Video 1. Experiment with the rotation of a lighter top.
Experimental data are shown in Table 1.
Table 1. Experimental data for the rotation of a lighter top. Time measurements are made for every 10th revolution.
Turnovers converted to distance
The graph of the mathematical model of speed is shown in fig. 3.
The graph of the mathematical model of the coordinate is shown in fig. four.
Rice. Fig. 3. Graph of the mathematical model of speed for the IDVSD of the top in the first experiment. Experimental velocity data are indicated by blue dots.
Rice. Fig. 4. Graph of the mathematical model of the coordinate for the IDVUSDD of the top in the first experiment. The experimental data coordinates are marked with blue dots.
3. Study of the second (heavier) top.
The movement (rotation) of the second top will be recorded by video recording with a frame rate of 600 frames per second.
Top weight: 0.015 kg.
The diameter of the top is 0.057 meters.
Rice. 5. General view of the second, heavier top.
Video 2. Experiment with the rotation of a heavier top.
Experimental data are shown in Table 2.
Table 2. Experimental data for the rotation of a heavier top. Time measurements are made for every 10th revolution.
The graph of the mathematical model of speed is shown in fig. 6.
The graph of the mathematical model of the coordinate is shown in fig. 7.
Rice. Fig. 6. Graph of the mathematical model of speed for the IDVSD of the top in the second experiment. Experimental velocity data are indicated by blue dots.
Rice. Fig. 7. Graph of the mathematical model of the coordinate for the IDVUSDD of the top in the second experiment. The experimental data coordinates are marked with blue dots.
4. Comparison of speed graphs for the first and second experiments.
Figure 8 shows two speed graphs - for a light top and for a heavier one.
The graph of the mathematical model of speed for a lighter top is plotted with green dots. The graph of the mathematical model of speed for a heavier top is plotted with blue dots.
Rice. 8. Graphs of speed for light and heavy tops. The experimental data coordinates are marked with blue dots.
Spinning tops (flywheels) still have many secrets. After all, that mat model that I brought is not the only option for the movement of tops (flywheels). You should continue to search, and explore tops from various materials and even magnets.
5. Research of a brass top - capstan.
The movement (rotation) of the brass top will be recorded by video recording with a frame rate of 600 frames per second.
To determine the distance traveled, we stick a red mark on the surface of the top disk.
Top weight: 0.104 kg.
The diameter of the top is 0.05 meters.
Rice. 9. General view of a brass top.
Video 3. Experiment with the rotation of a brass top.
Experimental data are shown in Table 3.
Table 3. Experimental data for the rotation of a brass top. Time measurements are made for every 10th revolution.
The graph of the mathematical model of speed is shown in fig. ten.
The graph of the mathematical model of the coordinate is shown in fig. eleven.
Rice. Fig. 10. Graph of the mathematical model of speed for the IDVSD of a brass top. Experimental velocity data are indicated by blue dots.
Rice. Fig. 11. Graph of the mathematical model of the coordinate for the IDVUSD of a brass top. The experimental data coordinates are marked with blue dots.
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- shoulder 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 of “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.
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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 higher than 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).
Probably, each of us in childhood had a spinning top toy. How interesting it was to watch her spin! And I really wanted to understand why a fixed top cannot stand vertically, and when you start it, it starts to rotate and does not fall, maintaining stability on one support.
Although the top is just a toy, it has attracted the close attention of physicists. Yula is one of the types of body, which in physics is called a spinning top. As a toy, most often it has a structure consisting of two half-cones connected together, in the center of which an axis passes. But the top can have another form. For example, the gear of a clock mechanism is also a top, like a gyroscope - a massive disk mounted on a rod. The simplest top consists of a disk, in the center of which an axis is inserted.
Nothing can make a spinning top stay upright when it is stationary. But one has only to unwind it, as it will stand firmly on the sharp end. And the faster the speed of its rotation, the more stable its position.
Why does the spinning top not fall
Click on the picture
According to the law of inertia, discovered by Newton, all bodies in motion tend to maintain the direction of motion and the magnitude of the 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 spins, the harder it is to overcome the force of inertia.
Top precession
Let's push the spinning 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 their original position. As a result, there will be a precessional force directed perpendicular to the direction of the push. The spinning top will turn to the left at an angle of 90 degrees with respect 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, similarly to other rotating bodies, it receives a moment of momentum (angular momentum). The magnitude of this moment depends on the mass of the top and the speed of rotation. A rotating force arises, which forces the axis of the top to maintain the angle of inclination relative to the vertical during rotation.
Over time, the speed of rotation 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 divergent spiral. This is the precession of the axis of the top.
The precession effect can also be observed if, without waiting for its rotation to slow down, one simply pushes the top, i.e., applies an external force to it. The moment of the applied force changes the direction of the angular momentum of the axis of the top.
It has been experimentally confirmed that the rate of change of the 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 sway and return to a vertical position. Moreover, if you throw it up, its axis will still retain its direction. This property of the top is used in technology.
Before humanity invented the gyroscope, it used different ways of orientation in space. These were a plumb line and a level, which were based on gravity. Later, the compass was invented, which used the Earth's magnetism, and the astrolabe, the principle of which is based on the position of the stars. But in difficult conditions, these devices could not always work.
