Friday post: almost trivial puzzles

    Hi Habr, here is Friday, I want something funny. A post came out last week that reminded me of some fun puzzles. Of course, I am not Perelman or Makovetsky , but I will try to make my contribution. In the end, they wrote every day, and I - only on Fridays.

    Only one request - for those to whom the solutions are obvious, remember that the mentioned luminaries not only posed interesting tasks, but could also explain their solution just as interestingly and clearly.

    # 1 Seven integrals and a magic mirror

    Probably everyone knows that our space is homogeneous and isotropic . This means that if I conduct some kind of physical experiment while sitting in a chair, and then transfer to another chair and repeat it, the results will be the same. This is the homogeneity of space and from it follows the law of conservation of momentum . In the same way, if I do not conduct this experiment now, but say in an hour, the result will not change anyway, it is the uniformity of time and the law of conservation of energy follows from it . (I always suspected that my attitude to life had some fundamental basis).
    And finally, if I lazily turn in the chair half a turn and repeat the same experiment, I will still get the same result, this is the isotropy of space and the law of conservation of angular momentum follows from it .
    Yes i forgot to say
    that I live far away - in deep space, all your gravity does not reach me and I basically never revolve around anything.

    Together, these symmetry / conservation laws give us the first seven integrals of the classical equations of motion.
    And now, having finished my experiments, I finally get out of the chair, go up to the mirror and smile dazzlingly at myself. However, excuse me, it’s not me in the mirror at all - he has a watch on his other hand, he winks at me with the other eye, and in general he has a heart on his right, in a word freak-freak . Even children know that in the mirror I see my resemblance, however symmetrically reflected relative to the vertical plane. And now, ALLOW, but after all, the space is IZO-TROPO-BUT , I just proved it irrefutably! Why is the VERTICAL plane so distinguished? And why, for a change, do not see myself in the mirror upside down, reflected relative to the horizontal plane?

    # 2 Spherical bicycle in vacuum

    No, we’ll take the most ordinary bike, but we’ll put the perfect tires on it - from infinitely thin, infinitely flexible and absolutely inextensible rubber (if you are inclined to experimental tests, look for alibabe). Pump it up and go! Stop, are you sure that you will go far?
    And what interesting force holds the bike virtually in the air? The pressure on the wheel rim is exactly the same from above and below, it looks like the rim miraculously just hung without touching the ground.
    At this point normally distributed chorus mumbling saying that the compressive pressure of the tire is increased and it increases energy and de-E / X for de will just be that Eph .... . However, FORCE IS WHAT? What exactly presses the bike from below?
    In short, one who cannot give a satisfactory explanation loses the right to travel on any wheeled vehicle with the exception of the tram.

    # 3 Seriously charged cat

    This is just the task for which I was encouraged to view the cartoon from the comments on the aforementioned post.
    For those who are too lazy to watch, a brief summary: a positively charged cat (I wonder what other exotic animals live at the authors?) Stands and silently looks at a wire with an electric current running past it. As you know, a wire consists of motionless positively charged ions and freely moving electrons, which create this very current. Then the cat apparently gets bored, and it just as silently begins to move parallel to the wire at the same speed as the electrons. In this case, the electrons are stationary relative to it, and the ions begin to move and, according to the theory of relativity, are compressed in the direction of motion, and the distance between them also decreases. Thus, the ion density becomes larger (from the point of view of the cat) and the wire receives a positive charge that naturally repels the cat.
    It turns out a very elegant explanation of the nature of the magnetic field, however ... But when the cat stood, because then the electrons moved? And they were also compressed, and they had to pull this same (FIXED - notice) cat?
    Somehow this is poorly consistent with my picture of the world. In general, these cats can not be understood, especially charged.
    And then what is the nature of the magnetic field?

    # 3.a Einstein from around the corner

    This is a logical continuation of the previous task, inspired by a dream, inspired by viewing the previous cartoon.
    In general, I am standing in the middle of an empty and long corridor , gloomy shadows .... The door opens and the Einsteins come in in a piece, slowly, in a foot and minting step, they end by an endless chain and hide at the end of the corridor, I see twenty of them all the time, as soon as one enters the corridor, the other exits from the other side. But these are Einsteins, they don’t know how to walk nonrelativistically, and the chain gradually accelerates so that Einsteins begin to experience a serious Lorentz contraction, and the distance between them also decreases. In case someone objects to me that only Einsteins themselves are shortened and not the distance between them, we will give everyone a rope in their left hand, let it connect them all and shrink with them, now the skeptics are confounded. It turns out that I'm starting to see more and more Einsteins at the same time, because the length of the corridor does not change. Well, let it be so, which you will not see in the fast phase of relativistic sleep.
    By the way, why exactly to the left? Because in the right one, a blaster suddenly appears from everyone from which they begin to shoot me together. Mother! But I also have a blaster, and I'm younger, and I once visited a military department! And most importantly - all of these Einsteins suffer from the Lorentz time dilation, so I could easily cope with twenty. But there are more and more of them, and there will be more and more until they overcome me, is there any paradox in this? By the way, how many of them can see me at the same time? While walking slowly it was 20, but now? Indeed, from their point of view, the corridor is getting shorter, so they can see less and less of their own, can I find myself outside the corridor at some speed?
    And in general, is it not a dream? Where did so many Einsteins come from? Well, 100, well, 200, but these already go past me for two hours and they all don't end. Guess they just fool me! The corridor loops around and they run past me in a circle! But then again ... if their finite number and I see more and more, then in the invisible part of the corridor there are less and less? But although the distances between them decrease but remain the same, then the invisible part of the corridor itself becomes shorter?
    Mysteriously all this, perhaps I should go back to the slow phase of sleep before it's too late.

    What did I want to say with this post? There is simply such a thing - a thought experiment . It seems Sir James Clerk Maxwell invented it , the one who invented the Maxwell demonguardian now BSD. So, following the masters of this business gives true pleasure and expands consciousness no worse than many other substances. And mastery of the technique of a thought experiment allows you to turn the most boring meeting into a magical carnival of the spirit.
    Not boring to everyone on Friday

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