Friday post: almost trivial puzzles
Hi Habr, that is Friday, I want something merry. Last week, a post came out that reminded me of several fun puzzles. Of course, I am not Perelman and not Makovetsky , but I will try to contribute. In the end, they wrote every day, and I only wrote on Fridays.
Only one request - for those to whom the solutions are obvious, remember that the above-mentioned leading figures not only posed interesting problems, but could also explain their solution in the same interesting and understandable way.
Probably everyone knows that our space is uniform and isotropic . This means that if I conduct a physical experiment sitting in a chair, and then move 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 conduct this experiment not now, but let's say in an hour, the result will not change anyway, this 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 around in a chair half a turn and repeat the same experiment, I still get the same result, it’s an isotropy of space and it follows the law of conservation of angular momentum .
Together, these laws of symmetry / conservation give us the seven first integrals of the classical equations of motion.
And now, having finished my experiments, I finally get up from my chair, walk up to the mirror and smile at myself dazzlingly. However, let me have it, I’m not in the mirror at all - he has a watch on his other hand, he winks at me with another eye and in general he has a heart on his right,with the word freak-freak . Even children know that in the mirror I see my similarity, but symmetrically reflected relative to the vertical plane. And now, once again, LET, but the space of I-ZO-TROP-BUT , I just proved it incontrovertibly! What is the way allocated to the vertical plane? Why should I, for a change, not see myself in a mirror upside down, reflected on a horizontal plane?
No, we will take the most ordinary bike, but we will put the tires on it ideal - from infinitely thin, infinitely flexible and absolutely inextensible rubber (who is inclined to experimental checks - look at alibabe). How should we pump and go! Stop, are you sure you will go far?
And what is the interesting power of keeping the bike actually in the air? The pressure on the wheel rim is exactly the same from above and below, it seems that the rim miraculously just hung without touching the ground.
At this point, a friendly boom is usually heard saying that when a tire is compressed, pressure builds up in it and energy grows, and de-E / de-iX will be just the effect that ..... However, what is the power of what? What exactly puts pressure on the bike below?
In short, the one who can not give a satisfactory explanation, loses the right to move on any wheeled transport with the exception of the tram.
')
This is exactly the task for which I was encouraged to view the cartoon from the comments to the above post.
For those who are too lazy to look, a brief summary: a positively charged cat (I wonder, what other exotic little animals live at home with the authors?) Stands and silently looks at the wire with an electric current running past it. As is known, the wire consists of stationary positively charged ions and freely moving electrons, which create this current itself. Then the cat apparently gets bored, and she just 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, contract in the direction of motion, and the distance between them also decreases. Thus, the density of ions becomes greater (from the point of view of the cat) and the wire receives a positive charge, which naturally repels the cat.
It turns out a very elegant explanation of the nature of the magnetic field, however ... And when the cat was standing, then did the electrons move? And, too, were compressed, and should have this same (IMMOBILE - notice) cat ATTRACT?
Somehow it is not consistent with my picture of the world. In general, these cats do not understand, especially charged.
And then what is the nature of the magnetic field?
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, dark shadows .... The door opens and comes in Einsteins, slowly, in step and chasing step, they walk past me endlessly 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 goes from the other side. But this is Einsteins, they do not know how to walk nonrelativistically, and the chain gradually accelerates so that the Einsteins begin to experience a serious Lorentz contraction, and the distance between them shrinks. In case anyone says to me that only the Einsteins themselves are reduced, but not the distance between them, we will give everyone a rope in my left hand, let it connect them all and crawl along with them, now the skeptics are confounded. It turns out that I am starting to see more and more Einsteins at the same time, the length of the corridor does not change. Well, let it be so, what you will not see in the fast phase of relativistic sleep.
By the way, why in the left? Because on the right everybody suddenly has a blaster from which they begin to fire at me in unison. Mama! But I also have a blaster, and I am younger, and I once attended a military department! And most importantly, all these Einsteins suffer from the Lorentz slowing down of time, so with twenty I would manage easily. 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? And by the way, how many of them can see me at the same time? While walking slowly was 20, and now? After all, from their point of view, the corridor becomes shorter, so they can see less and less of their own, can I be out of the corridor for them at some speed?
And generally, is this a dream? Where do so many Einsteins come from? Well, 100, well, 200, but these already pass by me for two hours and everything does not end. I guessed, I'm just being fooled! The corridor loops and they run past me in a circle! But then again ... if there is a finite number of them and I see more and more, then in the invisible part of the corridor they are less and less? But although the distances between them are decreasing, but remain the same, then the invisible part of the corridor itself becomes shorter?
