Little physics problems

I will cite several problems, mainly from physics. I like them. I hope you enjoy them too.

Forget about black holes, dark energy and matter; forget about the Schrödinger cat, the big bang and the evolution of the universe; forget about strings and superstrings; and even forget about fractals. In these topics, as in politics, the majority considers themselves possible to speak out. And speak out. And much was said and said sensible, and even more was said and said confusion and simply absurdity. I confess, I had a hand in this. And let's get back to the simplicity of classical physics and its understandable tasks. Sometimes it’s good to go down from heaven to earth.

For most tasks, I do not provide a solution. The most useful thing is to find the solution yourself. Of course, the tasks are not for a professional physicist, excluding the task of the tape and the gun.

Most of the tasks, one way or another, were discussed on the Internete. But time goes on and new generations come, and maybe for them the tasks will be new.

I am an IT specialist. Then why am I writing about physics? I am a physicist by training. When I changed my profession, I began to design and program with great enthusiasm. I dreamed that for twenty to thirty years programming would take such a step forward that it would not be necessary to implement the algorithm in detail, but only need to describe the statement of the problem. And what do I see after thirty years? I have reached the age when you can bobble. And I will do so. Judging by the “best publications” section of Habr, programming is in many ways marking time. The same eternal digging in programming languages, complicated by the fact that their number has grown by two orders of magnitude (offhand). The misfortune of verbiage of pseudo-Anglicisms was added:

sheherdit missed, features, bugs and property, hackathon, frontend and backend, hadup, lead, yard, developer, meeting, sawed, fuck, guide book, review, push; There is a code analyzer, but it doesn’t work by push (and even more so not at the time of writing the code), but with a great delay and checks at the wrong level that all sorts of sonar cube and other pvs. pool request; do nothing from the word at all, EGAIS, fix, offer, subject. By itself, BSP is a great thing (like BPO and BIP), mysta, fusion, ZUP, reptiloids, Klushkina ZiK, trash, of all typical conf, it was ZuP / ZiK and their modules in the soft starter / spacecraft that least of all always interfered with the code, the dude opens a pullrequest, the dude appoints himself as a reviewer, the dude writes LGTM, the dude holds the code, it will be deanon, skill, legacy, outstaff, ticket, I emailed task in two days, fetch;the task scope now allowed me to slightly refactor the swamp, full of shit in which I swim all the time; Submetod, almost a hundred people in outstaff (half of the whole company) turned out to be "waiting for a new project", consider them limbically unemployed.

I selected these pearls of the game in the form of just two Habr articles and comments on them. And, it seems, this style is approved by Habré. And it seems by default that the goof is the one who does not adhere to this style. Classics are not held in high esteem in postmodernism. Yes, the described style reminds me of a loud, humiliated semblance of the style of postmodernism. There is a wonderful book on this subject, “Intellectual Tricks. A Critique of Modern Postmodern Philosophy . ” Authors Jean Bricmont, Alain Sokal. Especially good in it is “Appendix A. Breaking the Boundaries: Toward a Transitive Hermeneutics of Quantum Gravity”. One would like to write an article “Breaking the Boundaries: Toward a Transformative Hermeneutics of Programming Languages”. But, alas, the talent is not the same.

But in physics and mathematics there are incomparably more reasons to saturate the language with terminological innovations (bordism, cobordism, homotopy, homology, cohomology ...). And in physics and mathematics, English, no doubt, is no less applied than in programming.

But for some reason in physics and mathematics there is no such anglicized diarrhea as in programming. Physicists and mathematicians preserve the human language and do not show off unnecessarily. And if a term is introduced, then it is clearly defined and its necessity justified.

In general, I am disappointed with the progress in programming. Maybe the money slows him down? It can be hindered by a very low entry threshold, when almost any youth quickly begins to “sculpt”, earn and teach others with “ease in unusual thoughts”. I repent, and it was with me. In less than a year of programming, I was already a project manager and thought that I grabbed God by the beard. But soon I returned to reality. Decades have passed, and I, unfortunately, do not see much progress in programming. "And nothing has changed". But in physics over the past thirty years, so much has been done: the discovery of the accelerated expansion of the Universe, the theory of strings and superstrings, quantum computing ... And, most importantly, a standard model has appeared.

Depressed by the stagnation in the basics of computer science, I prefer to write articles on physics, mathematics, rather than articles on programming.

