Kvant 1989, No. 1 · Pages

QUANTUM
Popular science
physical and mathematical journal

1 2 3 4 5 6 7 8 9 10 11 12

Ice - X

1989

YAX DARGENT

G. DECHAMP

1

In the room:

2 To our readers
3 Interview with Academician I. M. Gelfand
14 A. V. Zaretsky. Ice-X

From the history of science
13 L. N. Kryzhanovsky. Lightning rod, politics and… hats

Problem «Kvant»
21 Problems M1141—M1145, F1148—F1152
22 Solutions to problem M1116—M1120, F1128—F1132
29 List of readers who sent correct solutions

«Kvant» for younger students
30 V. L. Gutenkhacher. Fractions - camel - parquets

---

34 R. Bradbury. P means package

40 Kaleidoscope «Kvant»

School in «Kvant»
Physics 8, 9, 10:
45 Equilibrium of a mechanical system and the method of virtual displacements
47 How many states does a substance have?
49 A few changes regarding the photo effect

Math circle
52 S. M. Lvovsky, A. L. Toom. It is possible and it is not possible

Olympids
56 V. V. Vasilov, A. A. Fomin. XXIX International Mathematical Olympiad
59 O. F. Kabardin, V. A. Orlov. XIX International Physics Olympiad

Information
65 New admission to the All-Union Correspondence Mathematical School
66 Correspondence Physics and Technology School at MIPT
69 Correspondence Physics and Technology School at MISiSe
70 New admission to the correspondence department of Small Mechanics

72 Entrance exam options
75 Answers, directions, solutions

Our cover

1 Ice is a seemingly simple and well-known state of water. But from the point of view of physics, ice in its mode is a unique object. About this - to become A.V. Zaretskaya.

2 Do you think that the wrappers are tied to the Parisian fashionista hats of the second half of the 18th century? The hint is in the note “Lightning rod, politics and… hats.”

3 Chess page.

4 Puzzle “Naughty balls”.

© Nuka Publishing House.
Main editorial office of physical and mathematical literature, «Kvant», 1989

2

Dear readers!

Happy New Year! For the magazine «Kvant» this is the twentieth year of its existence. Many of those who, as schoolchildren, regularly read our magazine and reflected on its articles and tasks are now scientists, teachers, and engineers. We hope that our new subscribers will find a lot of interesting and useful things in the magazine.

Each issue of the magazine contains a wide range of materials. These are stories about the achievements of science and its applications, about the history of physics and mathematics and new discoveries. We introduce readers to major reporting and foreign scientists, their views and paths to science.

The magazine is filled with a variety of tasks that are necessary to train a drinking mind. These are challenging problems for beginners (“younger schoolchildren”), more serious selected school problems, exam problems for applicants that require even longer understanding of the Olympic data and research problems, cycles of problems for a mathematical circle. A special role is given to the “Problem Book «Kvant»” section, which holds traditional daily competitions for the best solution to a problem from this section. It must be said that most of the materials in our magazine are designed for server reading “with a pencil in hand.”

The magazine will find food for thought not only for “theorists”, but also for “experimenters” - those who love physical experiments.

Materials regularly appear in the magazine, telling about various olympids and competitions, about correspondence and evening physics and mathematics schools, about new books on mathematics and physics.

This year the volume of our magazine has increased by 16 pages. We would like to use this opportunity to better satisfy the wishes of our readers. Thus, those interested in computers and programmers will be addressed to articles on mathematical issues of programming, on the use of computers in science and in life. Much attention will be paid to various areas and achievements in space exploration. In particular, we will provide regular information about the work of the Weighty Youth Aerospace Society, recently created under the Komsomol Central Committee. On the pages of «Kvant» there will also be a place for works of fiction, primarily science fiction.

We would really like «Kvant» to become your friend, assistant, teacher, owner. And this is impossible without your direct participation, without test “feedback”. We are waiting for your letters!

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INTERVIEW WITH ACADEMICIAN
I. M. GELFANDOM

Gelfand’s mathematical seminar, which has been meeting on Monday evenings at Moscow University for 45 years, is universally famous. Both first-year students and famous scientists study at the seminar. Several generations of outstanding mathematicians were educated by this seminar.

Israel Moiseevich is the founder of the All-Russian Correspondence Mathematical School and the chairman of its Scientific Council. The main task of this school is to help those schoolchildren who, unlike residents of Moscow, Leningrad and other large cities, are practically deprived of mathematical literature and contacts with scientists. Created 25 years ago, the All-Use Correspondence Mathematics School was the first school of its type in our country; Subsequently, other correspondence schools were opened following its example.

The meeting with Academician Gelfand was always planned as a general interview «Kvant». We prepared questions that we thought would be interesting to both our reader and our interlocutor. Israel Moiseevich, having looked through the list of questions, noted that the questions were very interesting, but he… did not consider himself competent to answer them.

• You understand, I do not consider myself to have the right to express my opinion to your young reader. Let me tell you how we do mathematics at an age that coincides with the age of most of the readers of Quantum - from 13 to 17 years old. I’m not sure that I now remember all the tasks that I was doing during this time, but I remembered the tasks that I’m going to talk about very well. Here is the story of I.M. Gelfand.

1*

4

One of Graham Greene’s novels is called “The loser takes all”). My mathematical biography was surprisingly happy, and for many years it was the fulfillment of this statement. How am I lucky? In short, firstly, that I did not study at the university (or any educational institution at all), and, secondly, that as a result of certain life difficulties that befell my family, I found myself in Moscow without parents, unemployed, at sixteen and a half years old.

I. M. Gelfand talks with the outstanding French mathematician J.-P. Serre after his speech at Gelfand’s seminar at Moscow State University. April 1984.

I’ll try to understand the meaning of the phrase “the loser takes all” with the help of a story by another English writer, Somerset Mose. The main character of the story, a certain church servant, fails: during the certification of church personnel, it turns out that he is illiterate, and he is kicked out of work. He starts working as a seller of sirah, then buys out a table kiosk, then several other kiosks… and makes a brilliant commercial career, becoming the richest man in his city, its zer. And so they interview him - like you are interviewing me now - and he tells the journalist that he is illiterate. The needy journalist will exclaim: “What heights would you have reached if you were literate!” The answer immediately follows: “I would be a church minister.”

So in February 1930, at the age of 16 and a half, having arrived in Moscow to visit distant relatives, I was partly unemployed, took on any temporary work, and mostly sat in the Lenin Library and “acquired” the knowledge that I had not received at school and at an independent vocational school. In the library I met university students and began attending seminars. At the age of 18 I was already teaching, and at 19 I went to graduate school, and my further mathematical biography followed an ordinary, regular course, here it fits into the mainstream that is inculcated for mathematicians.

But I didn’t want to tell the reader «Kvant» about this part of my life, but about the previous one - about how I studied mathematics between the ages of 13 and 17. I want to talk about this for two reasons. Firstly, in my deep conviction, in the majority of future professional mathematicians, mathematical talent manifests itself precisely in the period from 13 to 16 years**), although, of course, there are fluctuations, from earlier development to much later development (from 20 to 30 and even 40 years), which is observed in some strong mathematicians. Secondly, this period formed

*) Loser takes all
**) Israel Monseevich spoke about this in great detail in an interview published in Izvestia on December 3, 1987.

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Val is my way of doing mathematics. The subject of research, of course, varies
was created, but the artistic image of mathematics that developed at that time
was the basis in the choice of tasks that have attracted me to this day.
Have a nice day. It seems to me that without understanding this motivation it is impossible
understand each algorithm of my ways of knowing and choosing topics
busy. In light of this motivation, they actually developed very differently.
therefore it is logical.

Gelfand’s seminar. For per-
second table from the left -
the growth of the seminar is famous
American mathematician
R. McPherson. 1981

The first thing I remember is from 5th-6th grade. I then realized that there was a task
questions in geometry that cannot be solved algebraically. I made a table
face the ratio of chord length to arc length every 5°. Only a lot of poses
I learned that there are trigonometric (not algebraic!) functions
and that in essence I then compiled trigonometric tables.

Around the same time, I worked through a problem in elementary algebra.
I didn’t have an algebra course, I didn’t know the theory, but sometimes I had to decide
difficult tasks, I use, for example, (unknown to me) abbreviated formulas -
multiplication. If I couldn’t decipher how to solve a given problem,
Well, I looked into the answer, learned from the formulation of the problem and the answers to restore
identify methods for solving them. In particular, I then understood and remembered completely
life, that a new area can be mastered by solving problems, and there is no shame in
look back, because when we solve any problem, we always
there is a hypothesis about the answer. Doing mathematics is generally similar to solving
problems in which something is known about the answer. These mathematical
bot differs from the training required, of course, to perform a competition
no work for entering a university.

At the beginning of 6th grade, I paid attention to geometry problems in which
a right-angled triangle often appeared with sides 3, 4, 5, and even
sides 5, 12, 13. I wanted to find all right triangles with
whole sides, and I found the promised formula for their sides, i.e. I found all Pi-
Fagorian triplets. (Of course, I didn’t know this term then.) Unfortunately,
I don’t remember now how I did it.

I studied mathematics during illnesses and holidays. I have now replaced
It’s amazing how much strong schoolchildren manage to do when they stay away due to illness
Houses. Therefore, after illness for several personal days, I
left it at home.

We go through geometry according to Kiselev, not yet spoiled by subsequent
processing. Some theorems were given in the form of a problem. I achieved my promise
notebook (in those years it was not so easy), wrote on each page

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condition of the theorem and filled almost all the pages with proofs over the summer. This is how I learned to write mathematical papers.

I’ll miss a big period. I will only mention a book on Davydov’s algebra with ingenious methods for solving maximum and minimum problems using elementary methods (i.e., without the use of differential calculus). For example, find the maximum of $ab$ if $a+b$ is given; find a rectangle of maximum area for a given period; find the maximum of the product of non-negative numbers $a_1a_2...a_n$, if their sum $a_1+a_2+...+a_n$ is given; squares are cut out of a square on a given side, and a box is made from the remaining part; How many squares should be cut to maximize the volume of the box?

I was greatly impressed by combinatorics and Newton’s biome, which I had been thinking about for a long time.

I lived in a small town with only one school. My mathematics teacher was a very kind, although seemingly stern man named Titarenko. He had a large Zaporozhye mustache. I have never met a better teacher, although I knew more than him, and he understood it. He loved me very much and encouraged me in every possible way. Encouraging is the most important thing for a teacher, isn’t it?

The supply of books on mathematics was significant. I saw in advertisements that there were books on higher mathematics, and I imagined higher mathematics as something very interesting. My parents could not order these books - there was no money. But here I was lucky again. At the age of 15, I was taken to Odessa to have appendicitis operated on. I told my parents that I would not go to hospital until they bought me books on higher mathematics. The parents agreed and bought Belyaev’s higher mathematics course for colleges in Ukrainian. However, there was only enough money for the first part, with a presentation of differential calculus and analytical geometry on the plane.

I was lucky that I didn’t start with an in-depth university course. It was a very basic book. The level of Belyaev’s book can be judged by the introduction to it. It said, in particular, that there are three types of functions: analytical, defined by formulas; empirical, given by tables; correlational. I learned what correlation functions are many years later from a student studying probability theory.

On the third day after the operation, I took up this book and read it, interspersed with reading the novels of E. Zola, for nine days (then after such operations it was 12 days). During this time I read it.

From the book I got two great ideas. Firstly, any geometric problem on a plane and in space can be written down by formulas. However, I suspected this before. I also learned about the existence of such wonderful funcurds, such as the ellipse.

The second idea - it revolutionized my worldview - was that there is a formula for calculating sinhus: sin $x=x-\frac{x^3}{3!}+\frac{x^5}{5!}$ -…

Before this, I believed that there are two mathematics - algebraic and geometric, and that geometric mathematics belonged to the “transcendent” for algebraic. Take, for example, the formula for circularity - there is a “geometric” number l. Or, let’s say, sinhus - it is determined purely geometrically.

When I discovered that sinhus can be written algebraically in the form of a series, the barrier fell, mathematics became unified. And now I perceive the various branches of mathematics together with mathematical physics as a single whole.

Of course, I was convinced that extreme problems are solved automatically, and although they lose their charm, you have in your hands a powerful apparatus for solving them.

