Of the thousands of people who played with a spinning top as a child, not many will be able to correctly answer this question. How, in fact, to explain the fact that a spinning top, placed vertically or even obliquely, does not tip over, contrary to all expectations? What force keeps him in such a seemingly unstable position? Isn't gravity acting on him?

There is a very curious interaction of forces here. The theory of the top is not simple, and we will not delve into it. Let us outline only the main reason due to which the rotating top does not fall.

On fig. 26 shows a top rotating in the direction of the arrows. Pay attention to the part BUT his rim and on part AT opposite to it. Part BUT tends to move away from you, part AT- to you. Follow now what movement these parts receive when you tilt the axis of the top towards you. With this push you force the part BUT move up part AT- way down; both parts receive a push at right angles to their own motion. But since the circumferential speed of the parts of the disk is very high during the rapid rotation of the top, the insignificant speed reported by you, adding up with the high circular speed of the point, gives the resultant, very close to this circular one, and the motion of the top remains almost unchanged. From this it is clear why the top, as it were, resists an attempt to overturn it. The more massive the top and the faster it rotates, the more stubbornly it resists tipping over.

Why doesn't the spinning top fall?

The essence of this explanation is directly related to the law of inertia. Each particle of the top moves in a circle in a plane perpendicular to the axis of rotation. According to the law of inertia, the particle at every moment tends to go from the circle to a straight line tangent to the circle. But every tangent lies in the same plane as the circle itself; therefore, each particle tends to move in such a way that it always remains in a plane perpendicular to the axis of rotation. It follows that all the planes in the top, perpendicular to the axis of rotation, tend to maintain their position in space, and therefore the common perpendicular to them, i.e., the axis of rotation itself, also tends to maintain its direction.

A spinning top, being tossed, retains the original direction of its axis.

We will not consider all the motions of the top that occur when an extraneous force acts on it. This would require too detailed explanations, which, perhaps, will seem boring. I just wanted to explain the reason for the desire of any rotating body to keep the direction of the axis of rotation unchanged.

This property is widely used by modern technology. Various gyroscopic (based on the property of the top) devices - compasses, stabilizers, etc. - are installed on ships and aircraft. [Rotation provides stability for projectiles and bullets in flight, and can also be used to ensure the stability of space projectiles - satellites and rockets - as they move. - Note ed.]

Such is the useful use of a seemingly simple toy.

The spinning top is amazing! You can look at this phenomenon for a long time, like at the fire of a fire, experiencing unquenchable interest, curiosity and some other incomprehensible feelings ... In understanding the theory of the classic spinning top and its adequate application in practice, perhaps the “dog is buried” ...

The use and conquest of gravity ... Or maybe we just sometimes want to think so when we see phenomena that we cannot immediately understand and give them an explanation.

Let's start answering the question in the title of the article. I have divided the text of the answer into short numbered paragraphs in order to make it as easy as possible for the perception of information with the possibility of distractions during the reading process and an easy subsequent return to the text and meaning of the article. Move on to the next paragraph only after understanding the essence of the previous one.

Let's turn to the picture, which shows a classic spinning top.

1. Fixed absolute coordinate system Ox 0 y 0 z 0 shown in purple in the figure. The center of a rectangular Cartesian coordinate system is a point O on which the spinning top rests.

2. Moving coordinate system Cxyz shown in the figure in blue. The axes of this system do not rotate with the top, but repeat all its other movements! The center of this rectangular coordinate system is the point C, which lies on the middle plane of the top disk and is its center of mass.

3. The relative movement of the top is the movement (rotation) relative to the moving coordinate system Cxyz.

4. Portable movement is the movement of the top along with the moving coordinate system Cxyz relative to the fixed system Ox 0 y 0 z 0 .

5. The vectors of forces and moments are shown in green in the figure.

6. The top disk has a mass m and weight G= m* g, where g- acceleration of gravity.

7. The fact that a non-spinning top falls on its side, as a rule, does not surprise anyone. The top falls on its side due to the overturning moment Mdef= G* P, which will inevitably arise for any slightest deviation of the axis of the top z from the vertical axis z 0 . Here P- arm of strength G, measured along the axis y.

