars are considered as material points and R (\displaystyle R) is the distance between their centers of mass. The force of universal gravitation acts as a connection between these bodies. F G: G M 1 M 2 / R 2 (\displaystyle (F_(G)):(GM_(1)M_(2)/R^(2))), Where G (\displaystyle G) is the gravitational constant. This is the only acting force here, it causes the accelerated movement of bodies towards each other.

  However, in the event that each of these bodies rotates around a common center of mass with linear velocities v 1 (\displaystyle (v_(1))) = ω 1 (\displaystyle (\omega )_(1)) R 1 (\displaystyle (R_(1))) And v 2 (\displaystyle (v_(2))) = R 2 (\displaystyle (R_(2))), then such a dynamic system will retain its configuration indefinitely if the angular velocities of rotation of these bodies are equal: ω 1 (\displaystyle (\omega _(1))) = ω 2 (\displaystyle (\omega _(2))) = ω (\displaystyle \omega ), and the distances from the center of rotation (center of mass) will be related as: M 1 / M 2 (\displaystyle (M_(1)/M_(2))) = R 2 / R 1 (\displaystyle (R_(2)/R_(1))), moreover R 2 + R 1 = R (\displaystyle (R_(2))+(R_(1))=R), which directly follows from the equality of the acting forces: F 1 = M 1 a 1 (\displaystyle (F_(1))=(M_(1))(a_(1))) And F 2 = M 2 a 2 (\displaystyle (F_(2))=(M_(2))(a_(2))), where the accelerations are respectively: a 1 (\displaystyle (a_(1)))= ω 2 R 1 (\displaystyle (\omega ^(2))(R_(1))) And a 2 = ω 2 R 2 (\displaystyle (a_(2))=(\omega ^(2))(R_(2)))

  Formulas

  Usually the concept of centrifugal force is used within the framework of classical (Newtonian) mechanics, which is the subject of the main part of this article (although a generalization of this concept can be quite easily obtained for relativistic mechanics in some cases).

  By definition, centrifugal force is the force of inertia (that is, in the general case, part of the total inertial force) in a non-inertial frame of reference, which does not depend on the speed of the material point in this frame of reference, and also does not depend on accelerations (linear or angular) of this frame itself. frames of reference relative to the inertial frame of reference.

  For a material point, the centrifugal force is expressed by the formula:

  - centrifugal force applied to the body, - body mass, - angular velocity of rotation of the non-inertial reference frame relative to the inertial one (the direction of the angular velocity vector is determined by the gimlet rule), - the radius vector of the body in the rotating coordinate system.

  The equivalent expression for the centrifugal force can be written as

  if we use the notation for a vector perpendicular to the axis of rotation and drawn from it to a given material point.

  The centrifugal force for bodies of finite dimensions can be calculated (as is usually done for any other forces) by summing up the centrifugal forces acting on material points, which are the elements into which we mentally divide the finite body.

  Conclusion

  It should be borne in mind that in order to correctly describe the motion of bodies in rotating frames of reference, in addition to the centrifugal force, one should also introduce the Coriolis force.

  There is also a completely different understanding of the term "centrifugal force" in the literature. This is sometimes called a real force applied not to a rotating body, but acting from the side of the body on the constraints limiting its movement. In the example discussed above, this would be the name of the force acting from the side of the ball on the spring. (See, for example, the link to TSB below.)

  Centrifugal force as a real force

  

  Centripetal and centrifugal forces when bodies move along circular trajectories with a common axis of rotation

  Applied not to connections, but, on the contrary, to a body being rotated as an object of its influence, the term "centrifugal force" (literally, the force applied to a rotating or rotating material body, causing it to run away from the instantaneous center of rotation), is a euphemism based on a false interpretation of the first law (Newton's principle) in the form:

  Every body resists change in its state of rest or uniform rectilinear motion under the action of an external force

  Every body seeks maintain a state of rest or uniform rectilinear motion until an external force acts.

  An echo of this tradition is the idea of ​​a certain strength, as a material factor that realizes this resistance or aspiration. It would be appropriate to talk about the existence of such a force if, for example, contrary to the acting forces, the moving body would maintain its speed, but this is not so.

  The use of the term "centrifugal force" is valid when the point of its application is not a body that experiences rotation, but a connection that limits its movement. In this sense, the centrifugal force is one of the terms in the formulation of Newton's third law, the antagonist to the centripetal force that causes the rotation of the body in question and is applied to it. Both these forces are equal in magnitude and opposite in direction, but are applied to different bodies and therefore do not compensate each other, but cause a really tangible effect - a change in the direction of movement of the body (material point).

  Remaining in an inertial frame of reference, consider two celestial bodies, for example, a component of a binary star with masses of the same order of magnitude and , located at a distance from each other. In the adopted model, these stars are considered as material points and there is a distance between their centers of mass. The force of universal gravitation acts as a connection between these bodies, where is the gravitational constant. This is the only acting force here, it causes the accelerated movement of bodies towards each other.

  However, if each of these bodies rotates around a common center of mass with linear velocities = and = , then such a dynamic system will retain its configuration indefinitely if the angular velocities of rotation of these bodies are equal: = = , and the distances from the center of rotation (center of mass) will be related as: = , and , which directly follows from the equality of the acting forces: and , where the accelerations are respectively: = and .

  Centripetal forces that cause the movement of bodies along circular trajectories are equal (modulo): =. Moreover, the first of them is centripetal, and the second is centrifugal, and vice versa: each of the forces, in accordance with the Third Law, is both one and the other.

