The small peak that we conquered by reading and understanding the previous chapter allows us to answer the question posed in the title.
Let's imagine some kind of top, for example the one 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.
Don't let the complexity of the drawing scare you, it is only apparent. After all, what is complex is just something that is not sufficiently understood. Some effort and attention - and everything will become simple and clear.
Fig.4.
Let's take rectangular system coordinates xyz 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 axis xyz will be parallel to the plane of the disk and lie inside it. Let's agree that the axes xyz participate in all movements of the top, except its own rapid rotation.
On the right top corner(Fig. 4, b) we depict the same coordinate system xyz. We will need it later to speak the “language” of vectors.
First, we will not spin the top, and we will try to place it with the lower end of the axis on a supporting plane, for example, on the surface of a table. The result will not disappoint our expectations: the top will definitely fall on its side. Why is this happening? The center of mass of the top (point CM) lies above its support point (point ABOUT). Weight force G the top, as we already know, is applied at the CM point. Therefore, any small deviation of the axis z top from the vertical B will cause the appearance of a shoulder of force G relative to the fulcrum ABOUT, that is, the appearance of a 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 Sh. Let, as before, the z axis of the top be tilted from the vertical B by a small angle, i.e. the same moment M acts on the top. What has changed now? As we will see later, a lot has changed, but the basis of these changes is the fact that now every material point i The disk already has a linear speed V, due to the rotation of the disk with angular velocity Sh.
Let us select one point in the disk, for example, point A, which has a mass m A and lies in the middle plane of the disk at a distance r from the axis of rotation (r is the radius of the disk). Let's consider the features of its movement per revolution.
So, in starting moment time, point A, like all other points on the disk, has a linear velocity, the vector of which V A lies in the plane of the disk. The top (and its disk) is acted upon by a moment M, which tries* to overturn the top, imparting linear velocities to the points of the disk, the vectors of which Wi are perpendicular to the plane of the disk.
Under the influence of moment M, point A begins to acquire speed W A . Due to the law of inertia, the speed of a material point cannot increase instantly. Therefore, in the initial position (point A is on the y-axis), its speed is W A =0, and only after a quarter of a revolution 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 influence of the moment M the rotating top rotates around the axis at, and not around the axis X(as was the case with the untwisted top). This phenomenon marks the beginning of unraveling the mystery of the top.
The rotation of the top under the influence of the moment M is called precession, and angular velocity rotation - the speed of precession, let's denote it y 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 Shch.
As a result of the portable movement, the relative vector linear speed V A material point A, already returned and starting position, will be turned towards the portable rotation.
Thus, a picture that is already familiar to us arises of the influence of portable motion on relative motion, the influence that gives rise to Coriolis acceleration.
The direction of the Coriolis acceleration vector of point A (in accordance with the rule given in the previous chapter) is found by rotating the vector relative speed V A of point A 90° in the direction of the portable (precessional) rotation of the top. The Coriolis acceleration a of point A, which has a mass mA, generates an inertial force FK, which is directed opposite to the acceleration vector a k and is applied to the material points of the disk in contact with point A.
Reasoning In a similar way, you can obtain the directions of the Coriolis acceleration and inertia force vectors for any other material point on the disk.
Let's return to point A. Inertia force 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 inertial force, is called gyroscopic.
Its value is determined using the formula:
M GA = r F k = m A r 2 Shch P = I A
Size I A = m Ar 2, depending 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 moment of inertia was introduced into mechanics by L. Euler.
Not only individual points, but also entire bodies have moments of inertia, since they consist of individual material points. With this in mind, let's create a formula for the gyroscopic moment MG created by the top's disk. 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 the angular velocities Shch and Shch P will remain the same, since all points of the disk (except for those that lie respectively on the wildebeest axes) rotate with the same angular velocities Shch and Shch P.
NOT. Zhukovsky, “the father of Russian aviation,” who was also involved in the study of 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 vector of the kinetic moment H with the vector of the angular velocity of the portable rotation u P along the shortest path.
In a particular case, the speed of portable rotation is the speed of precession.
In practice, a similar rule is also used to determine the direction of precession: precession tends to combine the kinetic momentum vector H with the momentum vector physical strength M along the shortest path.
These simple rules lie at the basis of gyroscopic phenomena, and we will use them widely in the future.
But let's return to the top. Why it does not fall, turning around the x axis, is clear - the gyroscopic moment is preventing it. But maybe it will fall, rotating around the y-axis as a result of precession? Also no! The fact is that, as it precesses, the top begins to rotate around the y-axis, which means that the weight force 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 with it. Therefore, in this case, 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, while the angular velocity own rotation The top U 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 rotation of the top. Allowing for some simplifications, we can assume that the end of the top axis, point K, moves in a circle and the axis of rotation itself z describes in space conical surfaces with vertices at a point ABOUT.