8. According to the figure, the fall of a non-rotating top occurs around the axis x!

Relative to the absolute fixed coordinate system Ox 0 y 0 z 0 axis x when falling, it moves in a plane-parallel manner along a cylindrical surface with a radius OC.

Axis y while rolling over a circle with a radius OC, changing direction in absolute space along with the axis z, which rotates around a point O.

Considering the fall of the top in absolute space with respect to the point C, we can conclude that the top and the coordinate system rigidly associated with it Cxyz rotates around an axis x in the direction of the overturning moment Mdef.

9. Consider the motion of an arbitrary material point belonging to the disk of a spinning top. To do this, select a point A, which has a mass m A and lying, for example, in the plane xy on the periphery of the disk at a distance R from the center of mass of the point C.

10. We assume that initially the point A has a linear velocity of relative motion VArel, due only to the rotational motion of the top around the axis z. Velocity vector VArel parallel to axis x.

11. Remember that a top spinning clockwise with a very high angular velocity ω rel around the axis z, the moment is still valid Mdef, resulting from the inevitable initial deviation of the axis z from the vertical.

12. A point with mass cannot instantly change its speed because for this it needs to be given an acceleration equal to infinity - which is considered impossible due to the law of inertia. This means that the increase in speed VAlane caused by the overturning moment Mdef, will occur for some time and the spinning top will have time to turn through a certain angle. To simplify the explanation of the process, we conditionally assume that the transfer speed of the point A VAlane reaches its maximum at the moment when the point A rotates 90° (¼ turn) and intersects the axis x.

13. In the figure, the vectors of the portable velocity of the point A VAlane at different times at different angles of rotation are shown in magenta, and the relative velocity vector VArel in the initial position of the point is shown in brown.

14. In accordance with the above, if you look at the figure, it becomes obvious that the top will start tipping not around the axis x, around the axis y!

15. Due to the resulting portable movement (overturning), when the point A by making a revolution around the axis z, will return to the initial position on the axis y, its absolute velocity vector VA will be turned down in the direction of capsizing, that is, in the direction of portable movement relative to the relative velocity vector VArel.

16. Any change in speed can only be due to the action of non-zero acceleration! In this case, this acceleration is called Coriolis acceleration. acore. It is directed along the line of action of the speed VAlane portable movement that caused it. Vector acore parallel to axis z.

17. Portable motion that caused Coriolis acceleration acore, gives rise, respectively, to the force of inertia Fcore, which acts in the direction opposite to the direction of the vector acore.

18. In turn, the Coriolis force of inertia Fcore creates a moment about the axis x Mgir= Fcore* R called the gyroscopic moment. It is the gyroscopic moment Mgir, counteracting the overturning moment Mdef, balances the system and does not allow the spinning top to fall on its side !!!

19. The spinning top, not having time to turn around one axis, begins to turn around the other, and so on, as long as there is rotation, while the kinetic moment acts H= ω rel* m* R 2 /2 !

Figuratively, we can say this: as soon as a spinning top begins to fall under the action of the moment of gravity Mdef, turning around a certain axis, so after a moment a gyroscopic moment arises around the same axis Mgir preventing this rotation. So these two moments “play catch-up” - one drops the top, the other keeps it from falling ...

20. Axis z, rigidly connected with the axis of rotation of the top, describes in the absolute coordinate system Ox 0 y 0 z 0 cone with apex at a point O. Such a circular movement of the axis z with speed ω lane called precession.

21. The vector diagram shown in the figure below shows, balancing each other, the overturning moment of gravity Mdef and gyroscopic moment Mgir.

Mdef= Mgir= H* ω lane

Gyroscopic moment Mgir tries to rotate the angular momentum vector along the shortest path H in the direction of the angular velocity vector of the translational rotation ω lane. In this case, the precession is a vector ω lane- seeks to rotate the same vector H and combine it along another shortest path with the vector of the overturning moment of gravity Mdef. These two actions determine the basis of the phenomenon, whose name is the gyroscopic effect.

As long as there is rotation ω rel≠0 ), the top has a kinetic moment H, which ensures the existence of the gyroscopic moment Mgir, which in turn compensates for the action of the moment of gravity Mdef, which gave rise to the gyroscopic moment Mgir…

Such is the story about “the house that Jack built”, only the circle is closed, and it exists while “the top is spinning - childhood fun”!

