MYSTERIES OF THE ORDINARY TOP
The spinning top is a simple-looking toy that children of all times and peoples had fun with. But it has a number of amazing and at first glance inexplicable properties!
J. B. Chardin. Boy with a wolf. 18 century.
In addition to the usual spinning top, there is also its complicated version - the top, which has a mechanism for unwinding.
"The behavior of a spinning top is extremely amazing! If it does not spin, then it immediately capsizes, and it cannot be kept in balance at the tip. But this is a completely different object when it is spinning: it not only does not fall, but also shows resistance when it is pushed , and even assumes a more and more upright position." - so the famous English spoke about the top scientist J. Perry.
Japanese tops
Spinning tops were brought to Japan from China and Korea about 1200 years ago. Spinning top is one of the favorite games in Japan." Some are very skillfully made: they descend from the mountain dance on a tightrope, shatter into pieces that keep spinning."
At present, there are about 1,000 in Japan. different types tops, the shapes of which can be very different - from ordinary spinning tops to products of a complex, bizarre shape. Their sizes range from 0.5 mm to 90 cm.
1. Energy layer
The new Achilles Z energy layer has become larger in diameter than the previous version. Along the edges are metal blades in the form of a scythe. They significantly add weight, and the peripheral location increases the centrifugal force of the rotation of the bay.
To the left and right are the small blue wings already familiar to everyone, which open during the battle, like the Voltraek B5 and serve as a break blocker. But it should be borne in mind that this block only increases resistance to breaking, and does not completely eliminate it.
A pair of blue wings are symmetrically located above and below. On the one hand, they slightly increase endurance during rotation. On the other hand, they smooth out the contour to make it harder for attacking opponents to hook on and blow the roof off the Super Z Achilles A5. In addition, any retractable elements absorb shock, which reduces rebound during collisions. Due to this, the beyblade will be more difficult to knock out of the arena;
2. Power disk
The new disc is really new. It was designated the number 00 (double zero). Previously, the zero disk was the heaviest of all, but now it has a competitor in weight. Already if you surprise with a new beyblade, then surprise in everything, they decided in Takara Tomy;
3. Driver (tip)
The updated driver is called Dimension (Dm). In fact, this is a modified Xtend driver of the previous version of Achilles. It also has two basic modes (attacking and defensive) and also changes its height. However, it has changed externally and the mechanism for switching modes has become different. Inside there is a black rod on which rotation takes place. In the old system, there was a ring on top that had to be pulled back and turned to set the desired mode. Now there is a third element. The ring itself has become geared for ease of rotation, and when turned, a small sleeve comes out of it, in which the rod is hidden;
4. The kit also includes a left-right trigger.
How do you not have such a top? So you missed a lot in your childhood ... Get it immediately! It will confuse the head, to whom you want ... I twist, I twist, I want to know a lot ... for example, about the dynamic properties of this wayward overturning top. Just look at these famous physicists W. Pauli and N. Bohr. What do you think they are into? ...
No one knows when the Chinese top was first launched, and who invented it. But it is known that for the first time the great physicist Lord Kelvin became interested in the unusual properties of the Chinese top during rotation.
Later, the Chinese spinning top acquired another name, "Thomson's spinning top", after the name of a scientist who studied gyroscopes. Since then, such tops "twist" all over the world!
Chinese spinning top- this is a ball with a cut off top, on the cut surface in the center there is a leg-axis. To see in the rotation of this top something that distinguishes it from an ordinary top, you need to observe one rule during its manufacture: the center of mass of the top should not coincide with the geometric center of the workpiece ball.
In steady state, i.e. in the equilibrium position, the Chinese spinning top is similar to "Roly-Vstanka". The center of gravity is located below the center of curvature of its surface.
Without rotation, the top, under the action of gravity, is installed so that the leg is extended vertically. The top rests on the plane by one point of its spherical surface. If it is strongly untwisted, then, rotating, it begins to bend over, turns over, and then stands up on its leg. The rotation does not stop. True, unbelievable? But, fact!
The main parameters of the top: O - the center of mass of the top, h - the distance from the center of mass to the fulcrum; K is the center of curvature of the top at the fulcrum, r is the radius of curvature.