A rotating top is an example of the movement of a body that has one fixed point (for a top it is point O). The problem of the nature of the movement of such a body played important role in the development of science and technology, many outstanding scientists devoted their works to its solution.
Of the thousands of people who played with a top as children, not many will be able to answer this question correctly. How, in fact, can we explain the fact that a rotating top, placed vertically or even inclined, does not tip over, contrary to all expectations? What force holds him in such a seemingly unstable position? Doesn't heaviness affect him?
There is a very interesting interaction of forces taking place here. The theory of the spinning top is not simple, and we will not go deeper into it. Let us outline only the main reason why the rotating top does not fall.
In Fig. 26 shows a top rotating in the direction of the arrows. Pay attention to the part A its rim and part IN, the opposite of it. Part A tends to move away from you, part IN- to you. Now observe what kind of movement these parts receive when you tilt the axis of the top towards you. With this push you force the part A move up part IN- down; both parts receive a push at right angles to their own movement. But since during the rapid rotation of the top the peripheral speed of the parts of the disk is very high, the insignificant speed you report, adding up to the large circular speed of the point, gives a resultant very close to this circular speed - and the movement of the top almost does not change. This makes it clear why the top seems to resist an attempt to topple it. The more massive the top and the faster it rotates, the more stubbornly it resists tipping over.
Why doesn't the 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, at every moment the particle tends to move from the circle onto a straight line tangent to the circle. But every tangent is located in the same plane as the circle itself; therefore, each particle tends to move so as to remain at all times in a plane perpendicular to the axis of rotation. It follows that all 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 thrown, retains the original direction of its axis.
We will not consider all the movements of the top that occur when an external force acts on it. This would require too much detailed explanations, which may seem boring. I just wanted to explain the reason for the desire of any rotating body to maintain the direction of the axis of rotation unchanged.
This property is widely used modern technology. Various gyroscopic (based on the properties of a top) devices - compasses, stabilizers, etc. - are installed on ships and aircraft. [Rotation ensures the stability of projectiles and bullets in flight, and can also be used to ensure the stability of space projectiles - satellites and rockets - as they move (Editor's note).]
That's how it is beneficial use a seemingly simple toy.
Of the thousands of people who played with a top as children, not many will be able to answer this question correctly. How, in fact, can we explain the fact that a rotating top, placed vertically or even inclined, does not tip over, contrary to all expectations? What force holds him in such a seemingly unstable position? Doesn't heaviness affect him?
There is a very interesting interaction of forces taking place here. The theory of the spinning top is not simple, and we will not go deeper into it. Let us outline only the main reason why the rotating top does not fall.
In Fig. 26 shows a top rotating in the direction of the arrows. Notice part A of its rim and part B opposite it. Part A tends to move away from you, part B towards you. Now observe what kind of movement these parts receive when you tilt the axis of the top towards you. With this push you force part A to move up, part B to move down; both parts receive a push at right angles to their own motion. But since during the rapid rotation of the top the peripheral speed of the parts of the disk is very high, the insignificant speed you report, adding up to the large circular speed of the point, gives a resultant very close to this circular speed - and the movement of the top almost does not change. This makes it clear why the top seems to resist an attempt to topple it. The more massive the top and the faster it rotates, the more stubbornly it resists tipping over.
Figure 26. Why doesn't the top fall?
Figure 27. A spinning top, when thrown, retains the original direction of its axis.
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, at every moment the particle tends to move from the circle onto a straight line tangent to the circle. But every tangent is located in the same plane as the circle itself; therefore, each particle tends to move so as to remain at all times in a plane perpendicular to the axis of rotation. It follows that all 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.
We will not consider all the movements of the top that occur when an external force acts on it. This would require too much detailed explanation, which would probably seem boring. I just wanted to explain the reason for the desire of any rotating body to maintain the direction of the axis of rotation unchanged.
This property is widely used by modern technology. Various gyroscopic (based on the property of a top) devices - compasses, stabilizers, etc. - are installed on ships and aircraft.
Such is the useful use of a seemingly simple toy.
The art of jugglers
Many amazing magic tricks Various programs of jugglers are also based on the property of rotating bodies to maintain the direction of the axis of rotation. Let me quote an excerpt from exciting book English physicist prof. John Perry's Spinning Top.
Figure 28. How a coin tossed with rotation flies.
Figure 29. A coin tossed without rotation lands in a random position.