Leonard Euler (Russia) laid the foundations for the theory of the top by solving the problem for a top with the center of gravity at the fulcrum. The theory was developed by Joseph Louis Lagrange (France), having solved the problem with a top whose center of gravity is on the axis of rotation, but not at the fulcrum. Sofya Vasilievna Kovalevskaya (Russia) advanced the most in solving the problem of the theory of the top, solving the problem for a top with a center of gravity not lying on the axis of rotation.

... Or maybe the rotation of the top occurs for completely different reasons, and not according to the above theory, which Lagrange told the world about? Maybe this model describes the process “correctly”, but the physical essence is different? Who knows ... but there is still no mathematical solution to the problem in general terms, and the spinning top has not yet revealed absolutely all its secrets to humanity.

Subscribe to announcements of articles in the boxes located at the end of each article or at the top of each page, and Do not forget confirm subscription .

P confirm subscription required by clicking on the link in a letter that will come to you at the specified mail (may come in a folder « Spam » )!!!

I will read your comments with interest, dear readers!

Page 3

Formula (92.1) shows that the angular velocity of precession coj is the smaller, the greater the angular velocity of rotation of the top around its axis of symmetry.

Formula (92.1) shows that the angular velocity of precession ω, the less, the greater the angular velocity of rotation of the top around its axis of symmetry.

The position of the axis of the figure (the axis of symmetry of the body) is easy to establish at any top and observe its movements during the rotation of the top. The instantaneous axis of rotation is, generally speaking, invisible.

Metal groups can be considered as symmetrical tops, which have two moments of inertia about the axes perpendicular to the main axis of rotation of the top.

Metal groups can be considered as symmetrical tops, which have two moments of inertia about the axes perpendicular to the main axis of rotation of the top. Often in a molecule, one can distinguish a rigid base, to which one or more rigid tops are associated.

Internal rotation /t/1/a, (VI. 152.

Metal groups can be considered as symmetrical tops, which have two moments of inertia about the axes perpendicular to the main axis of rotation of the top. Often in a molecule one can distinguish a rigid base, with which one or several rigid tops are connected.

The center of gravity of the top, the axis of which performs a rapid precession, practically stopped and again acquired some speed only in the last stage of motion, when the angular velocity of the top's rotation noticeably decreased.

In the absence of rotation about its own axis, its state of equilibrium with the vertical direction of the axis will be unstable (if the center of gravity is above the fulcrum); when the angular velocity of rotation of the top around the axis becomes sufficiently large, its state of merostatic rotation becomes stable (not only in the linear, but even in the strict sense), if only the weight force is considered as the acting force. But if air resistance is taken into account, then dissipative forces enter into the equations of small oscillations, and we theoretically find, as is the case in reality, that the angular velocity, albeit slowly, will decrease, so that in the end the top will fall. An exhaustive explanation of this phenomenon will be given in Chap.

An example of a rigid body, well, a fixed point, is a top, the pointed leg of which rests against a nest made in a stand, so that this end of the leg remains motionless when the top rotates.

For the entire molecule having mass M, including the rotating group in an equilibrium position, the main central axes of inertia 1, 2, 3 and the main moments of inertia about these axes / d, 1B, / s are found; then the coordinate axes of the top are drawn so that the axis 2 coincides with the axis of rotation of the top, the x-axis passes through the center of gravity of the top and is perpendicular to the z-axis, and the y-axis passes through the intersection point of the axes x, z and would be perpendicular to them. Top atoms lying on the rotation axis z are excluded from further consideration.

At a high rotation speed of the top, the precession rate is negligible. When the rotation of the top weakens, there is always a precession.

Turn on the electric motor and bring the speed of rotation of the top to 8000 rpm. When the top rotates, heavy minerals settle and get stuck in the grooves of the top 5, and the light ones are thrown along with the liquid onto the walls of the separating funnels 2 and 6 and through the outlet 3 enter the Buchner funnel. Since filtration is slow, the oil pump is turned on.