The work of the gyroscope, invented at the beginning of the 19th 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, through the center of which an axis passed. The whole structure was enclosed in a ring. But she had one significant drawback - her 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 aircraft, rockets, and submarines.
In a gyrocompass, a rotating wheel (rotor) is installed in a gimbal suspension, which is a universal hinged support in which a fixed body can freely rotate simultaneously in several planes. Moreover, the direction of the axis of rotation of the body will remain unchanged regardless of how the location of the suspension itself changes. Such a suspension is very convenient to use where there is pitching. After all, the object fixed in it will maintain a vertical position no matter what.
The gyroscope rotor maintains its direction in space. But the earth is spinning. And it will seem to the observer that in 24 hours the rotor axis makes a complete revolution. In a gyrocompass, the rotor is held in a horizontal position by means of a weight. Gravity creates torque, and the rotor axis always points due north.
The gyroscope has become an essential element of the navigation systems of aircraft and ships.
In aviation, a device called the attitude indicator is used. This is a gyroscopic instrument that determines the roll and pitch angles.
On the basis of the top, gyroscopic stabilizers were also created. A rapidly rotating disk prevents the axis of rotation from changing, "extinguishes" the pitching on ships. Such stabilizers are also used in helicopters to stabilize their vertical and horizontal balance.
Not only the top can maintain a stable position relative to the axis of rotation. If the body has the correct geometric shape, during rotation it is also able to maintain stability.
"Relatives" of the top
The top has "relatives". It's 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 are stationary, the bike falls on its side. And if they roll, then he keeps his balance.
And a bullet fired from a rifle also spins in flight, like a spinning top. It behaves this way because the barrel of the rifle has screw rifling. Sweeping through them, the bullet receives a rotational motion. And in the air, it retains the same position as in the trunk, with a sharp end forward. Cannon shells rotate in the same way. Unlike the old cannons that fired cannonballs, the flight range and accuracy of hitting such projectiles is higher.
Children are sometimes very curious and sometimes ask questions that are very difficult to answer. For example, why don't people fall from it? After all, it is round, rotates around its axis, and even moves in the vast expanses of the Universe among a huge number of stars. Why, at the same time, can a person walk calmly, sit on the couch and not worry at all? In addition, some peoples live “upside down”. Yes, and a sandwich that is dropped falls to the ground, and does not fly into the sky. Maybe something pulls us to the Earth and we can not come off?
Why don't people fall off the surface of the earth?
If the child began to ask such questions, then you can tell him about gravity, or in another way - about the earth's attraction. After all, it is this phenomenon that causes any object to strive towards the surface of the Earth. Thanks to gravity, a person does not fall and does not fly away.
Earth gravity allows the population of the planet to move freely on its surface, erect buildings and all kinds of structures, sled or ski down the mountain. Thanks to gravity, objects fall down instead of flying up. To test this in practice, it is enough to toss the ball. He will fall to the ground anyway. That's why people don't fall off the surface of the earth.
But what about the Moon?
Of course, gravity does not allow a person to fall from the Earth. But another question arises - why does the Moon not fall on it? The answer is very simple. The moon moves constantly in the orbit of our planet. If the Earth's satellite stops, then it will surely fall to the surface of the planet. This can also be verified by doing a little experiment. To do this, tie a string to the nut and unwind it. It will move in the air until it stops. If you stop spinning, then the nut will simply fall. It is also worth noting that the moon's gravity is about 6 times weaker than the earth's gravity. It is for this reason that weightlessness is felt here.
everyone has
Almost all objects have the power of attraction: animals, cars, buildings, people and even furniture. And a person is not attracted to another person just because our gravity is low enough.
The force of attraction directly depends on the distance between the individual bodies, as well as on their mass. Since a person weighs very little, he is attracted not to other objects, but to the Earth. After all, its mass is much larger. The earth is very big. The mass of our planet is enormous. Naturally, the force of attraction is great. Because of this, all objects are attracted to the Earth.
When was gravity discovered?
Children are not interested in boring facts. But the story of the discovery of gravity is quite strange and funny. was discovered by Isaac Newton. The scientist sat under an apple tree and thought about the universe. At that moment, a fruit fell on his head. As a result of this, the scientist realized that all objects fall exactly down, because there is an attractive force. continued his research. The scientist found that the force of gravity depends on the mass of bodies, as well as on the distance between them. He also proved that at a great distance objects are not able to influence each other. This is how the law of gravity came about.
Does everything fall down: a small experiment
In order for a child to better understand why people do not fall from the surface of the Earth, you can conduct a small experiment. This will require:
- Cardboard.
- Cup.
- Water.
The glass must be filled with liquid to the very brim. After that, the container should be covered with cardboard so that air does not get inside. After that, you need to turn the glass upside down, while holding the cardboard with your hand. It is best to experiment on the sink.
What happened? Cardboard and water remained in place. The fact is that there is absolutely no air inside the container. Cardboard and water are unable to overcome the air pressure from outside. It is for this reason that they remain in their places.