7

While studying differential calculus, I learned that there is also interral calculus related to areas and volumes. But I didn’t know what it consisted of: after all, I didn’t have the second volume of Belyaev’s course!

Here I want to remember one more task. When we studied the volumes of bodies of rotation at school the following fall, my classmate, the later famous mathematician D.P. Milman, drew my attention to the following problem: find the volume of a body formed by the rotation of a circle around a tangent

Presentation of an honorary doctorate degree from the University of Lyon to I. M. Gelfand. June 1984.

Noah. To solve this problem, I divided the circle into strips and calculated the sum of the differences in the volumes of the corresponding rotation cylinders. Thus I came to the need to find the sum

$$\cos \varphi + \cos 2\varphi + \cos 3\varphi + \dots + \cos n\varphi.$$

(*)

What happened next was, as always, a mixture of ingenuity and stupidity. I simplified the mismo of the elementary solution using general trigonometry, but used the formula $e^{i\varphi} = \cos \varphi + i \sin \varphi$ (this formula is called Euler’s formula, but I didn’t know that). I got this formula from the series for sine, cosine and $e^x$ that struck me deeply. Next, it remains to find the sum of the geometric progression $e^{i\varphi} + e^{2i\varphi} + \dots$ and get the sum (*) from it, which is what I did.

On this task, I got into the habit of thinking about the problem even after solving it. Namely: he moved the circle away from the straight line and realized that when rotated, a body is obtained that resembles a rubber circle, on which

*) $i$ — minimum unit ($i^2 = -1$); about complex numbers can be read, for example, in «Kvant» (1983, No. 2, p. 16).

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Adding these equalities, I got

$$F(n+1) - F(1) = S_0 + \frac{S_1}{2!} + \frac{S_2}{3!} + \dots,$$

where $S_0$ is the amount I’m interested in, and

$$S_1 = f'(1) + f'(2) + \dots + f'(n),$$

$$S_2 = f''(1) + f''(2) + \dots + f''(n),$$

Then I wrote down the system

$$F(n+1) - F(1) = S_0 + \frac{S_1}{2!} + \frac{S_2}{3!} + \frac{S_3}{4!} + \dots,$$

$$f(n+1) - f(1) = S_1 + \frac{S_2}{2!} + \frac{S_3}{3!} + \dots,$$

$$f'(n+1) - f'(1) = S_2 + \frac{S_3}{2!} + \dots,$$

i.e. an infinite system with an infinite number of unknowns $S_0, S_1, S_2$, … I already mentioned that Kagan’s book had infinite determinants, so I was able to use the “Cramer’s rule” to find $S_0$:

$$S_0 = \frac{1}{1}$$

I expanded the numerator of this “fraction” into the first column, obtaining

$$S_0 = B_0(F(n+1) - F(1)) + B_1(f(n+1) - f(1)) + B_2(f'(n+1) - f'(1)) + \dots,$$

(1)

where $B_0 = 1, B_1, B_2, \dots$ are numerical determinants of infinite order. The resulting expression is called the Euler-Maclaurin formula, but of course I didn’t know about it. To calculate it, it was necessary to find the coefficients $B_0, B_1, B_2, \dots$.

To do this, I applied considerations that today are called “functional”. Namely, taking advantage of the fact that the coefficients $B_0, B_1, \dots$ does not depend on $f$, I selected a function $f$ such that the left side of the system formed a geometric progression (which I knew how to sum). The function $f(x) = e^{\alpha x}$ is suitable for this, substituting it into formula (1), I got (do the calculations!)

$$B_0 + \alpha B_1 + \alpha^2 B_2 + \dots = \frac{\alpha}{e^{\alpha}-1},$$

i.e., he obtained the product of the function for the required numbers $B_0, B_1, B_2, \dots$ (they are called Bernoulli numbers, and the polynomial $S_0$ for $f(x) = x^p$ is called the Bernoulli polynomial).

From this period I remember two more tasks. The first one came from a problem in our algebra assignment: Express $x_1^2 + x_2^2$ and $x_1^3 + x_2^3$ in terms of the coefficients of a quadratic equation whose roots are $x_1$ and $x_2$. A NATURAL GENERALIZATION IS THE PROBLEM: Express the sums $x_1^2 + \dots + x_n^2$ and $x_1^3 + \dots + x_n^3$, where $x_1, \dots, x_n$ are the roots of the equation $x^n + a_1x^{n-1} + \dots + a_n = 0$, via

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ego coefficients. Here Bezout’s theorem helped me, which I learned from Davydov’s textbook. I went further and set myself a more general task: to express the sum of the $k$-th powers of the roots of the $n$-th degree algebraic equation through the coefficients of this equation. I solved this problem (the answer is called Newton’s formula).

The second problem I was solving then arose when I discovered that cos $ix$ is a real number, or

$$\cos ix = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots$$

I thought about this unexpected fact and proved the following promising theorem: every even real function takes real values ​​on the minimum axis.

To prove this, it was necessary to clarify what a “function” is. I thought about what to call a function, and accepted the following definition: a function is the sum of a convergent infinite degree of a series. After this, the proof of the theorem is almost obvious.

This task, apparently, was the last thing I thought about before my arrival in Moscow. I solved it in the summer of 1929, and in the next six months, which were very difficult for my family and for me, I had no time for mathematics.

The further period of my studies in Moscow, unlike what was described, was not a “pure experiment”. In Moscow I was already exposed to many different influences, and my development no longer followed its own path. During this period, as I already said, I studied on my own in the library named after. V.I. Lenin, surviving on official earnings. However, time was even a control in Leninka. I met some university mathematics students. One of them told me that my interest in expressions of the form $f(n+1)-f(n)$ is associated with a whole science called finite difference theory, and suggested that I need to read Nörlund’s book “Differenzenkalkül” on this topic. The book was in German, but I could understand it with the help of a dictionary.

I started going to university seminars, and then I found myself under heavy psychological pressure. I found that my style of doing math just wasn’t working. New things have appeared in mathematics - new requirements regarding the rigor of proofs, great interest in the theory of functions of a real variable (TFV). (I note that today both this level of strictness and the TFDP are considered old-fashioned and outdated, but then…)

Then I realized that it is very important that a function does not have to be wrong, that a continuous function does not have to be differentiable, that a differentiable function does not have to be twice differentiable, etc.; that even if a function has arbitrary ones of all orders, then its Taylor series does not have to converge and that, even if it converges, its sum does not have to coincide with the value of the function! If this coincidence takes place, then such a function is called analytic, and this class of functions, the adherents of the TFDP argued, is so narrow that it is outside the main intervals of mathematics. And I only considered such functions!

Under the pressure of this point of view, I read a “modern, rigorous” textbook on analysis by Vallée Poussin, similar to those textbooks that are used today at the Moscow State University Mechanics, but better than them. Therefore, I have the understanding of a sympathetic first-year mechanite who is allowed to the beauties of mathematical analysis only after a one-year probationary period, a test of “rigorous justification” of the analysis.

But even here I was lucky. I began to read the wonderful textbook by I. I. Pri-

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the clock makes the box (see figure). What size squares should be cut to maximize the volume of the box?

It is clear that you cannot cut out very small squares, since the box will turn out to be too small. You cannot cut out very large squares, as then the box will be too narrow. How much should you cut?

Note: box volume $V = x(a - 2x)^2$. Woo-
dem look for maximum $4V = (4x)(a - 2x)(a - 2x)$,
i.e., the maximum of the product of three numbers whose sum is 2a.

5. Zadchacha. Calculate the sums of $\cos x + \cos 2x + \dots + \cos nx$ and $\sin x + \sin 2x + \dots + \sin nx$.

6. While calculating the volumes and surface areas of a rotating body, I.M. Gelfand discovered Guldin’s remarkable theorems:

the volume of a body of rotation of a flat figure around an axis that does not intersect it is equal to the product of the area of ​​this figure and the length of the circle described by the center of the gravity;

The surface area of ​​the body of rotation is equal to the product of the length of the boundary of the figure and the length of the circle described by the center of gravity of this boundary.

Zadchacha. At what distance from the center are the centers of the gravity of a semicircle and a semicircle?

7. To get acquainted with the determinants, let’s solve a system of linear equations:

$$\begin{cases} a_1x + b_1y = c_1, \\ a_2x + b_2y = c_2. \end{cases}$$

If $a_1b_2 - a_2b_1 \neq 0$, its solution gives the formulas:

$$x = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1}, y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1}.$$

The quantity $a_1b_2 - a_2b_1$ in the denominator is denoted by $\begin{matrix} a_1 & b_1 \\ a_2 & b_2 \end{matrix}$ and is called the determinant. Numerals also contain determiners $\begin{matrix} c_1 & b_1 \\ c_2 & b_2 \end{matrix}$ and $\begin{matrix} a_1 & c_1 \\ a_2 & c_2 \end{matrix}$.

Similarly, when solving the system

$$\begin{cases} a_1x + b_1y + c_1z = d_1, \\ a_2x + b_2y + c_2z = d_2, \\ a_3x + b_3y + c_3z = d_3 \end{cases}$$

the denominator will contain third-order determinants. Determinant

$$\begin{matrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{matrix}$$

equalized $a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1)$. Note that there is a second-order determinant in parentheses. This notation of the determinant is called its decomposition in the first column.

It is possible to agree on which one-years of the variables $a_1, a_2, \dots, d_4$ the determinant of the fourth order consists of:

$$\begin{matrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ a_4 & b_4 & c_4 & d_4 \end{matrix}.$$

It is much more difficult to guess what the signs of these annuals are.

Square tables of numbers are called (square) matrices, and the corresponding determinants are called matrix determinants. It is interesting that now I.M. Gelfand is fascinated by the question: what should the determinant of a cubic matrix be equal to? When the answer is received, he will be found to tell about it on the pages «Kvant».

8. Zadchacha. Prove that

$$1^2 + 2^2 + \dots + n^2 = n(n+1)(2n+1)/6$$

And

$$1^3 + 2^3 + \dots + n^3 = n^2(n+1)^2/4.$$

It’s great that $1^3 + 2^3 + \dots + n^3 = (1 + 2 + \dots + n)^2$. I wonder if this can be proven directly?

Bernoulille numbers make it possible to calculate the total sum $1^p + 2^p + \dots + n^p$ as a polynomial in $n$ of degree $p+1$.

9. Problems. a) Express the sum of the squares of the roots of the equation $x^2 + px + q = 0$ in terms of its coefficients.

6. The same applies to the sum of cubes of the roots of this equation.

b) The same - for the sum of squares and cubes of the roots of the equation

$$x^n + a_1x^{n-1} + \dots + a_n = 0.$$

Hint: use Vieta’s formulas:

$$a_1 = -(x_1 + x_2 + \dots + x_n), \quad a_2 = x_1x_2 + x_1x_3 + \dots + x_n-1x_n, \quad a_3 = -(x_1x_2x_3 + x_1x_2x_4 + \dots + x_n-2x_nx_1x_n),$$ where $x_1, x_2, \dots, x_n$ are the roots of the equation.

Solving the general problem of how the sum of powers of the roots of an equation is expressed through its coefficients, I. M. Gelfand proved Newton’s formulas:

$$p_k + a_1p_k-1 + a_2p_k-2 + \dots + a_k-1p_k = 0, \quad k \leq n, \quad p_k + a_1p_k-1 + a_2p_k-2 + \dots + a_np_k-n, \quad k > n.$$

Here $p_k$ denotes the sum of the $i$ powers of the roots $x_i^1 + x_i^2 + \dots + x_i^n$.

Newton’s formulas can be considered as a system of linear equations for $p_i$ with coefficients $a_1, a_2, \dots, a_n$. If you solve this system, you will get the Variant formula, which directly expresses the sum of the degrees of the equations through its coefficients. In modern mathematics, this complexion has received far development.

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Lightning rod, politics and…
hats

A lover of science from Philadelphia (Pennsylvania), Benjamin Franklin, later a diplomat and a member of many academies of sciences, including St. Petersburg, in 1749 wrote down in his work journal the idea of ​​an experiment that was supposed to confirm the electrical nature of thunderstorm phenomena, and in 1750, in a letter to the English scientist Peter Collinson, he proposed a lightning rod - a grounded metal rod that rises above the protected object.