If any symmetrical top is brought into rotation around its geometric axis of symmetry and placed on a plane in a vertical position, then this rotation, depending on the shape of the top and the angular velocity of rotation, can be stable or unstable.
The behavior of the top during rotation will depend on the ratio of the moment of inertia about the geometric axis of symmetry to the moment of inertia about the main central axis perpendicular to the axis of symmetry, as well as the ratio of the distance from the center of mass to the fulcrum (h) to the radius of curvature of the top's cap (r).
With a strong untwisting of the top, some slight involuntary deviation from the vertical position occurs. With further rotation, the geometric axis of symmetry of the top occupies an increasingly inclined position relative to the vertical axis of rotation.
There is no constant point of support on the surface of the top. The shifting point of support on its surface, constantly approaching the cut of the ball, describes a curved line on the surface on which the top rotates.
The center of mass of the top, which is below the geometric center of the ball from which it is made, is displaced from the axis of rotation and begins to rotate around it.
As the rotation progresses, the axis of rotation and the geometric axis of the top shift more and more relative to each other. Friction at the fulcrum creates a torque, determined by the divergence of the axes of symmetry and rotation and directed downward. This leads to an even greater tilt of the top on its side. With a high angular velocity of rotation, the center of mass rises, and the top itself “falls over” more and more on its side.
After the top passes by inertia through the horizontal position, the torque changes its direction due to gravity and tries to turn the top over.
As soon as the top touches the surface on which the rotation takes place with the edge of the leg, the fulcrum passes to the edge of the leg, and the Chinese top, like the most ordinary one, begins to process around the vertical axis, describing a conical surface. Due to the action of the moment of friction force directed to the vertical, the top, in the end, will align its axis with the vertical, and we will see the vertical rotation of the top "upside down", i.e. on the leg.
Over time, due to the rise of the center of mass and friction losses, the angular speed of rotation of the top decreases.
It is interesting that if, for example, you run it clockwise, then after turning it over, the direction of rotation of it relative to its own geometric axis of symmetry remains unchanged (if you observe the rotation from only one side - for example, from above).
But if we analyze the rotation of the top, observing it all the time of rotation only from one side, for example, from the side of the leg, then we can see that after tipping onto the leg, the rotation of the top around the axis of symmetry will be opposite to the original one. This was noticed in experiments when the top rotated on the surface of carbon paper. The line drawn as a result of rotation on the surface of the top shows where, at what moment the change in the direction of rotation occurred
Where, at what moment does this change of direction of rotation, imperceptible to the eye, take place?
When the geometric axis of the top during rotation goes into a horizontal position, at this moment there is no rotation around the geometric axis of symmetry of the top! This is where the visually imperceptible direction of rotation changes.
Entertaining wolves. Experiments, competitions, productionA spinning top is a children's toy that, when rotating around its axis, keeps a vertical position, and falls when the rotation slows down. In addition, when rotating a painted top, one can observe the optical effects of mixing and even decomposition of colors into components.
Materials:
Cardboard, paints, toothpicks or even better skewers, glue (PVA) or plasticine.
Tops do not have to be made of cardboard, you can use thick paper or thin plastic. You can try to make a big top from a CD, or a top with a pencil or felt-tip pen as the axis - then you can see interesting traces of rotation.
Manufacturing process:
On cardboard or thick paper, draw a few circles with a compass, about 5 cm in diameter. Color according to the diagrams and cut out. If the child does not yet use the compass, you can use a round glass or coffee cup as a template, the main thing then is to find the center. You can make one circle template - find the center there by folding in half and again in half, pierce the middle, and then applying to the painted circles transfer the center to them.
In the center of the circle, a small hole is made with an awl (toothpicks break), into which a toothpick or a cut wooden skewer (necessarily with a sharp end) is inserted. We fix the stick with PVA glue (dries for a long time) or a piece of plasticine (it will be faster here).
It turned out to be a wolf.
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These are tops that we made from thick paper, drawing a pattern with watercolors and inserting toothpicks and skewers.
Experiments with color
The simplest schemes of tops are by sectors. The circle is divided into an even number of sectors and painted, for example, in yellow and blue or yellow and red. When rotated, we will see green and orange, respectively.
In this experiment, you can see how the colors mix.
Here you can experiment with the number of color sectors.