Mysteriously all this, perhaps it is time for me to return to the slow phase of sleep before it is too late.
What did I want to say with this post? Simply, there is such a thing - a thought experiment . Invented it seems Sir James Clerk Maxwell , the one who invented the demon Maxwell patron of the now BSD. So, following the masters of this business is a real pleasure and expands the consciousness not worse than many other substances. A mastery of the technique of mental experiment allows you to turn the most boring meeting into an enchanting carnival of the spirit.
Happy Friday everyone
.
Only one request - for those to whom the solutions are obvious, remember that the above-mentioned leading figures not only posed interesting problems, but could also explain their solution in the same interesting and understandable way.
# 1 Seven Integrals and Magic Mirror
Probably everyone knows that our space is uniform and isotropic . This means that if I conduct a physical experiment sitting in a chair, and then move 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 conduct this experiment not now, but let's say in an hour, the result will not change anyway, this 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 around in a chair half a turn and repeat the same experiment, I still get the same result, it’s an isotropy of space and it follows the law of conservation of angular momentum .
Yes i forgot to say
that I live far away - in deep space, all your gravity does not reach me, and in principle I never revolve around anything.
Together, these laws of symmetry / conservation give us the seven first integrals of the classical equations of motion.
And now, having finished my experiments, I finally get up from my chair, walk up to the mirror and smile at myself dazzlingly. However, let me have it, I’m not in the mirror at all - he has a watch on his other hand, he winks at me with another eye and in general he has a heart on his right,
# 2 Spherical bike in vacuum
No, we will take the most ordinary bike, but we will put the tires on it ideal - from infinitely thin, infinitely flexible and absolutely inextensible rubber (who is inclined to experimental checks - look at alibabe). How should we pump and go! Stop, are you sure you will go far?
And what is the interesting power of keeping the bike actually in the air? The pressure on the wheel rim is exactly the same from above and below, it seems that the rim miraculously just hung without touching the ground.
At this point, a friendly boom is usually heard saying that when a tire is compressed, pressure builds up in it and energy grows, and de-E / de-iX will be just the effect that ..... However, what is the power of what? What exactly puts pressure on the bike below?
In short, the one who can not give a satisfactory explanation, loses the right to move on any wheeled transport with the exception of the tram.
')
# 3 Seriously charged cat
This is exactly the task for which I was encouraged to view the cartoon from the comments to the above post.
For those who are too lazy to look, a brief summary: a positively charged cat (I wonder, what other exotic little animals live at home with the authors?) Stands and silently looks at the wire with an electric current running past it. As is known, the wire consists of stationary positively charged ions and freely moving electrons, which create this current itself. Then the cat apparently gets bored, and she just 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, contract in the direction of motion, and the distance between them also decreases. Thus, the density of ions becomes greater (from the point of view of the cat) and the wire receives a positive charge, which naturally repels the cat.
It turns out a very elegant explanation of the nature of the magnetic field, however ... And when the cat was standing, then did the electrons move? And, too, were compressed, and should have this same (IMMOBILE - notice) cat ATTRACT?
Somehow it is not consistent with my picture of the world. In general, these cats do not understand, especially charged.
And then what is the nature of the magnetic field?
# 3.and Einstein because of 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
By the way, why in the left? Because on the right everybody suddenly has a blaster from which they begin to fire at me in unison. Mama! But I also have a blaster, and I am younger, and I once attended a military department! And most importantly, all these Einsteins suffer from the Lorentz slowing down of time, so with twenty I would manage easily. 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? And by the way, how many of them can see me at the same time? While walking slowly was 20, and now? After all, from their point of view, the corridor becomes shorter, so they can see less and less of their own, can I be out of the corridor for them at some speed?
And generally, is this a dream? Where do so many Einsteins come from? Well, 100, well, 200, but these already pass by me for two hours and everything does not end. I guessed, I'm just being fooled! The corridor loops and they run past me in a circle! But then again ... if there is a finite number of them and I see more and more, then in the invisible part of the corridor they are less and less? But although the distances between them are decreasing, but remain the same, then the invisible part of the corridor itself becomes shorter?
Mysteriously all this, perhaps it is time for me to return to the slow phase of sleep before it is too late.
What did I want to say with this post? Simply, there is such a thing - a thought experiment . Invented it seems Sir James Clerk Maxwell , the one who invented the demon Maxwell patron of the now BSD. So, following the masters of this business is a real pleasure and expands the consciousness not worse than many other substances. A mastery of the technique of mental experiment allows you to turn the most boring meeting into an enchanting carnival of the spirit.
Happy Friday everyone
.
Source: https://habr.com/ru/post/230395/
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