In the spring of 1752, the experiment according to Franklin’s plan was successfully carried out in France - sparks were extracted from a metal pole installed in a booth on an insulator during a thunderstorm. Somewhat later (the same year) in Philadelphia, Franklin himself carried out an experiment with an “electric kite” - he extracted sparks from a string wet from the jedge, onto which a kite with a metal tip was launched and which Franklin, being in a shelter, held by a silk ribbon tied to it. The first lightning rods soon appeared, although their implementation proceeded relatively smoothly. Often by us, not by the sharp ones. A situation arose that filled the struggle between the blunt-ended people and the sharp-pointed ones from Jonathan Swift’s Gulliver’s Travels (the first edition of the book was published in 1726). The Royal Society of London (academy of sciences) opposed the mini-improvement proposed by Wilson. Nevertheless, King George III in 1777, during the war with the American colonies who were fighting for independence from England, ordered that pointed lightning rods be replaced with blunt ones so as not to use the enemy’s invention.

(Continued on p. 20)

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Snowflakes under a magnifying glass
seemed much larger
what they really were
and waited for luxurious flowers
or tenth stars.
See how well it has been done! —
Kai said. — Snowmen are proud
more interesting than real flowers!
And what accuracy!
Oh, if only
they didn’t hide!

G.-H. Andersen.
“The Snow Queen”

15

Ice has always seemed to people as a symbol of beauty, precision, and size. But soup-like beauty, beauty that carries within itself coldness, evil, and sometimes death. No wonder Dante imagined that ice should be in the very center of the Earth, at the last step leading to Hell, and Satan’s treasures were hidden in an ice chest.

Undoubtedly, the artist’s exploration of natural phenomena and the laws of the world goes through his emotional perception of the surrounding world. A scientist must be dispassionate, both when choosing an object of study and when analyzing numerous (and sometimes contradictory) data.

Why might ice be interesting from a physics point of view?

The chemical formula of ice is H₂O. When cooled, water freezes, or, more precisely, crystallizes. Currently, thirteen (!) different structural forms of ice have been discovered. Even if this number is considered unlucky by some, it puts ice in a special position - not a single substance, simply in its chemical composition, can be similar to such common phase transitions. This is the first reason to take a closer look at this object.

figure 1 shows almost all the phase diagram of H₂O. It has been studied quite well, although it is still rich in white spots. Suffice it to say that ice-X and ice-XI were discovered only a few years ago. The ice that we are all accustomed to is also indicated in the diagram - this is ice-I, or, more precisely, ice-Iₙ. At normal pressure and 0 °C, its specific density is 0.917×10³ kg m⁻³, it is lighter than water, i.e., during crystallization, the density of water decreases.

This completely unlearned property of ice (after all, crystallization generally leads to an increase in density) is of enormous importance for life on Earth. The ice shell that forms on the surface of the water creates a column of effective thermal insulation, so that reservoirs rarely freeze to their entire depth. I don’t even want to think what would happen to the underwater world if ice were even a little heavier than water!

Fig. 1. Fig. 2.

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nom on figure 3, b, leads to a violation of paragraph 3.

But it is known that ice is not an ideal dielectric, but has a completely different electrical conductivity, and proton conductivity. Read, protons can move! So, are the BFP rules wrong? The correct answer is this: they are almost always correct, but it is the exceptions to these rules that are responsible for the amazing electrical properties of ice.

At the end of the fifties, the Tsarist physicist Jacquard proposed a very elegant way to describe these electrical properties by the movement of free defects that arise precisely when the BFP rules are violated. This is a very unteachable mechanism of electrical conduction, so let’s look at it in more detail.

Let’s “spoil” the ice lattice by violating 2nd of the BFP rules: we remove a proton from one oxygen atom and add it to another - as in figure 3, b. We will get so-called ion defects (they are designated OH⁻ and H₃O⁺). Now let’s spoil the lattice by violating rule 3, as shown in figure 3, f. In this case, orientation defects will arise (the so-called L- and D-defects).

Defects are always present in the structure of ice, but there are very few of them - in chemically pure ice at a temperature of -10 °C, for every 3 × 10¹¹ water molecules there are about 10⁵ orientational defects and only a couple of ion defects.

At figure 4, a it is shown how the movement of protons can occur. Let’s consider the ion defect H₃O⁺. Proton 1 moves to the place indicated by the cross. Previously, the H₃O⁺-ion was at the oxygen atom I, but now it will be at the oxygen atom II. Next, proton 2 moves to the place also marked with a cross - now it is H₃O⁺ at atom III, etc. We see that each time different protons move, as if passing the baton to each other. And as a result, the H₃O⁺ ion “goes” all the way “by itself”. figure 4, b shows the ice lattice after the H₃O⁺ ion moves from position I to position X.

For clarity, we took a hypothetical flat square lattice of ice. In this lattice, each oxygen atom has clear equivalent oxygen neighbors, as in a three-dimensional lattice, and therefore all the conclusions obtained here are valid for the true lattice of ice. Try to figure out for yourself how OH⁻-ions, L- and D-defects move.

It turned out that in some other substances the conduction mechanism is the same. Nowadays, even a lot of blame follows the study of the mechanism of electrical conductivity of ice, since in a number of biological objects the transference is very similar to the movement of protons in ice.

Jaccard’s theory required a lot, but every year the number of experimental facts that require further deepening of existing ideas is growing. New, and sometimes very uneducated, hypotheses appear. According to one of these bold hypotheses, the true charge carriers in ice are electrons that “mount” defects moving protons. Whether this is so, time will tell.

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The bulk properties of ice are quite necessary and even extincious, but the properties of the ice surface are still surprising.

Wake up any person in the middle of the night and ask at what temperature does ice melt? “Of course, at 0°C.” But the question “at what temperature does the ice surface begin to melt?” cannot be answered unambiguously even by experts in ice physics.

From the point of view of fundamental science, studies of the ice surface are extremely interesting. But these studies are of equal, if not greater, practical importance. The movement of icebreakers, anti-icing of ships and aircraft, construction in the polar and polar regions, the movement of glaciers - in all these cases it is important to know what happens on the surface of the ice, i.e. at the interface between ice - metal, plastic, various groups, etc. It is necessary to know the mechanisms of adhesion of ice to various structural materials, the nature and magnitude of friction and similar characteristics. And how necessary are materials with a minimum friction coefficient for various sports equipment!

By the way, why do many materials glide so well on ice?

This question may seem simple to many. After all, good gliding is when there is a layer of water between the body and the ice. And when a body presses hard on ice, it melts - under pressure, melting begins at a lower temperature.

Let’s figure it out. As follows from the physical diagram, the melting temperature of ice increases by 1 degree with additional pressure Δp=10⁷ Pa. Let the mass of the ridge be M=60 kg, and the section of the ridge S=3·200 mm²=6·10⁻⁴ m²; then the additional pressure will be Δp=Mg/S=10⁶ Pa. This means that under the ridge, the melting point is only… 0.1 °C. But what about skiing at -10°C? Yes, the explanation for good gliding is apparently something else.

It is believed that in the extreme temperature range there is a thin quasi-id layer on the surface of crystalline ice (“quasi” means “almost” or “as if”). It is unlikely that this thin membrane is a common liquid. However, many of its properties are very close to those of water. In some experiments, the existence of a quasi-id layer can be traced even at -30 °C, i.e. 30 degrees before melting!

Perhaps the presence of a quasi-id layer causes a small friction coefficient? Such a proposal can explain a lot, but, unfortunately, the true nature is much more complex and remains to be understood.

By the way, the debate between the concepts of “quasi-id layer” and “pressure melting” has been going on for more than a century. At the origins of this dispute there are replacement physicists - Faraday and the Thomson brothers (one of them is known to everyone as Lord Kelvin). Yes, yes, these world famous scientists also studied ice.

I have no doubt that you all played snowballs. But have you ever wondered why it’s so easy to split a snowball? Snow is not dough, not plates, but small crystals (!) of ice. After all, you can’t make a “snowball” out of metal filings, but you can make a snowball out of ice. Don’t think so?

But Michael Faraday, more than a hundred years ago, apparently began to think. In 1850, he drew attention to the following fact: pieces of ice freeze if they are brought into contact at a temperature of about 0 ° C. Why? It was to explain this fact that Faraday suggested

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THAT THERE IS A SPECIAL - QUASI-LIQUID - LAYER ON THE SURFACE OF THE LBDA.

Let’s now look at figure 5, and I think it will immediately become clear to you why the presence of a quasi-liquid layer leads to the freezing of two pieces of ice.

It is by the presence of a quasi-liquid layer that I try to explain the very high plasticity of ice and the movement of glaciers.

Faraday’s authority was very great, but he had an opponent - James Thomson (brother of Lord Kelvin). He drew attention to the then known part of the phase diagram of ice and theoretically proved that ice should melt as it slides down. This means that if you squeeze two pieces of ice, they should fuse (look at figure 6).

Faraday conducted a series of experiments refuting the point of view

When Georg changed his lightning rods, a lightning rod had already appeared on the spire of the Peter and Paul Fortress in St. Petersburg (probably the first in our country).

It should be noted that in reality the shape of the lightning rod usually does not matter. But if the top of the lightning rod is at such a great distance from the ground that it is comparable to the height of a thundercloud (for example, in the case of lightning rods on the tallest skyscrapers), then it is possible to observe a quiet discharge from the tip of the lightning rod that does not turn into lightning, as Franklin assumed when he advocated a “quiet” lightning rod with a pointed top. Franklin wrote with amazing accuracy in 1767: “A rod thus sharpened either prevents the lightning from striking a cloud, or, when struck, conducts it into the ground without damage to the building.”

(For the end, see p. 33)

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Quests
M1141—M1145, F1148—F1152

M1141. A trapezoid is circumscribed about a circle. Prove that at least one of its diagonals forms an angle of no more than 45° with the base.

H. M. Sedrakin

M1142. The table $m \times n$ is filled with $mn$ numbers so that in each row and in each column these numbers form an arithmetic progression. The sum of the four numbers in the corners of the table is $s$. What is the sum of all the numbers in the table?

M1143. The mass of each of the 101 weights arranged around a circle is a natural number, and their total mass is 300 g. Prove that from this set one or more weights located in a row can be selected with a total mass of 200 g.

B. B. Prozovolyov

M1144. Let $a_1, a_2, \dots, a_n$ be non-negative numbers. Which number is greater:

$$\sqrt{a_1^{1988} + a_2^{1988} + \dots + a_n^{1988}}$$ or $$\sqrt{a_1^{1989} + a_2^{1989} + \dots + a_n^{1989}}$$

A. I. Shekhorsky

M1145*. From the point $P$ two tangents PB and PC are drawn to the circle, and $\angle BPC > 90^\circ$. Point A is taken on the smaller arc BC. Prove that the area of ​​the triangle cut off from angle BPC by the tangent to the circle at point A does not exceed the area of ​​triangle ABC.

V. Yu. Progasov

F1148. A cylinder with a thread wound around it, the second end of which is fixed, is placed on a smooth inclined plane making an angle $\alpha$ with the horizon, as shown in figure 1. At the moment when the thread was vertical, the angular velocity of rotation of the cylinder was equal to $\omega$. Determine what is equal at this moment: a) the speed of the cylinder axis; b) the speed of a point on the cylinder touching the inclined plane. The radius of the cylinder is $R$.

S. S. Krotov

F1149. Thin long bare copper conductors are randomly thrown onto the rails of a toy railroad, closing the rails. Estimate the resistance between the rails if the distance between them is $l = 5$ cm, the diameter of the conductor is $d = 0,2$ mm, its length is $h = 30$ cm, $N = 100$ conductors are thrown.

A. P. Zilberman

F1150. A body is supplied to an ideal monatomic gas enclosed inside an oil bubble. Find the heat capacity of the gas (per 1 mole) in this process, if the pressure of the bubble can be avoided.

A. A. Sheronov

F1151*. An electrostatic voltmeter is a flat capacitor, one of the plates of which

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Fixed motionless, and the other can move, remaining parallel to the first plate. The movable plate is attached to the wall using a spring with a stiffness of $k=10$ N/m. The initial distance between the plates is $d=3$ cm, the area of ​​each plate is $S=0,5$ m². Calculate the voltmeter scale. What is the maximum voltage that can be changed with it? Consider separately the case when the high training is very small and quite large.