Figure 30. A thrown hat is easier to catch if it has been given rotation around its axis.
One day I was demonstrating some of my experiments to an audience drinking coffee and smoking tobacco in a magnificent room concert hall"Victoria" in London. I tried to interest my listeners as much as I could, and talked about how a flat ring must be given rotation if one wants to throw it so that one can indicate in advance where it will fall; They do the same thing if they want to throw a hat to someone so that he can catch this object with a stick. You can always rely on the resistance that a rotating body exerts when the direction of its axis is changed. I further explained to my listeners that, having polished the barrel of a cannon smoothly, one can never count on the accuracy of the sight; As a result, rifled muzzles are now made, i.e., they are cut into inside gun muzzles are spiral-shaped grooves into which the protrusions of the cannonball or projectile fit, so that the latter must receive rotational movement, when the force of the explosion of gunpowder forces it to move along the cannon channel. Thanks to this, the projectile leaves the gun with a precisely defined rotational movement.
That was all I could do during this lecture, as I have no dexterity in throwing hats or discus. But after I had finished my lecture, two jugglers appeared on the stage, and I could not wish for a better illustration of the above-mentioned laws than that given by each individual trick performed by these two artists. They threw spinning hats, hoops, plates, umbrellas to each other... One of the jugglers threw into the air whole line knives, caught them again and again threw them up with great accuracy; my audience, having just heard the explanation of these phenomena, rejoiced with pleasure; she noticed the rotation that the juggler imparted to each knife, releasing it from his hands, so that he could probably know in what position the knife would return to him again. I was then amazed that almost without exception all the juggling tricks performed that evening were an illustration of the principle stated above.”
Probably each of us had a spinning top toy in childhood. How interesting it was to watch her spin! And I really wanted to understand why a stationary spinning top cannot stand vertically, but when you launch it, it begins to rotate and does not fall, maintaining stability on one support.
Although the spinning top is just a toy, it has attracted close attention from physicists. The spinning top is one of the types of body, which in physics is called a top. As a toy, it most often has a design consisting of two half-cones connected together, with an axis running through the center. But the top can have a different shape. For example, the gear of a clock mechanism is also a top, as is a gyroscope - a massive disk mounted on a rod. The simplest top consists of a disk with an axis inserted into the center.
Nothing can force a top to remain upright when it is stationary. But once you untwist it, it will stand firmly on the sharp end. And what faster speed its rotation, the more stable its position.
Why doesn't the spinning top fall?
Click on the picture
According to the law of inertia, discovered by Newton, all bodies in motion tend to maintain the direction of movement and the magnitude of speed. Accordingly, a rotating top also obeys this law. The force of inertia prevents the top from falling, trying to maintain the original nature of the movement. Of course, gravity tries to topple the top, but the faster it rotates, the more difficult it is to overcome the force of inertia.
Precession of a top
Let's push the top rotating counterclockwise in the direction shown in the figure. Under the influence of the applied force, it will tilt to the left. Point A moves down and point B moves up. Both points, according to the law of inertia, will resist the push, trying to return to initial position. As a result, a precessional force will arise, directed perpendicular to the direction of the push. The top will turn to the left at an angle of 90° relative to the force applied to it. If the rotation were clockwise, it would turn to the right at the same angle.
If the top did not rotate, then under the influence of gravity it would immediately fall to the surface on which it is located. But while rotating, it does not fall, but, like other rotating bodies, receives angular momentum (angular momentum). The magnitude of this moment depends on the mass of the top and the rotation speed. A rotating force arises, which forces the axis of the top to maintain an angle of inclination relative to the vertical during rotation.
Over time, the rotation speed of the top decreases and its movement begins to slow down. Its upper point gradually deviates from its original position to the sides. Its movement takes place in a diverging spiral. This is the precession of the top's axis.
The effect of precession can also be observed if, without waiting for its rotation to slow down, you simply push the top, i.e., apply it to it external force. The moment of the applied force changes the direction of the angular momentum of the top axis.
It has been experimentally confirmed that the rate of change of angular momentum of a rotating body is directly proportional to the magnitude of the moment of force applied to the body.
Gyroscope
Click on the picture
If you try to push a spinning top, it will swing and return to a vertical position. Moreover, if you throw it up, its axis will still maintain its direction. This property of the top is used in technology.
Before humanity invented the gyroscope, it used different ways orientation in space. These were a plumb line and a level, the basis of which was gravity. Later they invented a compass, which used the magnetism of the Earth, and an astrolabe, the principle of which was based on the location of the stars. But in difficult conditions these devices could not always work.