For starters, we start entirely with elementary problems and tasks not from physics. Their decision will give occasion to reflect on the quirks of intelligence.

Arnold Primer

Russian mathematician Arnold cited such a problem.

Masha didn’t have enough for buying the primer seven kopecks, and Misha had one kopek. They formed to buy one primer for two, but there was still not enough money. How much was the primer?

If the answer does not quickly come to mind, then you are no longer a child. And I was convinced that I was far from a child. Alas, the answer did not come to me instantly.

Arnold said that the more important and grave a person was, the slower he would solve this problem. Much knowledge seems to interfere with fast, direct thinking, and prefers to apply a pattern of experience.

Tolstoy and three hats

. 10 . , , 25 . 25 . . 10+10+5. 15 . - , 25 . , . . C ?

( , (). 30% , 20% 10% .)

Once upon a time, I talked about this task to programmers at my workplace. Whatever a person is, a different answer. One project manager fought for his wrong answer. And he developed accounting in his project. So, in the end, he compiled a journal of bank entries, knocked out a balance and came to the correct answer.
Well, and how do you like the level of the 2nd grade TSSP!

Now let's move on to physics.

Volts to amperes

From mathematics, we know that multiplication is determined through addition. For instance:
$$$4*3=4+4+4$

But why then in physics volts can be multiplied by amperes, but volts with amperes cannot be added. What physics is hindering this?

Moreover, you can divide the volt into amperes, but you can not add and subtract.

Similarly, the degree is determined through multiplication. For instance:$$$a^3= a*a*a$ . Then why can not you build a volt into an ampere?

Is such a situation possible: I energize$$$U$ on the black box, measure the current$$$I$ in the circuit, measure the given output and get that it changes as a function$$ ? The answer from the point of view of mathematics, I found in Bridgman’s book “Dimension Analysis” and in Kogan’s book “Dimension of Physical Quantities ” (written clearly following Bridgman’s tracks). To my shame, I did not know the answer given. But still, I can’t take it physically. Where, physical, not mathematical reasoning? Bridgman's reasoning is almost convincing. But, and if I put in a black box, a person with a battery and a rheostat ... (what else is needed?) And he in response to I and U at the input, he gives at the output$I^U$



$$. All this can be done. Does this refute Bridgman’s constructions? One could argue that this is not a natural example. Ok, we’ll remove the person from the black box and put the machine with the program in its place$I^U$$$ . Does this disprove Bridgman’s constructions? They will tell me that this is again an artificial device. But this, if you know the black box device. Or, perhaps, be satisfied with the answer that if we are dealing with a device, we must conclude that this is not a natural device, but an artificial one, which, therefore, is not subject to natural laws?$I^U$

Which pool is harder to play big or small?

And balls and cues and pockets and a table, billiards are reduced to the same extent. Billiard player has not changed. Which pool is harder to play big or small?

Earthquakes and Earth axis

Can the earth's axis move as a result of an earthquake? Can the length of the day as a result of an earthquake change?

Here is the information from the media.

11 2011 8,9 . 373 - , 24 .

(JPL) (Richard Gross) , 15 139- . 1,6 .

, , , 10 .

27 2010 8,8 . NASA , , . , , 1,26 . , 2,7 ( 8 ).

(), 6,8 , 7 .

And what does the physicist think about this? We consider a reference frame with a beginning in the center of the Earth and motionless relative to the stars. As we know from physics, it is quite inertial. Then the axis of rotation of the Earth coincides with the straight line along which the vector of angular momentum of the Earth is directed. From physics, the law of conservation of angular momentum is known. According to him, no internal perturbations in the system change the angular momentum of this system. This means that no internal earthly cataclysm changes the moment of momentum. Now, if it becomes external, then the matter is different. Examples of cataclysms of an external nature: the fall of an asteroid, the ejection of part of the Earth into space, the gravitational effect of the moon.

So, the axis cannot shift as a result of an earthquake on Earth, unless an ejection into space has occurred. But here the earth’s crust or ice at the pole can move. And if a flag was stuck at the pole, then as a result of an earthquake, it may already be stuck not exactly at the pole. But this did not move the axis, but the surface shifted.

Further, if the moment of momentum does not change, then the angular velocity will not change and, therefore, the length of the day will not change.