A. R. Zilberman

F1152. Behind the lens at a distance of $l=4$ cm (large focal length), a flat mirror is located perpendicular to the main optical axis. In front of the lens, also perpendicular to the main optical axis, there is a sheet of checkered paper (Fig. 2). On this sheet, images of its cells are obtained at two positions of the sheet relative to the lens. These positions differ by $l=9$ cm. Determine the focal length of the lens.

E. P. Kuznetsov

Problem solutions
M1116—M1120, F1128—F1132

Answer: a) 74; b) 6) $2S+2$.
A tract of size $m \times n$ cells contains $f=(m+1)(n+1)$ nodes. For a given area $mn=S$ the value

$$f=S+m+n+1=S+m+\frac{S}{m}+1$$

takes the most value $2S+2$ for $m=1$ or $m=S$ (for a rectangle of size $1 \times S$). For other $m$ ($1<m<S$), the value of $f$ is less than $2S+2$. This follows from inequality $m+\frac{S}{m}<S+1$, which is equivalent to the next inequality $(m-S)(m-1)<0$. Note that a more general statement is true: given $S$ (S is an integer or the resulting number), of all polygons with vertices at checkered paper nodes, the polygon that has no internal nodes contains the largest number of nodes. This follows from Pick’s formula for cell $S$ of such polygons*: $S=i+\frac{r}{2}-1$, where $i$ is the number of nodes lying inside the polygon, and $r$ is the number of nodes on its boundary: the total number of nodes is $i+r \leq 2i+r=2S+2$. (Inequality turns into equality only at $i=0$.)

V. L. Gugenacher

M1117. An arbitrary triangle is given. Prove that: a) it is possible to construct three circles with centers at its vertices, by

Points $K, L, M$ are the points of tangency of the sides of a given triangle $ABC$ with the inscribed circle (see figure; if $OK=OL=OM$ are the radii of the inscribed circle

*) See the article “Around the Pick Formula”, «Kvant», 1974, No. 12.

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druga touching each other in pairs (at points K, L, M); b) if through the middle of each arc KL, LM, MK lying inside the triangle, draw a tangent to it, then four triangles are formed, the area of ​​one of which (the central one) is equal to the sum of the areas of the other three.

The caital line drawn through the midpoint $C'$ for KL (with center C) cuts off from $\triangle ABC$ an equilateral triangle equal in area to the quadrilateral $CKOL$: the height $CC'$ cuts this equilateral triangle into two right triangles on the left, equal to $\Delta KCO = \Delta LCO$. Here it should also be noted that this tangent (at figure is red) always intersects sides CA and CB (angle COB is obtuse, since the half-sum of two angles B and C of triangle ABC is less than $90^\circ$, therefore the hypotenuse CO of triangle COK is less than CB; similar is $\angle COA > 90^\circ$, and therefore $CO < CA$). A similar rearrangement for the other two red tangents turns the equilateral triangles with altitudes $AA'$ and $BB'$ into quadrilaterals $ALOM$ and $BMOK$. Thus, the sum of the areas of three equilateral triangles with red bases is equal to the area of ​​triangle $ABC$. This implies the last statement of the problem: the area not covered by triangles is equal to the area covered by them twice.

A. A. Gorbachev

M1118. a) Prove that the equation $(x+y+z)^2=x^2+y^2+z^2$ (*) has infinite multisolutions in four numbers $(x, y, z)$ that do not have a common divisor greater than 1.
b) How many solutions are there for which $z=1988$?

a) Let us rewrite equation (*) in the form
$$xy+yz+zx=0.$$
Note that if one of the unknowns is 0, then some others are also 0; In what follows we will not consider such trivial solutions. Let $d$ be the most common divisor of $x$ and $y$, $x=da$, $y=ab$. then $z=-xy/(x+y)=-dab/(a+b)$, where $a, b$ and $a+b$ are pairwise coprime, so $d$ is divisible by $a+b:d=k(a+b)$. From here we obtain a general formula for nontrivial solutions:
$$x=ka(a+b), y=kb(a+b), z=-kab,$$
where $a, b, k$ are integers, and $a$ and $b$ are coprime. For $k=1$ we obtain all possible different solutions from numbers that do not have a common divisor greater than 1.
b) Answer: 16. The question boils down to how many ways can one represent 1988 as the product $-ab$, where $a$ and $b$ are coprime integers. Since 1988 = $4 \cdot 7 \cdot 71$, in the role of $a$ we can take the product of some numbers - 1, 4, 7, 71 (then $b$ will be the product of the rest; the product of the “empty set” can naturally be considered 1.)
S. G. Mamikonyan, N. V. Vasiliev

M1119. Let us call a $k$-star a figure on a plane consisting of $k$ rays with a common beginning of partition of the plane into $k$ equal

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ny angles (360°/k).
For what k>2 is the following statement true: for any k points of a generic plane (no three of which lie on the same line) there exists a k-star, each of the k corners of which contains exactly one of these k points?

Fig. 1.

Fig. 2.

Zadatnik, Kanma

For k=4, consider two cases. If the points lie at the vertices of a convex quadrilateral, then the center of the 4-star can be taken as the point of intersection of its diagonals, and the rays can be directed along the bisectors of the angles between them (Fig. 2). If one of the points D lies inside the triangle ABC with vertices in the other three and ∠ADB>90°, then as O we can choose the point of intersection of the segment DC and a certain semicircle, the diameter of which A₁B₁ lies entirely on the segment AB, and the rays can be directed along straight lines OA₁ and OB₁ (Fig. 3).

To prove that for k=5 (and k>5) the statement is false, it is enough to give an example; 5 points lying on one arc of value α<72° are suitable as such. In fact, if the arc α intersects four points of the rays of a 5-star with center O, then α>72°. (since the half-sum of arcs AB′ and A′B on figure 4 is equal to ∠AOB′=36°.

M. Khovanov

M1120. a) The sequence a₀, a₁, a₂, … is given by the Fibonacci correspondence a₀=0, a₁=1, a₂=1, a₃=2, …, given the conditions: aₙ+1=aₙ+aₙ-1, n=1,2,…

b) Prove a similar statement for the Fibonacci sequence a₀=0, a₁=1, a₂=1, a₃=2, …, given the conditions: aₙ+1=aₙ+aₙ-1, n=1,2,…

Let us prove the general statement: if the sequence a₀=0, a₁, a₂, … integers is such that

GCD(aₘ, aₖ)=GCD(aₘ-k, aₖ) (1)

for any m>k≥1, then

GCD(aₘ, aₖ)=aₖ, (2)

where d=GCD(m, k).

Indeed, for any pair of initial values ​​m, k, repeating the operation

(m, k)→(m-k, m)

(from the larger number of the pair, the smaller one is subtracted) leads to pale (d, 0), where d=GCD(m, k); This is a version of the Euclid algorithm for finding the most general divisor of the numbers m, n, presented in the textbook with information for grade 9 (usually the Euclid algorithm is called its “reduced” form, consisting of a sequence of divisions with a remainder.

Thus, from equalities (1) it follows that

GCD(aₘ, aₖ)=GCD(aₘ, a₀)=GCD(aₘ, 0)=aₖ

• this is (2).

a) Let P(P(P(…(P(x))…))=Pₙ(x); this is a polynomial with integer coefficients, and aₘ=Pₘ(a₀) and aₘ=Pₘ-k(aₖ) for m>k. Note that Pₙ(x)=aₙ+xQₙ(x), where Qₙ(x) is also a polynomial with by integer coefficients. Let’s check (1): for m>k≥1

GCD(aₘ, aₖ)=GCD(Pₘ-k(aₖ), aₖ)=GCD(aₘ-k+

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F1128. A uniform thin hoop of mass M and radius R rests on a horizontal surface. The horizontal diameter of the hoop is a light, smooth tube in which a ball of mass m is placed, attached to the hoop by two springs of stiffness k (Fig. 1). Keeping the hoop motionless, the ball was deflected to the left by amounts x, after which the system was left to its own devices. I will clear the acceleration of the center of the hoop at the initial moment of time. There is no slipping of the hoop.

figure 2 shows the forces acting on the ball and the hoop.

Let the acceleration of the center of the hoop be equal to a. Then the equation expressing Newton’s second law for the construction motion of the hoop (in projections to the horizontal direction) will look like:

$$Ma = 2kx - F_{тр}$$

Here 2kx is the total strength force of both springs, $F_{тр}$ is the friction force from the horizontal surface.

If the acceleration of the center of the hoop is equal to a, then the vertical acceleration of the ball will be equal to $ax/R$, and the equation of motion of the ball (in projections to the vertical direction) is written as:

$$ma \frac{x}{R} = mg - N.$$

Here N is the reaction force on the side of the smooth tube. Finally, let’s write down the equation that describes the rotational motion of the hoop around the center:

$$MRa = F_{тр}R + Nx.$$

Solving the system of three equations, we get

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F1129. A coil shaped like a narrow pole (Fig. 1) is put on a cylindrical permanent magnet in the region of its pole. If the path of the coil is such that it oscillates harmoniously along the axis OO’ with an amplitude of $A=1$ mm (which is much smaller than the dimensions of the magnet and coil) and a frequency of $f=1000$ Hz, then an induced emf with an amplitude of $\mathcal{E}_0=5$ V is induced in it. What force will act on a stationary coil if a current of $I=200$ mA is passed through it?

When the coil moves along the axis of the magnet, the flux of magnetic induction through the coil changes. As a result, an induced emf $\mathcal{E}=N\Delta\Phi/\Delta t$ appears in it, where $N$ is the number of turns of the coil, $\Phi$ is the flux of magnetic induction through one turn.

To calculate $\Phi$ we proceed as follows. Let’s stretch some surface onto a coil and count the number of magnetic field induction lines $\bar{B}$ intersecting this surface. In this case, it is convenient to use a surface in the form of a cylinder with a lid (Fig. 2). In this case, we will keep in mind that when mixing the coil, the cover remains in place, but the distance between it and the coil changes. Let the coil move down by $\Delta x$. Then a “belt” with an area of ​​$2\pi R\Delta x$ is added to the side surface of the cylinder, where $R$ is the radius of the coil. Corresponding to $\Phi$ is added $\Delta\Phi$ - magnetic flux through this “belt”:

$$\Delta\Phi=2\pi R\Delta x\cdot B_r,$$

where $B_r$ is the radial (i.e., perpendicular to the side surface of the cylinder) projection of the vector $\bar{B}$. Thus,

$$\mathcal{E}=N\cdot 2\pi R\frac{\Delta x}{\Delta t}B_r=N\cdot 2\pi RvB_r,$$

where $v$ is the coil speed. With harmonic oscillations, the velocity amplitude

$$v_0=\omega x_0=2\pi fA,$$

And

$$\mathcal{E}_0=N\cdot 2\pi R\cdot 2\pi fAB_r.$$

On the other hand, the required force is equal to

$$F=\sum\Delta F=\sum N\cdot I\Delta l\cdot B_r=N\cdot 2\pi RIB_r.$$

From the last two equalities we finally obtain

$$F=\frac{\mathcal{E}_0I}{2\pi fA}\approx 0,16\text{ H.}$$

M. M. Tsyplin

F1130. The oscillatory circuit consists of a vacuum capacitor with a capacity $C$, the distance between the plates of which is $d$, and an inductor. The natural part of the circuit oscillations is equal to $\omega_0$. What will be your own

Let us write down the energy of the system at an arbitrary moment in time:

$$W=\frac{LI^2}{2}+\frac{CU^2}{2}+\frac{mv^2}{2}-\frac{qUx}{d},$$

where $L$ is the inductance of the coil, $I$ is the current in the coil, $U$ is the voltage on the capacitor, $v$ is the speed of the particle, $x$ is its coordinate. The equation of particle motion in the electric field of a capacitor has the form:

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Frequency if a free precise frequency of mass $m$ and charge $q$ is placed between the plates of a capacitor? There is also no strength. The marginal effects and strength of the “electrostatic device” are claimed.

$$ma = \frac{qU}{d}.$$

If $U$ changes according to the harmonic law $U = U_0 \cos \omega t$, then
$$v = \left( qU_0 / (md\omega) \right) \sin \omega t,$$
$$x = -\left( qU_0 / (md\omega^2) \right) \cos \omega t,$$
$$I = -CU_0 \omega \sin \omega t.$$

According to the law of conservation of energy
$$W = U_0^2 \left( \frac{L}{2} (C\omega)^2 + \frac{m}{2} \left( \frac{q}{m\omega d} \right)^2 \right) \sin^2 \omega t +$$
$$+ U_0^2 \left( \frac{C}{2} + \frac{q^2}{m(\omega d)^2} \right) \cos^2 \omega t = \text{const.}$$

This is only possible if the coefficients of $\cos^2 \omega t$ and $\sin^2 \omega t$ are equal. From here we get
$$\omega^2 = \omega_0^2 / 2 + \left( \left( \omega_0^2 / 2 \right) \right)^2 + \left( q\omega_0 / d \right)^2 / (mC)^{1/2}.$$

Let us evaluate under what conditions it is possible to apply with one “electrostatic choice”* (as was done above). This force, equal in order of magnitude to $q^2 x / (\varepsilon_0 d^3)$, should be much less than the “ordinary” electrostatic force $qU/d$, whence
$$\frac{q^2}{\varepsilon_0 d^3 m\omega^2} \ll 1.$$

Since the area of ​​the capacitor plates is $S \gg d^2$ and $C = \varepsilon_0 S / d$, formula (*) is valid if the introduction of a particle into the capacitor does not greatly change the frequency of the oscillations in the circuit.