Impetus Benedetti characterizes the direction, considering it as a kind of rectilinear element. So, he explains the rotation of the top by the straightness of the horizontal and tangential impetuses, which balance the severity of the parts to which they are attached. As long as the speed of the top is high, this allows it to maintain its position. When consumed, the impetuses give way to gravity, which leads to the fall of the top. Based on these considerations, Benedetti shows that there can be no perfect natural motion (and it is only eternal and uniform circular motion).

The small pinnacle, which we conquered by reading and assimilating the previous chapter, allows us to answer the question posed in the title.

Imagine some top, for example, what is described at the beginning of the book - a thin brass disk (gear) mounted on a thin steel axis. This version of the top is shown in Fig. 4.

Do not be afraid of the complexity of the drawing, it is apparent. After all, the complex is just not well understood. Some effort and attention - and everything will become simple and clear.

Fig.4.

Let's take a rectangular coordinate system xz and place its center at the center of mass of the shelf, that is, at the CM point. Let the axis z passes through the axis of its own rapid rotation of the top, then the axes xz will be parallel to the plane of the disk and lie inside it. We agree that the axes xz participate in all the movements of the top, except for its own rapid rotation.

In the upper right corner (Fig. 4, b) we will depict the same coordinate system xz. We will need it in the future to talk in the "language" of vectors.

First, we will not spin the top, and we will try to put it with the lower end of the axis on the reference plane, for example, on the surface of the table. The result will not deceive our expectations: the top will surely fall on its side. Why is this happening? The center of mass of the top (point CM) lies above its fulcrum (points O). weight force G top, as we already know, is applied at the CM point. Therefore, any small deviation of the axis z top from vertical B will cause the appearance of a shoulder of force G about the fulcrum O, that is, the appearance of the moment M, which will knock down the top in the direction of its action, that is, around the axis X.

Now let's spin the top around the z-axis to a high angular velocity Ω. Let the z-axis of the top deviate from the vertical B by a small angle, i.e. the same moment M acts on the spinning top. What has changed now? As we will see later, a lot has changed, but these changes are based on the fact that now every material point i The disk already has a linear velocity V, due to the rotation of the disk with an angular velocity Ω.

Let us select one point in the disk, for example point A, having a mass m A and lying in the middle plane of the disk at a distance r from the axis of rotation (r is the radius of the disk). Consider the features of its movement in one revolution.

So in initial moment point A, like all other points of the disk, has a linear velocity, the vector of which V A lies in the plane of the disk. A moment M acts on the top (and its disk), which tries * to overturn the top, giving the points of the disk linear velocities, the vectors of which W i are perpendicular to the plane of the disk.

Under the action of the moment M, point A begins to acquire speed W A . By virtue of the law of inertia, the speed of a material point cannot instantly increase in any way. Therefore, in the initial position (point A is on the y axis), its speed W A \u003d 0, and only after a quarter of a turn of the disk (when point A, rotating, will already be on the axis X) its speed W A increases and becomes maximum. This means that under the action of the moment M, the rotating top rotates around the axis at, not around the axis X(as it was with the unspun spinning top). In this phenomenon, the beginning of the solution to the mystery of the spinning top.

The rotation of the top under the action of the moment M is called precession, and the angular velocity of rotation is called the precession speed, we denote it by s p. Precessing, the top began to rotate around the y axis.

This movement is portable in relation to the own (relative) rotation of the top with a high angular velocity Ω.

As a result of the portable movement, the vector of relative linear velocity V A of the material point A, which has already returned to its initial position, will turn in the direction of the portable rotation.

Thus, there is already a familiar picture of the influence of the portable motion on the relative, the influence that gives rise to the Coriolis acceleration.

The direction of the Coriolis acceleration vector of point A (in accordance with the rule given in the previous chapter), we find by turning the relative velocity vector V A of point A by 90 ° in the direction of the portable (precessional) rotation of the top. The Coriolis acceleration of the point A, having a mass mA, generates an inertial force FK, which is directed opposite to the acceleration vector a to and is applied to the material points of the disk that are in contact with the point A.

Arguing in this way, one can obtain the directions of the vectors of the Coriolis acceleration and the force of inertia for any other material point of the disk.

