[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 527
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Every body with an asymmetric shape (non-spherical, non-cylindrical, etc. - in short, asymmetric) has three dedicated axes of inertia passing through its centre of gravity. They are called principal axes. Each such axis has its own value of moment of inertia - minimum, average and maximum.
As far as I remember, if I'm not mistaken, the rotation of the body along any axis that does not coincide with the major axes is unstable. This was explained at the level of theory about 200 years ago, almost under Euler. It is also unstable when rotating around the major axis.
Unstable in itself means that any however small imbalance (error) in the rotation relative to the axis will cause it to grow rapidly. That is, even if you start the rotation very precisely, but on one of these unstable axes, a flip will still occur.
Of course, all the laws of conservation apply - of momentum, momentum and energy. That is why there is a reversal strictly by 180.
Another thing is that it is not so easy to detect this effect in practice. It took weightlessness.
Another thing is that it is not easy to detect this effect in practice. It takes weightlessness.
It is easy and simple to detect it - go to the circus or figure skating.
By the way, a cylinder can do the same trick; the main requirement here is total three-dimensional mass symmetry of the body relative to its own centre of mass, geometry has nothing to do with it.
A gymnast or a figure skater is grouped to accelerate rotation and unfolds to slow it down. It's the same here, but in reverse order and without a forced stop at the moment of touchdown.
Unfolding the cylinder, firing it perpendicular to the plane of rotation. Almost perpendicular,- therefore, a slight runout is created due to the rotation of the centre of mass around the axis of rotation.
The angular velocity of the rotation is reduced (the gymnast "unfolds"), but the position is unstable because the inertia of the centre of mass shifting forward as the "shot" proceeds around the intersection of the two axes: rotation and mass. As the centre of mass follows this inertia, the rotation speed decreases to zero, the cylinder overturns and begins to "group", automatically increasing the rotation speed already in the opposite direction. At the point of convergence of the two previously mentioned axes, stabilisation of the rotation parameters takes place, but the same inertia still leads to the repetition of the half-cycle. At the stabilisation point the angular velocity of rotation is maximum and the "flip" velocity is minimum, so the movement is most stable. At the "opening" point, it is the other way round. There is nothing much to simulate here, imho :)
SZZ messed up a bit :) about mass symmetry relative to the centre of mass. Relative to the axis of rotation, of course. Accordingly, the smaller the relative elongation of a body with uniform density relative to the axis of rotation, the more noticeable the effect will be. That's why it was the impeller, and not a simple nut, or, for that matter, a bullet, that As hand-jointed :) By the way, Janibekov's main phenomenon is his inordinate dynamic spatial imagination; there is not a single individual in the world capable of even a pathetic imitation of him.
What's in the video:
1. The rotation of a body is stable about the axes of both the greatest and smallest principal moments of inertia.
Under earthly conditions, an example of stable rotation about the axis of smallest moment of inertia: the rotation of a flying bullet is stable. Stable rotation around this axis, if I am not mistaken, is only true for an absolutely rigid body. A bullet can be considered as absolutely rigid.
In terrestrial conditionsan example ofstable rotation around the axis of the greatest moment of inertia: the gyroscope. By the way,stable rotation around this axis is alsotrue for a not absolutely rigid body. In short, under ideal conditions this rotation is stable for any body for unlimited time. Therefore, only this rotation is used, for example, to stabilize satellites with a significant non-rigid structure.
2. Rotation around an axis with an average moment of inertia is always unstable. Similar states of instability (in terms of energy) have a pendulum at the top or a ball at the top of a mountain.
The rotation will tend to go to a decrease in rotational energy. An analogy: a pendulum and a ball tends to reduce its potential energy. In doing so, the different points of the body will begin to experience variable accelerations. If these accelerations result in variable deformations (not abs. rigid body) with energy dissipation, then eventually the axis of rotation will coincide with the axis of maximum moment of inertia. An example would be a small, long piece of paper released from a height. No matter how you twist it, its rotation will stabilise around the axis with the maximum moment of inertia. If there is no deformation and/or no dissipation of energy (perfect elasticity) then you get an energetically conservative system. Figuratively speaking, the body will tumble around forever trying to find a 'comfortable' position, but every time it will bounce around and search for it all over again. The simplest example is the ideal pendulum. The bottom position is energetically optimal. But it will never stop there. Thus, the axis of rotation of a perfectly rigid and/or perfectly elastic body will never coincide with the axis of maximal moment of inertia, unless it originally coincided with it. The body will eternally perform complex techno-dimensional vibrations, depending on the parameters and initial conditions. You have to put a "viscous" damper or somehow actively dampen the vibrations. The Americans tried to damp these vibrations on their satellites by orientation system 10-15 years after us, wasting an enormous amount of fuel, until ours told the whole world about this effect.
3. If all main moments of inertia are equal , the vector of angular velocity of body rotation will not change either in magnitude or in direction. The example with the cube in the video. Roughly speaking, around which axis you twist, around which axis it will rotate.
What's on the video:
The theme of boobs is not covered
Another thing is that it is not so easy to detect this effect in practice.
Come on :) I found it on my passport cover at work right now (I taped it shut beforehand).
We don't work in orbit if anything :) .
They just do not seem to pay much attention.
It is easy and simple to spot it - go to the circus or figure skating.
Yes, but why is it named after Janibekov then? So it wasn't noticed before - even though theoretically it was all predicted a long time ago.
The trick is that it's a very good illustration of the Earth's pole shift. Very visual and frightening. And it turns out that such a change happens in a day or two, not in thousands of years.