A. V. Andrianov, D. A. Kuptsov

F1131. It is difficult to determine the wind speed in the smerya using conventional meteorological instruments (since the smerya is not large in size and moves) and is not zonal. It is proposed to measure it from a distance using a portable radio (since there is a lot of dust and small objects inside the smert, it reflects radiovolts). The radar emits radiovolts at frequency $f_0 = 10^{10} \text{ Гц}$. The spectrum of reflection from the mertz is shown in figure ($\Delta f = f - f_0$). Find the maximum wind speed in Smereta.

The frequency of the reflected signal differs from the frequency of the emitter due to the Doppler effect. The maximum frequency shift will occur if the speed of the object reflecting radiovolts is directed exactly towards the radar (or away from it).

We will assume that the radar sends short radio pulses with a repetition frequency $f_0$, and the frequency of the pulses it receives is $f_1$. The time between arrivals at the radar antenna of the $n$th and $(n+1)$th pulses is equal to
$$\frac{1}{f_1} = \frac{1}{f_0} + T_{n+1} - T_n,$$
where $T_n$ is the propagation time of the $n$ pulse from the radar to the reflecting object and back, $T_{n+1}$ is the propagation time of the $(n+1)$ pulse.

Let us denote by $L_n$ the distance from the radar to the reflecting object at the moment of emission of the $n$ pulse, and by $L_{n+1}$ the same distance at the moment of emission of the $(n+1)$ pulse. It’s obvious that
$$T_n = \frac{2L_n}{v + c}, \quad T_{n+1} = \frac{2L_{n+1}}{v + c},$$

*) You can read about how this force can be determined for some specific cases in the note “Method of Electrostatic Isolations” (Kvant, 1987, No. 3, p. 39). (Editor’s note)

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where $v$ is the desired wind speed in the tornado, $c$ is the speed of propagation of radio waves (speed of light). The difference $L_n - L_{n+1}$ is equal to the displacement of the reflecting object in a time equal to $1/f_0$:

$$L_n - L_{n+1} = \frac{v}{f_0}.$$

From here we get

$$f_1 = f_0 \frac{(c + v)}{(c - v)} \approx f_0 \left(1 + \frac{2v}{c}\right),$$

And

$$v \approx \frac{c}{2} \frac{\Delta f}{f_0} \approx 100 \text{ m/c.}$$

D. A. Kuptsov

F1132. It has recently been established that the extinction of animal species on Earth is particularly intense during periodic periods of time lasting $T = 6,2$ million years. This pattern is explained by the hypothesis proposed by the existence of the star Nemesis, which is a satellite of the Sun. This faint and therefore invisible star moves in an orbit that is located within the so-called “Oort belt”, containing a supply of comets (Fig. 1). By disturbing the movement of comets, Nemesis causes a “comet house” on Earth, the duration of which practically coincides with the time of Nemesis’s stay inside the Oort belt.

Determine the greater half a of the orbit of Nemesis and the period of its revolution around the Sun, if it is assumed that the perihelion of the orbit is at a distance of 0.15a from the Sun. The area of ​​the ellipse is $S = \pi ab$, where $a$ and $b$ are the ellipse.

Over a time of $T = 6,2$ million years, the radius vector of Nemesis occupies the area protected by figure 2. This is the half of the area of ​​the ellipse minus the area of ​​the triangle with base $2b$ and height 0.85 $a$. The area of ​​this figure, as it is easy to calculate, is $(\pi ab/2 - 0,85ab)/\pi ab = 0,23$ the area of ​​the entire ellipse. Ta-

Fig. 1.

Fig. 2.

What part of the period is the time of Nemesis inside the Oort belt (this follows from Kepler’s second law). The total period of Nemesis’ orbit around the Sun is therefore $T_x = 27$ million years.

Comparing, with the help of Kepler’s third law, the motion of the Earth and Nemesis, we obtain a greater semi-presence of the invisible satellite of the Sun:

$$a = \left( T_x/T_0 \right)^{2/3} R_0 = \left( 27 \cdot 10^6 \right)^{2/3} a.e. = 9 \cdot 10^4 a.e. \approx 13,5 \cdot 10^{12} \text{km.}$$

Here $T_0$ is the Earth year, and $R_0$ is the radius of the Earth’s orbit, equal to one astronomical unit (1 AU), or 150 million km.

V. E. Belonuchkin

29

List of readers who sent correct solutions

Most of the readers who submitted solutions to problem M1091—M1115, F1103—F1117, coped with problems M1091, M1093, M1094, M1096, M1097, M1101, M1102, M1105, F1104, F1106, F1109—F1111, F1114—F1116. Below we publish the names of those who wrote the correct solutions to the remaining problems (the numbers after the last names are the last numbers of the numbers of the solved problems).

Mathematics
D. Aladzhanov (Ermolino village, Kaluga region) 04; N. Andrianov (Electrostal) 92, 98, 00, 03, 04; A. Akimov (Evpatoria) 98, 04; A. Hakobyan (Yerevan) 04; Sh. Aleksayan (Lininakan) 04; I. Arzhantsev (Kyiv) 92, 98, 04; V. Baranovsky (Omsk) 92, 98, 03, 04; S. Bashmakov (Sevastopol) 04; N. Bulusak (village Menyava, Ivano-Frankivsk region) 98; A. Boldanov (Stayy Oskol) 92, 03, 04; P. Borodin (Kirov) 98; J. Brener (Sergevka) 04; S. Butvina (Zhitomir) 92, 95, 98, 03, 04; Yu. Vavulov (Safonovo) 04; V. Wanchun (Moscow) 04; Ts. Vasiliev (Sofia, NRV) 04; Yu. Velikina (Dnepropetrovsk) 04; S. Voloshin (Podgorodnoe) 04; K. Volchenko (Donetsk) 03; M. World (Leningrad) 92; M. Vyborov (Kyiv) 92; E. Gabrilovich (Minsk) 98; Yu. Gold (Odesca) 04; P. Grigoriev (Riga) 92; Yu. Grinfeld (Moscow) 92, 98, 00, 03, 04; A. Gurman (Odesca) 04; A. Davydov (village Elfimovo, Gorky region) 04; Kh. Jafarov (village of Turkba AzSSR) 98; D. Tarsier (Reytov) 92; V. Drobovtsev (Moscow) 92; V. Dubrov (Minsk) 92, 98, 03, 04; E. Duda (Kharkov) 92; A. Efimuk (Minsk) 03, 04; V. Zhilinskaya (Vilnos) 92, 98-00; I. Zelenko (Saratov) 03, 04; S. Zelik (Kramatorsk) 92, 98, 04; I. Zuk (Essen, Germany) 00, 04; V. Ivleev (Dzhezkazan) 92; M. Iznagev (Moscow) 04; S. Izmalkov (Kharkov) 04; A. Imashova (Leningrad) 98; I. Iopne (Moscow) 92, 98, 00; D. Kabysh (Moscow) 98; E. Ghazaryan (Tilisi) 92; V. Kaloshin (Kharkov) 92, 95, 98, 00; L. Kaminstein (Kyiv) 92, 03, 04; D. Kamushadze (Tilisi) 04; N. Kaputkina (Moscow) 92, 98, 03; S. Kasamyan (Oktemberg district of Armenian SSR) 92; K. Kasimov (Tselinograd) 04; I. Kirillov (Ust-Kamenogorsk) 04; S. Kovatsenko (Vinnitsa) 92, 98, 03, 04; O. Kovrizhkin (Maikop) 92; D. Kozhevnikov (Onutin) 92; A. Kozachko (Vinnitsa) 92, 98, 03, 04; I. Kokoev (Tilisi) 04; G. Kolesnichkiy (Tilisi) 04; M. Kolede (Omsk) 92, 98, 04; V. Koloshin (Kharkov) 03, 04; P. Kononenko (Ivanovo) 92; D. Korzhnev (Kyiv) 04; D. Korzhin (Ivanovo) 03, 04; A. Korotky (Dneprodzerzhinsk) 04; A. Korshkov (Mozyr) 92, 98, 03, 04; G. Kropin (Leningrad) 98, 04; A. Kryukov (Saratov) 98, 04; T. Kulikova (Zvenigorod) 92, 95, 98, 03, 04; N. Lapusta (Ternovopol) 04; V. Levshin (Moscow) 98; S. Lesik (Donetsk) 92, 95, 98, 03, 04; Yu. Litvinova (Kyiv) 98; M. Loba (Lviv) 04; E. Lomovtseva (Belorets) 92; G. Magleridze (Rustavi) 04; A. Maly

(Continued on page 51)

30

FRACTIONS - CAMELS -
PARQUETS

Candidate of Physical and Mathematical Sciences
V. L. GUTENMACHER

It often happens that the same mathematical problem naturally arises in modern situations - from puzzles to applied calculations. Today we are almost talking about one of these tasks.

Arithmetic problem

Find all quadruples of natural numbers (p, q, r, s) for which the equality on the right is

$$\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s} = 1.$$

We will assume that $p \leq q \leq r \leq s$. Then the smallest number - $p$ - is not more than 4 (otherwise $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s} < 1$) and, of course, not less than 2. Therefore, we need to consider three cases: $p=2$, $p=3$, $p=4$.

Let $p=2$. Then $\frac{1}{q} + \frac{1}{r} + \frac{1}{s} = \frac{1}{2}$. Let’s estimate the smallest number again - $q$. It cannot be more than 6 (otherwise $\frac{1}{q} + \frac{1}{r} + \frac{1}{s} < \frac{1}{2}$) and must be at least 3.

Now into the equality $\frac{1}{q} + \frac{1}{r} + \frac{1}{s} = \frac{1}{2}$ we substitute four possible

32

be equal to $\frac{1}{2}, \frac{1}{3}, \frac{1}{7}$ respectively*). The judge adds one of his camels (it turns out 42) and separates 21, 14, 6 camels for the brothers respectively (41 in total), and takes his camel back.

Practical problem

The simplest of the “regular” parquets is a division of the plane into squares (Fig. 1). It is interesting to find out how many such parquets there are: each vertex of the parquet is adjacent to four regular polygons and all vertices of the device are the same (the latter means that the parquet can be moved so that any
*) The first four stands out among all the others in that it contains the most number - 42, with 42 = 2·3·7. Among the triplets of numbers whose reciprocal sum is 1, the beginning of our quadruple has the same property: (2, 3, 7). It turns out that among the paterok there will be (2, 3, 7, 43, 1806), where 1806 = 2·3·7·43; among sixes - (2, 3, 7, 43, 1807, 3263443), and then it will be the same. This is a very difficult theorem, which our older readers encountered in the 7th issue of «Kvant» for 1987 in the article “Partitioning of Units”.

If its given vertex goes to any other given vertex and all lines coincide). This is a completely practical task.