If you divide the top into seven parts and paint them (very pale watercolor) in accordance with the arrangement of colors in the spectrum, then when rotated, the top should turn white. We will observe the process of "gathering" colors, since white is a mixture of all colors.
This effect is difficult to achieve, my daughter and I did not succeed, apparently we painted the top (pictured) very brightly. Maybe we didn’t get the white color, but we got a beautiful rainbow effect, and even with some kind of three-dimensionality.
The most interesting patterns come from spiral patterns. They look especially fascinating when the rotation of the toy slows down.
Explanation of what you see: This optical illusion is due to the fact that the brain mistakenly reproduces the areas of black and white colors as colored (first experience). As we said above - white is a mixture of all colors. Black is the absence of color. When the eye sees a smeared combination of black and white, it perceives it as colored. The color depends on the proportion of white and black and on the speed of rotation.
Explanation from the book: Interesting Experiments with Paper by Stephen W. Moyer
Interesting: The property of a top to take a vertical state during rotation is widely used in modern technology. There are various gyroscopic(based on the rotational property of the top) devices - compasses, stabilizers and other useful devices that are installed on ships and aircraft. Such is the useful use of a seemingly simple toy.
Active games for children
Games with spinning tops not only contribute to the development of fine motor skills of the child, but can also amuse and keep the children's company on holiday. We play and compete with children.
Competitions for children's holidays:
- The players simultaneously launch all the tops. Whose spinning top spins the longest is the winner.
- Or organize obstacles on the table in the form of small objects - you need to try not to hurt them or, on the contrary, knock them down, depending on the condition.
- Draw a playing field with sectors. Each participant has his own sector, whose spinning top flies out of the sector - he lost.
- Or also a game on the playing field: whose top knocks down the rest of the tops and remains alone - he won.
A symmetric top is a molecule in which two main moments of inertia are equal ( I B = I C for an elongated top or I A = I B for a flattened top). The third moment of inertia is not equal to zero and does not coincide with the other two. An example of an elongated symmetrical top is the FCH 3 methyl fluoride molecule, in which three hydrogen atoms are tetrahedrally bonded to the carbon atom, and the fluorine atom is at a greater distance from the carbon atom than the hydrogen. The rotation of such a molecule around the C axis – F (the axis of symmetry of the molecule) is different from rotation about the other two axes perpendicular to this one. The moments of inertia about the other two axes are I B= I C. Moment of inertia about the direction of connection С – F( I A), although small, it cannot be neglected. The contribution to the rotation about this axis (it coincides with the axis of symmetry of the molecule) is made by three hydrogen atoms located outside this axis.
The energy levels of a symmetrical top can be found in terms of the squares of the corresponding moments of momentum
For a symmetrical prolate top I x= I y, a Iz< I y . Axis Z coincides with the axis of least moment of inertia
Formula (2.40) can be rewritten as follows:
in formula (2.40) we have added and subtracted the expression ). The first term of expression (2.41) includes the square of the total moment p 2 , which is quantized and equal to b.j.(J+ 1) (see 2.2), and the second term includes the projection of the square of the moment on the axis Z, which is the axis of symmetry of the top. Moment Projection P z is quantized and takes the values P z= ћk. Thus, the quantized expression for the rotation energy will have the form:
Introducing rotational constants, we obtain
(A>B), (2.43)
(J= 0, 1, 2, ...; k= 0, ±1, ±2, ...).
For the case of an oblate top, the axis Z is the axis greatest moment inertia I C and given that I A =I B, can be written
, (C<B) (2.44)
(J= 0, 1, 2, ...; k= 0, ±1, ±2, ...).
In these formulas, the rotational constant B corresponds to the moment of inertia about the axes perpendicular to the axis of symmetry.
What values can take the values k and J. According to the laws of quantum mechanics, both quantities can be equal to either an integer or zero. The total moment of inertia of the molecule (quantum number J) can be quite large, i.e. J can take values from 0, 1, 2 ,..., ¥. However, infinite J difficult to achieve, since a real molecule at a high rotation speed can fall apart. If the value J selected, then on the number k restrictions immediately apply: k cannot exceed J as J characterizes the total moment. Let be J= 2, then for k values can be realized k= 2, 1, 0, -1, -2. The more energy is accounted for by rotation around an axis perpendicular to the axis of symmetry, the less k . Since the energy depends squarely on k, then k can also take negative values. From visual representations to positive and negative values k clockwise and counterclockwise rotation about the axis of symmetry can be assigned.