Let's go back to point A. The force of inertia F K on the shoulder r creates a moment M GA acting on the top around the x-axis. This moment, generated by the Coriolis force of inertia, is called gyroscopic.

Its value is determined using the formula:

M GA = r F k \u003d m A r 2 Shch P \u003d I A W W W

the value I A = m A r 2 , which depends on the mass of the point and its distance from the axis of rotation, is called the axial moment of inertia of the point. The moment of inertia of a point is a measure of its inertia in rotational motion. The concept of the moment of inertia was introduced into mechanics by L. Euler.

Moments of inertia are possessed not only by separate points, but also by whole bodies, since they consist of separate material points. With this in mind, let's make a formula for the gyroscopic moment M G created by the disk of the top. To do this, in the previous formula, we replace the moment of inertia of the point I A at the moment of inertia of the disk I D, and let us leave the angular velocities W and W P the same, since all points of the disk (with the exception of those that lie respectively on the axes hU) rotate with the same angular velocities W and w P.

NOT. Zhukovsky "the father of Russian aviation", who also studied the mechanics of tops and gyroscopes, formulated the following simple rule for determining the direction of the gyroscopic moment (Fig. 4, b): the gyroscopic moment tends to combine the angular momentum vector H with the angular velocity vector of the translational rotation the shortest path.

In a particular case, the speed of translational rotation is the speed of precession.

In practice, they also use a similar rule to determine the direction of the precession: the precession tends to combine the vector of the kinetic moment H with the vector of the moment of physical forces M along the shortest path.

These simple rules underlie gyroscopic phenomena, and we shall make extensive use of them in what follows.

But back to the wolf. Why it does not fall, turning around the x-axis, is clear - the gyroscopic moment prevents it. But maybe it will fall, turning around the y-axis as a result of precession? Also no! The fact is that, while precessing, the top begins to rotate around the y-axis, which means that the force of the weight G begins to create a moment acting on the top around the same axis. This picture is already familiar to us; we began our consideration of the behavior of a rotating top from it. Therefore, in this case, too, a procession and a gyroscopic moment will arise, which will not allow the top to tilt around the y axis for a long time, but will transfer the movement of the top to another plane, and in which its phenomena will be repeated again.

Thus, as long as the angular velocity of the top's own rotation W is large, the moment of gravity causes precession and gyroscopic moment, which keep the top from falling in any one direction. This explains the stability of the axis r top rotation. Allowing some simplifications, we can assume that the end of the axis of the top, the point K moves around the circle, and the axis of rotation itself z describes in space conical surfaces with vertices at a point O.

A spinning top is an example of the movement of a body that has one fixed point (for a top, this is point O). The problem of the nature of the motion of such a body played an important role in the development of science and technology, and many outstanding scientists devoted their works to its solution.

So, the giant Matif, in order to accomplish his feat, it was enough to pull the rope with a force of only 24 pounds!

Do not think that this figure of 24 pounds is only theoretical and that much more effort will actually be required. On the contrary, we got a result that is even too significant: with hemp rope and wooden pile effort required ridiculously negligible. If only the rope was strong enough and could withstand tension, then even a child, thanks to Euler's formula, could, by winding the rope 3-4 times, not only repeat the feat of the Jules Verne giant, but also surpass it.

What determines the strength of knots?

In everyday life, we often take advantage of the benefits that Euler's formula points out to us. What is, for example, any knot, if not a twine wound on a roller, the role of which in this case is played by another part of the same twine? The strength of any kind of knots - ordinary, "gazebo", "marine", - any kind of ties, bows, etc. depends solely on friction, which here is amplified many times over due to the fact that the lace wraps around itself, like a rope around pedestals. This is not difficult to verify if you follow the bends of the lace in the knot. The more these bends, the more times the twine wraps around itself - the greater the "winding angle" in Euler's formula, and therefore, the stronger the knot.

Unconsciously uses Euler's formula and the tailor when sewing on a button. He winds the thread many times around the part of the cloth captured by the stitch and then breaks the thread. For the strength of sewing, he can be calm: if only the thread is strong, the button will not come off. Here the rule already familiar to us is applied: with an increase in the number of revolutions of the thread in an arithmetic progression, the sewing strength increases exponentially.