We know that the sum of the angles of a regular $n$-gon is $180^\circ(n-2)$, and its one angle is equal to

$$\frac{180^\circ(n-2)}{n} = 180^\circ - \frac{360^\circ}{n}.$$

Let the corners of four regular polygons be combined at the vertex of the parquet: $p$-gon, $q$-gon, $r$-gon and $s$-gon. The sum of these four angles should be $360^\circ$. Let’s write this condition:

$$180^\circ - \frac{360^\circ}{p} + 180^\circ - \frac{360^\circ}{q} + 180^\circ - \frac{360^\circ}{r} + 180^\circ - \frac{360^\circ}{s} = 360^\circ.$$

It is easy to understand that this equality leads to the familiar correspondence

$$\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s} = 1.$$

33

If we assume that $p \leq q \leq r \leq s$, then we know all four of these - there are 14 of them. Since we are talking about polygons, we need to discard those four where $p=2$. There will be four four left:

(4,4,4,4), (3,4,4,6), (3,3,6,6), (3,3,4,12).

The first quadrangle corresponds to a parquet of identical squares (each vertex is adjacent to 4 regular quadrangles - see figure 1).

The second four (3,4,4,6) represents two possibilities for constructing the top (Fig. 2, $a$, $\delta$), but for the correct parquet it is possible to complete only the parquet at figure 2, $a$ - it turns out figure 2, $\delta$.

The third four (3,3,6,6) also corresponds to two polygon arrangements:

Lightning rod, politics and…
hats

(For the beginning, see p. 13)

Czech communications officer Prokop Diviš according to the rules of the unofficial tipula “magus of electricity” (magus electricus). In 1754, he built the first lightning rod in Europe, which he invented independently of Franklin. But let’s fast forward a few years ago.

In 1750, Viennese professor Joseph Franz decided to show Divish his electrical experience. Franz had a static electricity generator typical of that time - a doctorable glass ball with a conductor (outlet) in the form of a metal tube or rod suspended on silk cords. To test any electrical experiment, it was necessary to rotate a glass ball, which was then electrified by friction and transmitted to the parquet.

Divish came to Franz in a wig, as fashion demanded. But the wig had a secret: it was covered all over with scraps of wire. It has long been known that the absence of metal “takes away electricity” (this is one of the phenomena on which the action of a lightning rod is based). By laying his head on the conductor, Divish thereby diverted the electric train. Probably, Divish experienced unpleasant mistakes, sending a couple into the ground through his body, but why not endure it for the sake of a good joke!

ki! Franz was unsafe: he could not get a charge on the conductor. He never managed to surprise the “magician of electricity.”

If Divish had carried out borrowing from wire cuttings (in a hidden way, of course), he would have been spared from embarrassing mistakes. An idea of ​​this kind arose in the second half of the 18th century. We are talking about a lightning rod hat. figure (see page 2 of the cover) shows Parisian models with personal lightning protection in the form of a middle chain that goes down to the toes from a metal rod mounted on the hat. In the 18th century this was no joke.

L. N. Krizhanovsky

35

The story of the remarkable American science fiction writer is addressed primarily to those for whom space flight is the main dream of life, and their favorite picture is the view of the starry sky. However, in a word, to all those for whom “R means rocket.”
This is what the new section of our magazine will be called. In it we will publish various materials devoted to the exploration and exploration of outer space, the work of the All-Union Youth Aerospace Society.

R IS FOR ROCKET

R. BRADBURY

This huge thing, which we pressed our faces to, and felt the wind getting hot, and pressed ourselves even harder against it, forgetting who we were and where we came from, dreaming only about who we could be and where we could go…

But we were boys - and we liked being boys; and we lived in a small Florida city - and we liked the city; and we went to school - and we certainly liked school; and we climbed trees and played football, and we also liked our mothers and fathers…

And yet sometimes - every week, every day, every hour in that minute or second when we thought about the flame, and the stars, and about the limit beyond which they expected us - sometimes we liked the rockets better.

Fences. Rockets.
Every Saturday morning…
The guys gathered near the wet house.
The sun has barely risen, and they are already standing, screaming, until the neighbors come out with stun guns from their windows - they say, shut up now, or we’ll freeze for an hour, then blame yourself!

• Oh, go to the rocket, stick your head in the kyuzu! - the guys shouted back. They shouted and hid safely behind our production: after all, old Wickard from the neighboring house shoots without missing a beat.

On this cool, mild Saturday morning, I was lying in bed, thinking about how I had failed the semantics test the day before, when the head of Vatga suddenly came to me. It was not yet seven, and the wind was carrying thick fog from the Atlantic, and the weather service selectors placed on all corners had just begun to buzz, dispersing this mess with their rays: you could hear them howling gently and secretly.

I dragged myself to the window and looked out.

• Okay, space pirates! Turn off the engines!

• Hey! - Ralph Priory shouted. - We just found out: the launch schedule has been changed! Lunar, with the new X-L-3 engine, starts in an hour!

• Buddha, Muhammad, Allah and other real and semi-mythical figures! - I said and jumped away from the window with such agility that the guys fell onto the grass from the impact.

I immediately pulled on a jumper, quickly put on my shoes, put nutritional capsules in my back pocket - today we won’t have time to eat, swallow the pills, your stomach will churn - and took the vacuum elevator from the second floor down to the first.

On the lawn, the guys, all five of them, were biting their lips and jumping up and down with impatience, making angry horns.

“The last one to add to the monorail,” I shouted, rushing past them at a speed of 5 thousand miles per hour, “will be the bug-eyed Martian!”

Sitting in the cabin of the monorail, whistlingly carrying us to the Cosmodrome, twenty miles from the city - just a few minutes’ ride - I felt like bugs were crawling in the pit of my stomach. Give a fifteen-year-old boy one more launch. Almost every week, small continental cargo rockets came and went on schedule, but this launch… A completely different matter - strength, power… The Moon and beyond…

“My head is spinning,” Priory said and hit me on the arm.

I hit him back.

• Me too. Well, tell me, is there a Saturday in the week?

We exchanged big, knowing smiles. Mentally we went through all the stages of preparedness. The other pirates were good guys. Sid Rossen, Mac Leslie, Earl Marney - they, like all the guys, jumped, ran and also loved rockets, but for some reason I thought that they were unlikely to do what

36

what Ralph and I will do one day. Ralph and I dreamed of the stars; they were more desirable to us than a handful of blue-white diamonds of the purest water.

We bawled along with the bawlers, laughed along with the laughers, but in our souls it was still quiet; and now the barrel-shaped cabin, rustling, stopped, we jumped out and, shouting and laughing, ran, but we ran calmly and even somehow slowly: Ralph was ahead of me, and everyone pointed in one direction, to the treasured hedge, and split up places along the wire, hurrying up those who were lagging behind, but not looking back at them; and finally everything was assembled, and the mighty rocket came out from under the plastic dome, burned onto a huge interstellar circus tent, and went along shiny rails to the launch point, escorted by a huge portal crane, reminiscent of a dystorically winged lizard, which fed this fiery monster, cooled and nurtured it, and now its birth is about to take place in the red-hot brought in a flash in the sky.

I stopped breathing. I didn’t even take a breath until the rocket reached the concrete patch, accompanied by beetle tractors and large, bulky vans with people, and all around, fiddling with the mechanisms, mantis mechanics in abset suits were chirping, humming, croaking at each other, radiophones visible to us and inaudible to us, but we heard everything in our minds, in our hearts, in our souls.

“Lord,” I finally said.

“Almighty, all-merciful,” said Ralph Priory, standing next to me.

The other guys also said something in this vein.

And how can you not admire it! Everything that people have dreamed about for centuries was dismantled, sifted through and forged into one - the most cherished, the most wonderful, the most winged dream. Whatever the outline - a hardened flame, an impeccable form… A frozen fire, ready for secret ice djali there, in the middle of the concrete prairie; a little more, and it will wake up with a roar, and rise up, and this thoughtless, magnificent, mighty head of the Milky Way will become poor, so that the stars will fall down like a meteorite firefall. And if a Coal Sack gets in the way, by God, as soon as he gives a sigh, he will immediately jump to the side!

She hit me right under the breath, hit me so hard that I mistook an acute attack of jealousy, and envy, and longing, as if from something unfinished. And when finally a self-propelled carriage with astronauts walked across the field, surrounded by silence, I was with them, dressed in outlandish white helmets, spherical pressure helmets and a kind of majestic negligence - just like a tape recording team introducing itself to the public before a training meeting at some local tape recorder. But they were flying to the moon - now a rocket went there for each place - and only onlookers gathered around the huge space, we, the boys, were the only ones who were more for a safe launch and departure.

“Damn it,” I said. - I wouldn’t give anything just to sing with them. Can you imagine…

“I’d give my annual pass for that,” said Mack.

• Yes… I wouldn’t regret anything.

Needless to say, what a great event this was for us kids, as if suspended in the middle between our morning game and the powerful and internal afternoon fireworks that were soon expected for us.

And now all preparations are completed. The rocket ran out of fuel, and people ran away from it in different directions, like ants running away from a metal idol. And the Dream opened its window, roared, and rushed into the sky. And now she disappeared along with the uterine howl, and all that was left of her was a hot zone in the air, which was transmitted through the ground to our feet, and up the legs to the very heart. And where she stood there was now a black, melted pit and a cloud of rocket smoke, as if a cumulus cloud had been nailed to the ground.

• She’s gone! - Priory shouted.

And we all began to breathe quickly again, prepared for the spot, as if we had been stunned by some monstrous parapistol.

“I want to grow up as quickly as possible so that I can sing on such a rocket.”

I bit my lip. Where am I, the green end; Moreover, in space, work according to the application is not accepted. Wait until they take you away. Finally someone, like Sidney, said:

• Okay, now let’s go to the TV show.

Everyone agreed - everyone except Priory and me. We said no, and the guys left, laughing and talking, only Priory and I were left looking at the place where the spaceship had recently stood.

It took away our taste for everything else, this start. Because of him, I failed semantics on Monday.

37

And I didn’t care at all.
At such moments I said thank you to the one who invented the concentrates. When we have a bundle of nerves in our stomachs, the last thing I want to do is sit down at the table and finish a three-course meal. Without an appetite, a few concentrate tablets perfectly replaced both the first and more creative ones.
All day long and until late at night I was relentlessly, stubbornly haunted by the same thought. It got to the point where every night I had to resort to a sleep-inducing massage combined with the quiet melodies of Tchaikovsky in order to close my eyelids at least for a little while.
“For mercy, young man,” my teacher said that Monday, “if this continues, we’ll have to give you an overall assessment at the next meeting of the psychological committee.”
“Sorry,” I replied.
He looked at me intently.
— Do you have some kind of congestion in your head? Obviously, something is very simple, and, moreover, conscious.
I’ve lived my life.

• That’s right, sir, conscious, but not simple. And it’s very complicated. But in general, we can say in one word - packages…
  He smiled.
• P means package, right?
• That’s right, sir, something like that.
  “But we cannot allow this, young man, to affect your performance.”
  “Do you think, sir, that I should be subjected to a hypotypical internal test?”
• No no. “The teacher sorted through the sheets of paper, at the top of which my last name was written in large letters.
  My stomach tightened. He looked at me again.
• After all, Christophe, you are number one in our class, our favorite, that’s what he said.
  He closed his eyes, thinking.
  “We need to think carefully about this,” he concluded. He patted me on the shoulder and added: “Okay, keep borrowing.” And there is no need to grieve.
  He walked away from me.
  I tried to concentrate on my studies, but I couldn’t. Until the end of the lessons, the teacher kept looking at me, leafing through my tables and thoughtfully biting his lips. Around two o’clock he dialed a number on his audiophone and talked to someone for about five minutes.
  I couldn’t spill what he was saying.
  But when he put the phone down, he looked at me very, very strangely.