The operation of the gyroscope, invented in early XIX century by the German astronomer and mathematician Johann Bonenberger, did not depend on bad weather, shaking, pitching or electromagnetic interference. This device was a heavy metal disk with an axis passing through the center. This whole structure was enclosed in a ring. But it had one significant drawback - its work quickly slowed down due to friction forces.
In the second half of the 19th century, it was proposed to use an electric motor to accelerate and maintain the operation of the gyroscope.
In the twentieth century, the gyroscope replaced the compass in airplanes, rockets, and submarines.
In a gyrocompass, a rotating wheel (rotor) is installed in a gimbal, which is a universal articulated support in which a fixed body can freely rotate simultaneously in several planes. Moreover, the direction of the body’s rotation axis will remain unchanged regardless of how the location of the suspension itself changes. This type of suspension is very convenient to use where there is movement. After all, an object fixed in it will maintain a vertical position no matter what.
The gyroscope rotor maintains its direction in space. But the Earth rotates. And it will seem to the observer that in 24 hours the rotor axis makes full turn. In a gyrocompass, the rotor is held in a horizontal position using a weight. Gravity creates torque, and the rotor axis is always directed due north.
The gyroscope has become the most important element navigation systems of aircraft and ships.
In aviation, a device called an artificial horizon is used. This is a gyroscopic device with which the roll and pitch angles are determined.
Gyroscopic stabilizers have also been created based on the top. A rapidly rotating disk prevents changes in the axis of rotation and “quenches” pitching on ships. Such stabilizers are also used in helicopters to stabilize their balance vertically and horizontally.
Not only the top can save stable position relative to the axis of rotation. If the body has the correct geometric shape, when rotating, it is also able to maintain stability.
"Relatives" of the top
The top has “relatives”. This is a bicycle and a rifle bullet. At first glance they are completely different. What unites them?
Each of the wheels of a bicycle can be considered as a top. If the wheels don't move, the bike falls on its side. And if they roll, then he also maintains balance.
And a bullet fired from a rifle also spins in flight, just like a top. It behaves this way because the rifle barrel has screw rifling. As the bullet rushes through them, it receives a rotational motion. And in the air it maintains the same position as in the barrel, with the sharp end forward. Cannon shells rotate in the same way. Unlike old cannons that fired cannonballs, the flight range and accuracy of such projectiles is higher.
A good top should spin easily. To do this, it is necessary to correctly place the center of gravity. At high speed, the rotating top strives to maintain the position of its axis unchanged and does not fall. Gradually, due to friction, the rotation speed decreases. And when the speed becomes insufficient, the axis of the top spirals away from the vertical, followed by a fall.
Of the thousands of people who played with a top as children, not many will be able to answer this question correctly. How, in fact, can we explain the fact that a rotating top, placed vertically or even obliquely, does not tip over contrary to all expectations?
What force holds him in such a seemingly unstable position? Doesn't heaviness affect him? There is a very interesting interaction of forces taking place here. The theory of the spinning top is not simple, and we will not go deeper into it. Let us only outline main reason, as a result of which the rotating top does not fall.
The figure shows a top rotating in the direction of the arrows. Notice part A of its rim and part B opposite it. Part A tends to move away from you, part B towards you. Now observe what kind of movement these parts receive when you tilt the axis of the top towards you.
With this push you force part A to move up, part B to move down; both parts receive a push at right angles to their own motion. But since during the rapid rotation of the top the peripheral speed of the parts of the disk is very high, the insignificant speed you report, adding up to the large circular speed of the point, gives a resultant very close to this circular speed - and the movement of the top almost doesn't change.
This makes it clear why the top seems to resist an attempt to topple it. The more massive the top and the faster it rotates, the more stubbornly it resists tipping over.
spinning top, being thrown, maintains the original direction of its axis.
The essence of this explanation is directly related with 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, at every moment the particle tends to move from the circle onto a straight line tangent to the circle.
But every tangent is located in the same plane as the circle itself; therefore, each particle tends to move so as to remain at all times in a plane perpendicular to the axis of rotation.
It follows that all planes in the top, perpendicular to the axis rotations 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.
We will not consider all the movements of the top that occur when an external force acts on it.
This would require too much detailed explanation, which would probably seem boring.
I just wanted to explain the reason for the desire of any rotating body to maintain the direction of the axis of rotation unchanged. This property is widely used by modern technology. Various gyroscopic(based on the property of a top) instruments - compasses, stabilizers, etc. - are installed on ships and airplanes. Such is the useful use of a seemingly simple toy.
Rotation ensures the stability of projectiles and bullets in flight, and can also be used to ensure the stability of space projectiles - satellites and rockets - as they move.