  Therefore, strictly speaking, the use of each of the discussed terms is redundant, since they do not denote any new forces, being synonymous with the only force - the force of gravity. The same is true for the operation of any of the links mentioned above.

  However, as the ratio between the considered masses changes, that is, as the discrepancy in the motion of the bodies possessing these masses becomes more and more significant, the difference in the results of the action of each of the considered bodies for the observer becomes more and more significant.

  In a number of cases, the observer identifies himself with one of the participating bodies, and therefore it becomes motionless for him. In this case, with such a large violation of symmetry in relation to the observed picture, one of these forces turns out to be uninteresting, since it practically does not cause movement.

  see also

  Notes

  Links

  • Matveev A. N. Mechanics and Theory of Relativity: Textbook for university students. - 3rd edition. - M .: LLC "Publishing House" ONYX 21st Century ": LLC "Publishing House" World and Education ", 2003. - p. 405-406

  Previously, the characteristics of rectilinear motion were considered: movement, speed, acceleration. Their counterparts in rotational motion are: angular displacement, angular velocity, angular acceleration.

  • The role of displacement in rotational motion is played by corner;
  • The angle of rotation per unit of time is angular velocity;
  • The change in angular velocity per unit of time is angular acceleration.

  During uniform rotational motion, the body moves in a circle with the same speed, but with a changing direction. For example, such a movement is made by the clock hands on the dial.

  Suppose a ball rotates uniformly on a thread 1 meter long. In doing so, it will describe a circle with a radius of 1 meter. The length of such a circle: C = 2πR = 6.28 m

  The time it takes for the ball to make one complete revolution around the circumference is called rotation period - T.

  To calculate the linear speed of the ball, it is necessary to divide the displacement by the time, i.e. circumference per rotation period:

  V = C/T = 2πR/T

  Rotation period:

  T = 2πR/V

  If our ball makes one revolution in 1 second (rotation period = 1s), then its linear speed:
  V = 6.28/1 = 6.28 m/s

  2. Centrifugal acceleration

  At any point of the rotational motion of the ball, the vector of its linear velocity is directed perpendicular to the radius. It is easy to guess that with such a rotation around the circle, the linear velocity vector of the ball is constantly changing its direction. The acceleration that characterizes such a change in speed is called centrifugal (centripetal) acceleration.

  During uniform rotational motion, only the direction of the velocity vector changes, but not the magnitude! So the linear acceleration = 0 . The change in linear speed is supported by centrifugal acceleration, which is directed to the center of the circle of rotation perpendicular to the velocity vector - a c.

  Centrifugal acceleration can be calculated using the formula: a c \u003d V 2 / R

  The greater the linear velocity of the body and the smaller the radius of rotation, the greater the centrifugal acceleration.

  3. Centrifugal force

  From rectilinear motion, we know that the force is equal to the product of the body's mass and its acceleration.

  With uniform rotational motion, a centrifugal force acts on a rotating body:

  F c \u003d ma c \u003d mV 2 / R

  If our ball weighs 1 kg, then to keep it on a circle, centrifugal force is required:

  F c \u003d 1 6.28 2 / 1 \u003d 39.4 N

  We encounter centrifugal force in Everyday life at any turn.

  The friction force must balance the centrifugal force:

  Fc \u003d mV 2 /R; F tr \u003d μmg

  F c \u003d F tr; mV 2 /R = μmg

  V = √μmgR/m = √μgR = √0.9 9.8 30 = 16.3 m/s = 58.5 km/h

  Answer: 58.5 km/h

  Please note that the speed in the turn does not depend on body weight!

  Surely you have noticed that some turns on the highway have some inclination into the turn. Such turns are "easier" to pass, or rather, you can pass at a greater speed. Consider what forces act on the car in such a turn with an inclination. In this case, we will not take into account the friction force, and the centrifugal acceleration will be compensated only by the horizontal component of the gravity force:

  
  

  F c \u003d mV 2 / R or F c \u003d F n sinα

  The force of gravity acts on the body in the vertical direction F g = mg, which is balanced by the vertical component of the normal force F n cosα:

  F n cosα \u003d mg, hence: F n \u003d mg / cos α

  We substitute the value of the normal force in the original formula:

  F c = F n sinα = (mg/cosα)sinα = mg sinα/cosα = mg tgα

  Thus, the angle of inclination of the roadway:

  α \u003d arctg (F c /mg) \u003d arctg (mV 2 /mgR) \u003d arctg (V 2 /gR)

  Again, note that body weight is not included in the calculations!

  Task #2: on some section of the highway there is a turn with a radius of 100 meters. average speed passage of this section of the road by cars 108 km/h (30 m/s). What should be the safe angle of inclination of the roadbed in this section so that the car does not “take off” (neglect friction)?

  α \u003d arctan (V 2 / gR) \u003d arctan (30 2 / 9.8 100) \u003d 0.91 \u003d 42 ° Answer: 42°. Pretty decent angle. But, do not forget that in our calculations we do not take into account the friction force of the roadway.

  4. Degrees and radians

  Many are confused in understanding the angular values.

  In rotational motion, the basic unit of measurement for angular displacement is radian.

  • 2π radians = 360° - full circle
  • π radians = 180° - half circle
  • π/2 radians = 90° - quarter circle

  To convert degrees to radians, divide the angle by 360° and multiply by 2π. For example:

  • 45° = (45°/360°) 2π = π/4 radians
  • 30° = (30°/360°) 2π = π/6 radians

  The table below shows the basic formulas for rectilinear and rotary motion.