Dependence, delight, and regret - everything mixed together in this look. A little sadness and a lot of joy. Yes, the eyes were expressive.
I sat and didn’t know whether to laugh or cry.
That same day, Ralph Priory and I went home early from school. I told Ralph what had happened, and he frowned: that’s his habit.
I became alarmed. And we began to stir up this anxiety together.
“What, Chris, do you think they’ll send you somewhere?”
The monorail cabin pinched. This was our stop. We left. And they walked slowly towards the house.
“I don’t know,” I answered.
“That would be disgusting,” said Ralph.
“Maybe I need to see a psychiatrist so he can read my brain, Ralph?” But it’s also impossible for your studies to go to waste.
We stayed at my house and looked at it for a long time. Then Ralph said one page thing:

• There are no stars during the day, but we still see them, right, Chris?
  “True,” I said. “We see.”
  “We’ll stay tight, okay, Chris?” I can’t, damn them, remove you from school now. We are friends. It wouldn’t be fair.
  I didn’t answer because a lump was tightly blocked in my throat.
• What’s wrong with your eyes? - asked Priory.
  “Oh, nothing, I’ve been staring in the sun too long.” Let’s go to the house, Ralph.
  We groaned under the spray of water in the shower, but somehow without much enthusiasm, even when we turned on the ice-cold water.
  While we stood in the dry air, blown by hot air, I had time to think. Literature, I reasoned, is full of people who struggle with harsh, irreconcilable opponents. The brain, the muscles - they all turn to fight against all sorts of obstacles until they win or lose. But I have no signs of external conflict. The fact that it burdens me with sharp teeth burdens me from the inside, and besides me, only a psychologist will smooth out all my scratches. Of course, this doesn’t make anything any easier for me.
  “Ralph,” I said as we began to get dressed, “I’m caught up in a war.”
  -Are you alone? he asked.
  “I can’t involve you,” he explained.

38

or I. - Because this is a completely personal matter. How many times did mom say, “Chris, not much yet, your eyes are bigger than your stomach”?

• A million times.
• Two million. Now let’s rephrase that, Ralph. Let’s put it another way: “Don’t imagine so much, Chris, your imagination is too big for your body.” So, the war is between the imagination and the body, which cannot keep up with it.
  Priory nodded reservedly.
• I understand you, Christophe. I understood what you said about personal war. In this sense, there is also a war going on within me.
  “I know,” I said. — For other guys, it seems to me that it will pass. But for you and me, Ralph, I think this will never go away. I think we’ll be waiting the whole time.
  We sat down in the sun on the roof of the house, laid out our terraces and started doing our homework. Nothing worked for Priory. Me too. Priory said dryly what I couldn’t muster the courage to say.
  — Chris, the Cosmonautics Committee is selecting people. Those interested do not apply. They are waiting.
• I know.
  “They’ve been waiting since the day when their hearts first skipped a beat at the sight of the Moon Pack, they’ve been waiting for years, from place to place every month they’ll find everything, that one fine morning a blue helicopter will descend from the sky, land on the lawn in their garden, a neat, smart pilot will climb out of the cockpit, quickly climb onto the porch and press the bell button. This helicopter is waiting until it turns one year old. And on your 20th birthday they drink a glass of wine and throw it away with a loud laugh: they say, well, to hell with it, it’s not really necessary.
  We sat in silence, weighing the weight of his words. They sat and were silent. But then he spoke again:
  “I don’t want to disappoint you like that, Chris.” I’m fifteen years old, just like you. But if I turn one, and the astronaut still doesn’t ring the doorbell of our boarding school, where I live, I…
  “I know,” I said, “I know.” I talked to someone who wasted their time. If this happens to us, Ralph, then… then we’ll have a drink together, and then we’ll go and get hired as loaders on the transport rocket of the European line.
  Ralph collapsed and turned pale.
• For movers…

Someone quickly and softly walked along the porch, and we saw my mother. I smiled.

• Hello, lady!
• Hello. Hello Ralph.
• Hello, Jen.
  Looking at her, I would not have given her more than twenty-five to twenty-six years, although she gave birth to and expressed me and has been serving for many years in the State Statistical Office. Thin, graceful, smiling: I imagined how much her father must have loved her when he was alive. Yes, at least I have a mother. Weedy Ralph was brought up in a boarding school…
  Jen came up to us and put her hand on Ralph’s love.
  “You look really bad,” she said. “Is something wrong?”
  Ralph faked a smile.
• No, no, everything is fine.
  Jen didn’t need any prompting.
  “Staying to stay with us, Priory,” she suggested. “We miss you.” Isn’t that right, Chris?
• What a question!
  “I should have been sent back to the boarding school,” Ralph objected, though not very confidently. “But since you’re asking, and Chris needs help with semantics, then I’ll help him.”
  “Very great,” I said.
  “But first I have some things to do.” I quickly go there and back on the monorail, return in an hour.
  When Ralph clicked, the mother of many significant people looked at me, then with a gentle movement of her fingers she looked at my hair.
• He’ll prescribe something, Chris.
  My heart sank; it wanted to be silent for a while. It was waiting. I opened my mouth, but Jen continued:
• Yes, he will prescribe something somewhere. I got two calls at work today. Your teacher called first. Then… no, I can’t say. I don’t want to say anything until it happens…
  My heart spoke again, slowly and hotly.
• In that case, don’t tell me, Jen. These calls…
  She looked at me silently. She squeezed my hand with soft, warm palms.
  “You’re still so young, Chris.” Very very young.
  I sat in silence.
  Her eyes brightened.
  -You’ve never seen your father, Chris. Terribly bitter. You know who he was, right?

39

“Of course I know,” I said. “He worked in a chemical laboratory and almost never left the dungeon.

“Yes, he worked deep underground, Chris,” Mom confirmed. And for some reason she added: “And I’ve never seen the stars.”

My heart screamed in my chest. He screamed loudly and piercingly.

• Mom… mom…

For the first time in many years, I called her mom out loud.

When I woke up the next morning, the room was flooded with sun, but the couch on which Priory, our guest, had promised to sleep, was empty. I listened. No one was splashing in the shower, and the dryer was not humming. Ralph was not in the house.

I found a note pinned on the door.

See you at school this afternoon. Your swearing made me want to do something for her. She got a call this morning and she said she needed my help. Hello. Priory.

Priory runs errands for Jen. Strange. Jen got a call early this morning. I walked back to the couch and sat down.

I was still sitting when I heard shouts from outside:

• Hey, Chris! I fell asleep!

I looked out the window. Several guys from our gang were standing on the lawn.

• I’ll be down now!
• No, Chris.

Mom’s voice. Quiet and with some unusual shade. I turned around. She stood in the doorway behind me, her face pale, haggard, as if something was tormenting her.

“No, Chris,” she repeated softly. - Tell them to go without you, you won’t interfere at school… today.

The visa guys probably continued to make noise, but I didn’t hear them. At that moment, only my mother and I existed for me, so thin, pale, tense… Far away, the vibrators of the weather service began to hum and rumble.

I slowly turned around and looked at the guys. All three of them looked up - lips parted in a careless half-smile, rough fingers holding notebooks on semantics.

• Hey! - one of them shouted. It was Sidney.

• Sorry, Sydney. Sorry guys. Stomp on without me. I won’t be able to eat at school today. See you later, okay?

• Okay, Chris!

• What, are you sick?

• No. Just… In a word, walk without me. Then I met.

I stood there as if stunned. Finally, from the questioning faces turned upward, he glanced at the door. Mom was not there. She had already gone down to the first floor. I said as the guys, having noticeably quieted down, headed towards the monorail.

I did not use the taste-lift, but slowly walked down the leaf.

“Jen,” ​​I said, “where is Ralph?”

Jen pretended to be engrossed in combing her long brown hair with a vibrating comb.

• I got it. I needed him to leave.

— Why didn’t I go to school, Jen?

• Please, Chris, don’t ask.

Before I could say anything, I made a sound in the air. He pierced the fairly dense walls of our house and entered my raft, swift and thin, like an arrow from skilled music.

I took a sip. All my fears, hesitations, and doubts instantly disappeared.

As soon as I identified that sound, I thought of Ralph Priory. Oh, Ralph, if only you could be here now. I didn’t believe myself.

I listened to this sound, listened not only with my ears, but with my whole body, with my whole soul, and didn’t believe it.

Closer, closer… Oh, how I was afraid that he was starting to move away. But he didn’t leave. Lowering its tone, it began to descend near the house, scattering light and shadows with huge rotating petals, and I knew that it was a sky-colored helicopter. The humming stopped, and in the silence that followed, my mother leaned forward, released the vibrating comb from her hands, and took a deep breath.

In the silence that followed, I heard footsteps on the porch. The steps I’ve been waiting for for so long.

Someone pressed the bell.

I knew who.

And he stubbornly thought about one thing: “Ralph, why did you absolutely have to leave now that this happened? Why the hell?

Looking at the pilot, one would think that he was born in his uniform. She fit him like a spitting image, like a second skin—silver leather: a blue stripe here, a blue circle there. Strict and impeccable, as a form should be, and at the same time - the personification of cosmic power.

40

Kaleysky, Klim"

1989
1989

This “Kaleisky” is New Year’s: it is made up of a task where the number 1989 is the main “personal action”. We were offered tasks III. S. Abasov, N. K. Antonovich, I. Gshurbekov, D. F. Grigori, A. P. Savin, Y. Faizullaev, D. V. Fuks.

41

1. Can the sum 1 + 2 + … + n, for some n, end in 1989?
2. While making confetti, Vasya cut a sheet of paper into 4 parts, some of the resulting pieces he again cut into 4 parts, etc. Can there be 1989 pieces at some point?
3. The sides of the correctly 1989-gon unbounded continued, as a result of which the plane was divided into many parts. Are these parts more than a million or less?
4. In 1989 I turn one hundred years old, the sum total of the numbers of the year of my birth. What year was I born?
5. Solve the arithmetic puzzle:
   XYZT
   • XYZ
     X
     1989
6. A rectangular box, horizontal and vertical, was divided into 1989 cells with planks, and balls (possibly more than one) were placed in some of the cells. Is it possible to do this so that there are 300 balls in each horizontal row, and 100 balls in each vertical row?
7. Seven nines were written in a row: 9999999. Place the nimus between them so that the best expression equals 1989.
8. The triangle was cut into two polygons using a lightning cut, one of the resulting polygons was again cut into two, etc. What is the smallest number of cuts that should be made so that the total number of vertices of the resulting polygons is 1989?
9. What digit does the number 1989 end with?

42

P means package

(For the beginning, see p. 34)

His name was Trent. He spoke confidently, with unforced smoothness, without beating around the bush.
I stood silently, and my mother sat in the corner looking like a confused girl. I stood and listened.
Of everything that was said, I only remember some oddities.
-…excellent grades, high IQ. Perception A-1, curiosity AAA. The necessary passion for real and patient pursuit is eight long years…

• Yes, sir.
  —…we talked with your teachers of semantics and psychology…
• Yes, sir.
  -…and don’t forget, Master Christopher…
  Mister Christopher!
  -…and don’t forget, Master Christopher, no one should know that you were selected by the Cosmonautics Committee.
• Nobody?
• Your mate and teacher, of course, know about this. But besides them, no one should know. Did you understand me well?
• Yes, sir.
  Trent smiled discreetly, resting his hands on his sides.
• He wants to ask you - why is this? Why can’t you share with your friends? I’ll announce. This is a kind of psychological defense. Every year we select about ten thousand young people from the billion-dollar population of the Earth. Three thousand of them graduate from astronautics studies in eight years, with one specialty or another. The rest comes and returns home. They dropped out, but those around them have no need to know about it. In general, dropouts will occur in the first half of the year. It’s not very pleasant to return home, meet friends and report to them that the most substitute job in the world is too tough for you. So we do everything so that the return is painless. There is another reason. Also psychological. It is so important for boys to be in charge, to be superior to their comrades in some way. By strictly forbidding you to tell your friends that you have been selected, we deprive you of half the satisfaction. And in this way we check what is most important to you: petty ambition or space itself. If you’re only thinking about standing out, you’re on your way. If space is your recognition, if it is everything to you, then welcome.
  He nodded to his mother.
• Thank you, Mrs. Christopher.
  “Sir,” I said. “One question.” I have a friend, Ralph Priory. He lives in a boarding school…
  Trent nodded.
• Naturally, I cannot tell you his details, but he is registered with us. Is this your best friend? And of course you want him to be with you. I’ll check his file. Let’s try it out in a boarding school, you say? This is not very good. But… we’ll see.
• If possible, please. Thank you.
• Come see me at the Kostodrome on Saturday at five o’clock, Mr. Christopher. Until then, not a word to anyone.
  He saluted. And he left. And I took off into it in my helicopter, and at that very second my mother stood very close to me.
  “Oh, Chris, Chris…” she kept repeating, and we clung to each other, and whispered something, and said something, and my mother said how important this is for us, especially for me, how substitutable, and how very, sort of like in the old days, when a person fasted, and took a vow of silence, and didn’t talk to anyone, just prayed and tried to become worthy, and went to some monastery, somewhere in the gouache, and then returns to people, and serves as a model, and brings goodness to people. So now, she said, she concluded, she asserted, this is also a kind of high order, and I will become, as it were, a particle of it, I will no longer contribute to it, but I will contribute to the Universe, I will become everything that my father dreamed of becoming, but could not, did not live…
  “Of course, of course,” I muttered. “I’ll put it up, my word, I’ll put it up…” I faltered. “Jen, how… how do we tell Ralph?” How should we deal with him?
  “You’ll find out, that’s all, Chris.” Tell him so. Short and clear. Don’t tell him anything at all. He will understand.
• But, Jen, you…
  She smiled tenderly.
• Yes, Chris, I will be lonely. But I still have my job and Ralph remains.
• You want to say…
• I’ll pick him up from the boarding school. He will live here when you leave. That’s what you wanted to hear from me, Chris, right?
  I nodded, everything inside me seemed to melt.
• Yes, that’s exactly what I wanted to hear.