If there were no friction, we could not tie two strings or tie shoelaces; we could not have used buttons too: the threads would have unwound under their weight, and our suit would have been left without a single button.

Chapter Three

Rotational movement. Centrifugal force

Why doesn't the spinning top fall?

It can be said without exaggeration that out of a thousand people who amused themselves in childhood by spinning a top, hardly at least one will be able to correctly answer this question. Indeed, isn't it strange that a spinning top, placed vertically or even obliquely, does not tip over, contrary to all expectations? What force keeps him in such a seemingly unstable position? Isn't gravity acting on this little object?

Of course, no exception is made to the laws of nature for the spinning top. There is only an extremely curious interplay of forces here.

Rice. 22. Why does the top not fall?

On fig. 22 shows a top rotating in the direction of the black arrows. Pay attention to the part BUT ahead of the top and on the part AT, which is diametrically opposed to it. Part BUT tends to move from right to left, does not fall? part AT- from left to right. Now watch what movement these parts get when you push the axis of the top away from you. With such a push you force the part BUT move up part AT- down, i.e. both parts receive a push at right angles to their own movement. But since, with a rapidly rotating top, the initial speed of the parts of the disk is very high, it is quite understandable that the top, as it were, resists an attempt to overturn it. The more massive the top and the faster it rotates, the more stubbornly it resists tipping over.

So, we already know what reason prevents the top from tipping over, despite the fact that it is, it would seem, in an unstable position. This is well-known to us inertia - the main property of matter, which consists in the fact that any material particle tends to keep the direction of its movement unchanged. We will not consider here all the motions of the top that occur when an external force acts on it. This would require very detailed explanations, which, perhaps, will seem boring to most readers. We only wanted to explain the reason for the main desire of any rotating body - to keep the direction of the axis of rotation unchanged. This property explains a number of phenomena that we encounter in everyday life. The most skillful cyclist would not have sat on his steel horse for a minute if the rapidly rotating wheels did not strive to keep their axes horizontal: after all, the wheels are the same tops, only their axes are not vertical, but horizontal. And that's why it's so hard to ride a bike slowly: the wheels are no longer spinning tops. A child rolling his hoop unconsciously uses the same property of rotating bodies: while the hoop is in rapid rotation, it does not fall. The diabolo game is entirely based on the same principle: first, with the help of a string, we bring the double cone of the diabolo into a rapid rotational movement and then throw it high up; but, flying up and then falling down, the rotating diabolo does not cease to maintain the horizontality of the axis of rotation - that is why it is so easy to catch it on an elongated string, throw it up again, catch it again, etc. If the diabolo did not rotate, all this would be impossible even for the most skilled juggler.

Rice. 23. Diabolo is easy to catch just because it does not stop spinning during takeoff and fall.

The art of jugglers

Speaking of jugglers: almost all the most amazing "numbers" of their varied program are again based on the desire of rotating bodies to maintain the direction of the axis of rotation. Let me quote here an extract from a fascinating book by a modern English physicist, prof. John Perry's Spinning Top:

“I once showed some of my experiments in front of an audience drinking coffee and smoking tobacco in the magnificent premises of the Victoria Concert Hall in London. I tried to interest my listeners as much as I could, and talked about the fact that a flat ring must be given rotation if it is desired to be thrown so that it could be indicated in advance where it will fall; they act in the same way if they want to throw a hat to someone so that he can catch this object with a stick. You can always count on the resistance that a rotating body exerts when the direction of its axis is changed. I went on to explain to my listeners that once the muzzle of a cannon had been polished smoothly, one could never count on the accuracy of the sight; that the rotation into which an ordinary cannonball enters depends primarily on how the cannonball touches the opening of the cannon at the moment when it flies out of it; as a result, rifled muzzles are now made, i.e., spiral grooves are cut on the inside of the muzzle of cannons, into which protrusions of the core or projectile fall, so that the latter must receive rotational movement when the force of the explosion of gunpowder makes it move along the muzzle of the gun. Thanks to this, the projectile leaves the cannon with a precisely defined rotational movement, about which no doubt can arise. Rice. 26 indicates the kind of movement that the projectile then makes: just like a hat or a ring, its axis of rotation remains almost parallel to itself.