Yula in a bowl

I occupy a one-year-old grandson, launching a yule. She runs away under the sofa, then under the table. Tired of crawling after her. I start to let her in a round bowl so that the yule does not run away.


I find the effect of the yule flying out of the bowl: if you swing the bowl around a little, the yula starts climbing up the wall with acceleration and sometimes quickly flies out of the bowl. Own moment goes into orbital. What is the mechanism of this transition?
It is advisable to take the bowl with the bottom smoothly mating with the walls. To get the effect, you usually need to try several times.

Stone breaks ice

Early winter. Thin, fresh, smooth ice is on the pond. I take a stone and throw it on the ice to punch it. If a stone breaks through ice, then the effect of a circular air wave is visible - a divergent ring-bubble, eventually breaking up into individual bubbles.


How and why is a bubble ring formed? Stone trapping behind you? No. I experimentally checked this: even if a stone gets stuck in ice and prevents the penetration of air, the picture remains the same as when a stone penetrates through ice.

Related issues:

• When will the circle break up into bubbles?
• How fast is the circle expanding?

Spoon of sugar

Any physical problem for its practical solution requires some simplification. So when moving a stone thrown by hand, you can neglect the force of friction of the stone on the air. Further, to clarify, we can introduce friction proportional to the speed of the stone. And then you can enter a term quadratic in speed. And already such a model is easier to explore on a computer. And you also need to take into account the change in gravity with height. And you also need to take into account the change in air density with height ... Therefore, we can talk about a hierarchy of simplifications . For each accuracy, its own hierarchy of simplifications is required.

But for some tasks I could not determine this hierarchy.

Here are two examples.

How much sugar can be scooped up with a spoon practically?Theoretically - endlessly. But physics is an account of reality: flowability, trembling of the hand, vessel, air, tilt of the spoon, shape of sugar particles ... I could not offer a simple realistic model. Can you?


A similar task: a stack of dominoes in which each knuckle is shifted relative to the bottom. It turns out a stack with an inclination towards the shift, so that the knuckles move more and more, relative to the bottom. Theoretically, the shift of the upper knuckle can be made infinite with respect to the lower knuckle.


But in reality, what is the maximum possible shift ?

“Just rotate the Earth wherever they want, our interchangeable companies on the march”

Unforgettable first year physics department of BSU. Hostel. Evening in 113 rooms. We went to bed. A chatter began on all sorts of topics. There is such a task. The grasshopper jumped. So he lands, but does not stop, but immediately pushes off again, etc. And all in one direction. Each time he pushes the ground like a Vysotsky company. The question is, how much can he spin the earth with an arbitrarily long time of his jumps? Well, then you can go to Vysotsky’s companies.

We discuss for a long time, argue, then a voice is heard: “But what about maintaining the moment of momentum for a closed system?” And it became clear to everyone.

However, what is the power mechanism that interferes with the endless promotion?

Downwind faster than wind or use from wealth

Can a yacht sail faster than the wind?

This question was asked by telephone to my nephew businessman. I was indignant and said that this is nonsense. And in response: my friend has a yacht and he claims that you can sail faster than the wind. I soon realized what was happening.

This incident made me think about this. It turns out to the rich some things are practically accessible that the poor theorist seems absurd. Having become rich, my nephew began to think more relaxed, wider. He has become more interesting. Loose thinking led his business to Germany, where he offered the company a tricky tax evasion scheme. The Germans looked at him in amazement and twisted a hand at the temple: the law must not be circumvented!

Germany soon bothered him and he returned to Belarus. Now he speaks about the Germans for some reason, with the epithet “square”.

Sharp wave on the water or use from contemplation

Anyone who was fishing or just on the shore of the lake and watching its surface on a windless day, probably saw the effect of a sharp wave: sometimes there is excitement on the surface of the water and in the form of a sharp wedge (Mach cone?), Runs along the water straight or with a curvature , and disappears in a moment. Sounds like that? how a fish runs away from a pike along the surface of the water. But no matter how I tried to find something objective on the edge of the wedge, I did not find anything. The speed of the wedge tip is ten times greater than the speed of an ordinary circular wave on water. I did not find a description of the effect in any monograph on hydrodynamics. So when and how is a wedge-shaped wave formed on water?