Thus, for a given value J the following values can be realized k:
k = J, J– 1, J– 2, ..., 0, ... ,– (j– 1) ,–J,
i.e. total 2 J+ 1 values.
The first term in formulas (2.43) and (2.44) coincides with the energy expression (2.16) for a linear molecule ( k squared enters formulas (2.43) and (2.44)).
Each level of rotational energy with a given value J with degeneration multiplicity 2 J+ 1 splits into J+ 1 component in relation to the absolute value | k|, which takes values from 0 to J. Since energy depends on k 2 , then for the quantity k indicate its absolute value. Degree of degeneracy of levels with given values J and k equals 2(2 J+ 1), and levels with a given value J and with k= 0 equals 2 J+ 1. For levels k = 0, only the degeneracy associated with the independence of the energy from the quantum number is preserved mJ host 2 J+ 1 values. other levels ( k ¹ 0) are doubly degenerate with respect to k .
Distance between levels with different k(with a given J) depends for an elongated top on the value A - B, and for an oblate top on the value With – AT, i.e., it is the greater, the stronger the corresponding moments of inertia differ. For an elongated top, the energy levels are the higher, the more ( A - B> 0), and for an oblate top the levels are the lower, the more k (C - B< 0). On fig. 2.11 shows the arrangement of rotational energy levels and transitions between them for an elongated top with k from 0 to 3 ( AT = With= 1.0 cm -1, BUT= 1.5 cm–1, left side of the figure) and for an oblate top (B = A = 1.5 cm–1, C = 1.0 cm–1, right side of the figure). The energy levels of an asymmetric top are marked between them (A = 1.5 cm–1, B = 1.25 cm–1, C = 1.0 cm–1).
In the example considered, the rotational constants do not differ very much from each other, therefore, for a given J levels with different k are close to each other. With a large difference in the moments of inertia, which is often the case for real molecules, the normal order of levels with different J may be violated. For example, for an elongated top, the level with J= 3, k= 0, will lie below the level c J= 2, k= 2.
To obtain the IR absorption spectrum of a symmetric rotator, it is necessary to know the selection rules for quantum numbers J and k. Calculations show that for dipole absorption and emission we have D J= ±1 (selection rule similar to that for a diatomic molecule) and D k = 0. Last relation for D k=0 says that during transitions the projection of the angular momentum on the axis of the top should not change. This is true both for absorption and emission spectra and for Raman spectra. In Fig. 2.11, the arrows show the transitions in absorption and emission.
The position of the lines of purely rotational spectra can be determined if, using formula (2.43) or (2.44), we take the energy difference E vr between adjacent levels
For IR absorption D J = 1, J"=J""+1,J"=J"", then
Thus, in absorption and emission, a series of equally spaced lines is obtained, similarly to a current, as was the case for a diatomic molecule.
For CR, possible transitions are determined by the following selection rules
D J= ± 1, ±2, (2.46)
what gives (with J"=J""+ 1,J"=J""+ 2, J" = J) the following series of lines
at D J= 2 (J= 1, 2, ...) and
at D J= 1 (J = 1, 2, 3, ...).
In the latter case, the transition J""= 0 ® J"= 1 is prohibited by additional selection rules. Indeed, the selection rules D k= 0, means that the change in the angular momentum for rotation around the axis of symmetry ( k is the rotational quantum number for axial rotation) does not lead to a change in the polarizability, i.e., there is no Raman spectrum during this rotation. Availability for states with k= 0 only transitions from D J= ±2 means that in the transitions D J= ±1 the ground state cannot participate ( J= 0). For all non-zero J number k may be different from zero and transitions D J= ±1 are allowed.
Thus, in the Raman spectrum we obtain two series of lines, one of which (2.48) coincides with a similar series for a diatomic molecule (), and, accordingly, the second series (the lines of which are located twice as often as the lines of the first series. The lines of the second series coincide after one with the lines of the first series, which leads to an alternation of intensities.This alternation should not be confused with the alternation of intensities due to the nuclear spin.
As we can see, formulas (2.43 and 2.44) imply that they contain only one rotational constant AT. Therefore, by the distance between the rotational lines of a molecule of the type of a symmetrical top, one can determine the moment of inertia about the axes perpendicular to the axis of symmetry of the top. The moment of inertia about the axis of symmetry of the prolate (constant BUT) or oblate (constant With) top can not be determined. An example of molecules that have characteristic rotational absorption spectra and which are modeled by symmetrical tops are the molecules NH 3 , PH 3 , etc.
It should be taken into account that the obtained formulas (2.43 and 2.44) are approximate and do not take into account changes in the spectra that occur as a result of centrifugal stretching. For a symmetrical top, the centrifugal tension depends not only on the quantum number J, but also on the number k. When centrifugal tension is taken into account in formulas (2.43) and (2.44), terms of the fourth order with respect to J and k. Formulas (2.43) and (2.44) contain terms depending on [ J (J+ 1)] 2 , from k 4 and from J (J+ 1) k 2. Taking these terms into account, the rotational energy of a symmetrical prolate top is obtained by the formula
Permanent D J, D k and D J,k too small compared to AT, BUT and With. With IR absorption (D J= 1,D k) for possible transitions we have the formula
The second term in the formula causes only a slight change in line spacing, the last term depending on k, causes splitting of lines J® J+ 1 on J+ 1 components corresponding to the values k from 0 to J. To estimate the values of the constants D J and D J,k we present their values obtained by Gordy for the methyl fluoride FCH 3 molecule: AT= 0.851 cm -1 D J = 2.00 × 10 -6 cm -1, D J,k\u003d 1.47 × 10 -5 cm -1.
Despite the fact that D J,k small (10–4 ¸ 10–6 V), the indicated splitting can be observed for rotational lines due to the high resolution of modern spectrometers used.
2.3.4. Energy levels and spectra of type molecules
asymmetric top
To obtain a picture of the location of the energy levels of an asymmetric top, it is necessary to consider the energy levels of tops that are close to the two simplest extreme cases - an elongated and flattened symmetric top. General expression rotation energy has the form:
In the case of an asymmetric top, all three constants ( BUT, AT and With) are different. If they are arranged in descending order, then A> B> C(for I A<I B< I C). An elongated symmetrical top corresponds to the case when AT = With, and flattened - when BUT = AT. different meanings AT between BUT and With correspond to different degrees of top asymmetry. If a AT differs from BUT and With by a small amount, then the top can be called slightly asymmetric. Rice. 2.11 shows the change in energy levels when changing AT from With before BUT. The levels on the left correspond to an elongated symmetrical top ( AT = With), while the levels on the right are flattened ( AT = BUT). The presence of a slight asymmetry leads to a splitting of energy levels with opposite signs k (k- and k +). These levels are degenerate for symmetrical tops. The doubly degenerate levels of rotational energy of symmetric tops correspond to pairs of very close levels of asymmetric tops. The latter can be called components of doublet levels. In this case, the rotational levels of the oblate symmetric top correspond to the lower doublets of the asymmetric top, for which t< 0 (t = k-–k +), and to the levels of an elongated symmetrical top - the upper doublets of the asymmetric top, for which t ³ 0 (t.= - J, –J + 1, ..., +J). So the lowest level will be J–J, and the topmost J+J. For the special case when BUT\u003d 1.5 cm -1, AT\u003d 1.25 cm -1, With= 1.0 cm -1 ( c= 0) the corresponding arrangement of the levels is shown in fig. 2.11 in the center. As we can see, with increasing at characteristic is the proximity of the two lower levels and the two upper levels. For J= 2 lower level corresponds to level c k= 0 for an elongated top and level c k= 2 for an oblate top, i.e. denoted as 2 02 . Index t equal to the difference k-1 and k 1 can be used to designate the levels of an asymmetric top. For example, for levels J= 2 the symbols 2 02 = 2 -2 , 2 12 = 2 -1 , 2 11 = 2 0 , 2 21 = 2 +1 and 2 20 = 2 +2 will be used.
In table. 2.3 shows the rotational levels of the water molecule (H 2 O - A\u003d 27.79 cm -1, AT=14.51 cm–1. With= 9.29 cm–1) as the first case of interpretation of the rotational structure of the asymmetric top type.
Table 2.3
Energy values of the rotational levels of the H2O molecule, cm–1