Gun and general relativity

This is Feynman’s task. A shell with an atomic clock is shot vertically upward from the cannon from the Earth. In the end, he falls to Earth. They take out a watch and check with the earth. What clocks will lag due to GR effects? Neglect the acceleration time during the shot.

Some doctors of science began to calculate, to calculate ...

For non-physicists I will say that it is necessary to attract the principle of general relativity and the proven fact that the stronger the gravitational field, the slower the clock goes.

What is the collision mechanism under which the Maxwell distribution is established?

In thermodynamic equilibrium, the velocities of gas molecules are distributed according to Maxwell. And this distribution says that there can be molecules with arbitrarily high speeds. What absurdity, though unlikely? And even if we recognize the existence of a limit of the individual energy of the molecule, then how to understand the mechanism of uneven distribution? For example, we start from the same speeds. Collisions begin, as a result of which, allegedly, a redistribution of energy occurs. Usually consider elastic bumps. But with an elastic impact, energy cannot change. And with inelastic energy can only be lost. So what is the collision mechanism at which the Maxwell distribution is established? How do the high speeds of individual molecules arise?

Speed ​​vector

Speed ​​is a vector. If a point participates in two speeds, then the resulting speed is the vectorial sum of these two speeds. Galileo took advantage of this when determining the speed of a body thrown at an angle to the horizon. This movement is a superposition of two movements - vertical, controlled by gravity and horizontal - by inertia.
Now consider the movement of the arrow in the bow. The arrow rests in the middle of the bowstring. Half of the bowstring reaches for one top of the bow, and the other half reaches for the other top of the bow. So the movement of the base of the arrow is a superposition of the two indicated movements.

Accordingly, the resulting speed of the base of the arrow (and the entire arrow) is the vector sum of the speeds of movement to the ends of the bow. Now let's imagine that we have stretched the bow very much, so we can assume that the halves of the bowstring are almost parallel and their speed is v. Then the resulting speed is 2v, which means that the arrow must come off the bow. Absurd. So what's up?

Where did the energy come from?

My son and I drive to the village by car. My son is asking me a question. “Here we go with speed v. Let in my hands a stone of mass m. I throw it forward with speed v. Before the cast, the stone had kinetic energy$$$mv^2/2$ relative to the ground. When throwing, I added the same energy to him$$$mv^2/2$ . So, based on the energy balance, he will have a total kinetic energy$$$mv^2/2 + mv^2/2 = mv^2$ . At the same time, we calculate the kinetic energy by the definition of this energy:$$$m(2v)^2/2 =2mv^2$ . Difference with the first approach$$ . Paradox. The balance approach is not consistent with the kinetic approach from the definition. What's the matter? Explain how a physicist by education ” I could not explain the paradox right away. I was surprised myself. Moreover, having solved the problem later formally, that is formally,I cannot agree with the solution at the level of intuition. This situation is not uncommon (for the average level of thinking, apparently). So, for example, the motion of a gyroscope follows from Newton's equations. But still, it is very difficult to explain why the top does not fall on the basis of the force point of view, and not on the basis of the law of motion of the kinetic moment. And this is only a small step from the direct application of Newton's laws to the application of the consequences of these laws. And what then to speak about statistical physics, for example.$mv^2$

Expansion of the universe

A strip of rubber is attached to the wall. $$, they begin to stretch it at a constant speed without breaking to infinity, by the time the stretching begins, a point snail begins to crawl along it (from the end of the tape to the anchor point) at a speed$l_0=1$$$ lower tensile speed$v$$$$V$ , the question is: will it creep to the edge and how much time?)

This task appeared so. There was a conference on particle physics. Dinner. Theoretical physicist L.B. Okun offers various theorists to solve the given problem in order to check the quick thinking of theoreticians. HELL. Sakharov answered in two minutes.

It took me about two hours to outline a solution, which, however, I was not sure about. Then I doubted the correctness of the decision and spent another two days bringing the solution to a neat appearance. However, I am not 100% sure of the correctness of the decision now.
Isn't this a task similar to the task of moving a space rocket in an expanding universe? By the way, on the stretching tape at any point, the effect of expansion will be observed - all surrounding points run away from it.

Two days ago, he picked up the book of Gardner, the master of entertaining mathematics, and discovered this problem in a discrete version. Yes, nothing is new under the moon. "

And if the cochlea moves from the anchor point to the other end?