43

• He’ll be a good son, Chris. Almost as good as you.
• Excellent!
  We told Ralph Priory. They said that I was going to study in Europe for a year and my mother wanted him to live with us and be her son until I returned home. We blurted it all out as if the words were burning our tongues. When we finished, Ralph first shook my hand, then kissed my mother on the cheek and said:
• I’ll be glad. I’ll be very happy.
  It’s strange, Ralph didn’t even try to find out why I still knew where exactly and when I was planning to return it. He only said:
  — We played great together, right? - and sat down, as if he was afraid to continue the conversation.
  It was Friday night, Priory, Jen and I went to a concert at the Green Theater in the center of our community complex, then, mixture, returned home and began to get ready for bed.
  I didn’t have anything packed. Priory noted this in passing, but did not ask why. But the fact is that for the next eight years, others took care of my personal equipment. There is nothing to pack.
  The semantics teacher called and briefly and affectionately wished me all the best, smiling.
  Finally, we went to bed, but I couldn’t sleep for an hour, I kept thinking that this could be my last night with Jen and Ralph. Last night.
  And I’m just a fifteen year old boy…
  I had already begun to fall asleep when Priory, in the darkness, gently turned on his couch to face me and solemnly walked:
• Chris?
  Pause.
• Chris, are you still awake? - Deaf, like a distant eco.
  “I’m not sleeping,” I answered.
• Do you think so?
  Pause.
• Yes.
  “You… you’ve stopped waiting now, huh, Chris?”
  I understood what he meant. And he couldn’t answer.
  “I’m terribly tired, Ralph,” I said.
  He answered, lay down on his back and said:
• I thought so. You are no longer waiting. Oh, damn, that’s great, Chris. Great.
  He reached out and lightly hit me on the bicep.

Then we both fell asleep.
Saturday morning has arrived. Outside the window, the boys’ heads rolled in the seven o’clock fog. I saw how old man Wickard’s window knocked and the buzz of his parapistol began to creep up on the boys.

• Shut up now! - he shouted, but completely without evil. It was a general Saturday game. You could hear the guys scurrying away in response.
  Priory woke up and asked:
  — Should I tell them, Chris, that you won’t go with him today?
  “No way,” Jen walked from the door to the open window, and the light halo of her hair dispelled the fog. “Great, watgaga!” Ralph and Chris will be out now. Delay launch!
• Jen! - I exclaimed.
  She came up to Ralph and me.
• Spend your Saturday, as usual, with the guys!
  “I was thinking about hitting it with you, Jen.”
• Is there a day of rest for this?
  She quickly fed us breakfast, kissed us on the cheek and sent us out the door into the arms of the Vatga.
• Let’s not go near the Cosmodrome today, guys.
• What are you, Chris… Why?
  Their faces reflected a whole range of feelings. For the first time in history, I refused permission to the Cosmodrome.
• You’re the people, Chris.
• Of course, he’s playing the fool.
  “No,” said Priory. “He does it on the server.” I don’t want to go there either. We go every Saturday. Tired of it. We’d better go next week.
• Oh well…
  The unnies were unhappy, but they didn’t want to go without us. They said it wouldn’t be interesting without us.
• Well, okay… Let’s go next week.
• Certainly. Now what do we do, Chris?
  I told them.
  On this day we played “beat the can” and other games that we had long abandoned, then we went to a small approach to the rusty tracks of an old, abandoned railway, wandered through the forest formed by some birds, swam naked, and I kept thinking about one thing: today is the last day.
  Everything we ever ate on Saturdays, we remembered it all. All sorts of jokes and pranks. And, except for Ralph, no one suspected my answer, and with each

44

Every minute the cherished “five o’clock” was getting closer and closer.
At four I told the guys “to witness.”
“Are you leaving already, Chris?” Well, what about in the evening?
“Come in at eight,” I said. - Let’s go see the new picture with Sally Ghiberti!

• That’s right.
• Key to start!
  And Ralph and I went home.
  Mom was not at home, but on my bed there was an audio film on which she left a part of herself - her smile, her voice, her words. I inserted the video into the projector and put it on the wall. Soft brown hair, mother’s white face, her quiet voice:
• I don’t like to beg, Chris. I’ll go to the laboratory and work there. Happy birthday to you. Hugged tightly. When I see you again… you will be a man.
  That’s all.
  Priory waited outside the door, and I played the video for the fourth time.
• I don’t like to beg, Chris. I’ll go… work… happily. Tightly… hugged…
  I also recorded a video the night before. Now I put it in the projector and left it - two or three stray words.
  Priory kept me waiting until halfway through. It’s impossible for him to go with me to the Spaceport. At the monorail station I shook his hand firmly and said:
• We had a great day today.
• Yeah. Now, what, until next Saturday?
  “I wish I could answer yes.”
• Answer “yes” anyway. Next Saturday - forest, gang, packages, old Wickard with his trusty parapistol. We laughed at each other.
• Agreed. Next Saturday, early in the morning. And you take care… take care of our mother, okay, Priory, do you promise?
  “What a stupid question, you idiot,” he said.
• Exactly, you idiot.
  He took a sip.
  — Chris.
• Yes?
• I’ll be waiting. Just like you waited, and now you don’t have to wait anymore. Will wait.
  “I don’t think you’ll have to wait long, Ralph.” I hoped it wouldn’t be long.
  I lightly hit him once on the arm. He responded in kind.
  The monorail door closed. Kabi rushed forward and Priory remained behind.
  I got off at the Kosmoport stop. The control building was about five hundred meters away. I walked this stretch for ten years.
  “When I see you again, you will be a man…”
  “Not a word to anyone…”
  “I’ll be waiting, Chris…”
  All this is like a cork in the heart, and you just don’t want to leave, and it floats before your eyes…
  I thought about my dream. Moon rocket. Now she will no longer be a part of my soul, my dream. Now I will become a part of it.
  I walked and walked and walked, feeling completely insignificant.
  The very minute I approached the controls, the evening London rocket took off. She slipped onto the ground, and glided in and filled my heart with sweet trembling.
  And I immediately began to grow terribly fast.
  I followed the rocket with my eyes until someone’s heels clicked in greeting next to me.
  I’m petrified.
  — K. M. Christopher?
• Yes, sir. Called, sir.
• This way, Christopher. These questions…
  The fence to which a week ago we pressed our faces, and felt the wind getting hot, and pressed ourselves even harder against it, forgetting who we are, where we come from, dreaming only about who we could be and where we could go…
  The fence, against which a week ago we pressed our faces, who liked being boys, liked living in a small Florida town, and certainly liked school, and liked playing football, and they liked dads and mothers too…
  Boys who, every week, every hour, minute, incessantly thought about the flame, and the stars, and the vastness beyond which all this awaited them… Boys who liked more.
  Mom, Ralph, we’ll see you. I’ll come back.
  Mother! Ralph!
  And I went through the gate and entered inside the fence.

This story is reprinted from the collection of fantasy stories by Ray Bradbury (M.: Children’s Literature, 1973). Translation from English by N. Gal and E. Kabalevskaya.

45

Physics 8, 9, 10

The note “Equilibrium of a mechanical system and the method of virtual displacements” published below is intended for eight-year-olds, the note “How many states does a substance have?” - for ninth-graders, “A few changes regarding the photo effect” - for tenth-graders.

Equilibrium of a mechanical system and the method of virtual displacements

He said that for the construction of one of the cathedrals in Switzerland, its architect came across blocks that allowed him to lift particularly heavy loads to great heights. He designed a very complex chain hoist, but got confused in the numerous tension forces of the cables and was never able to calculate how many people would need to be hired to service it. For help, the architect approached the famous scientist Juhann Bernoulli (1667-1748). Imagine the architect’s surprise when Bernoulli, barely looking at the drawing, immediately gave the answer. The amazed architect asked to reveal to him the secret of the pillar calculation…

Let us also try (following Bernoulli) to discuss the equilibrium conditions of a mechanical system from a slightly different position than was done in the next chapter of the school textbook “Physics 8”.

Obviously, in order to find out the conditions of equilibrium of a system, the reaction forces of mechanical connections are driven *). Connections are usually called those restrictions that are imposed on the position of individual parts of the system, or on the possibility of their movement.

*) See the article by A. I. Chernoutsan “Kinematic connections in diamnic problems” in the second issue of the journal for 1988.

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Examples of mechanical connections include a thread on which a load is suspended, a hinge connecting individual parts of a mechanism, a plane on which the body of the system is located, etc. The more connections there are in a system (for example, threads in a pulley block designed by our architect), the more difficult it is to monitor the action of all the reaction forces arising in them.

In many cases, mechanical connections have some remarkable property, which Bernoulli used as the basis for his simple and elegant method of finding the equilibrium conditions of a mechanical system. This property lies in the fact that the total work of all reaction forces arising in the connection of the system for any sufficiently small possible deviations from the equilibrium position is equal to zero.

Here it is necessary to specifically stipulate what is meant by “any sufficiently small possible deviations” of the system from the equilibrium position. First of all, of course, the movements of the points of the system that determine such deviations should not contradict the mechanical connections (the threads should not fray, the hinges should not break). Such movements are called emergent, or virtual (which means mental). They should be distinguished from true movements that occur under the influence of applied forces. Virtual displacements are not relative to the process of motion of the system - they are introduced only in order to identify the correspondence of forces existing in the system and establish equilibrium conditions. A small amount of displacement is needed so that the reaction forces can be considered unchanged.

Connections for which the sum of the elementary work of the reaction forces for any virtual displacements is equal to zero are called ideal. We emphasize that not all connections are ideal. So, if the bodies of the system lie on a plane with friction, then this connection (plane) is no longer ideal.

Let us consider how the property of ideal connections is fulfilled using the example of a system of points forming an absolutely rigid body. Any point $A$ of such a body is connected to all other points by reaction forces. Let us select some point $B$ and denote the reaction forces acting between the points by $\vec{F}_{AB}$ and $\vec{F}_{BA}$ (the first of them is applied to point $A$, the second - to $B$). Let’s forget for a moment about all other points of the body, except for the two highlighted ones. We are then left with a system of two point materials connected by an incompatible rigid rod. According to Newton’s third law,

$$\vec{F}_{AB} = -\vec{F}_{BA}.$$

Let us now perform some virtual movements of these points $\Delta s_A$ and $\Delta s_B$. We can assume that $\Delta s_A$ is the translational displacement of the rigid body as a whole, common to all points, and $\Delta s_B = \Delta s_A + \Delta s'$, where $\Delta s'$ is the displacement corresponding to the expression of point $B$ relative to point $A$. The total work done by the reaction forces during these virtual displacements is

$$\Delta A = (\vec{F}_{AB} \cdot \Delta s_A) + (\vec{F}_{BA} \cdot \Delta s_B) = \left( (\vec{F}_{AB} + \vec{F}_{BA}) \cdot \Delta s_A \right) + (\vec{F}_{BA} \cdot \Delta s').$$

The first term on the right side of this equality is equal to zero due to condition (*). The second term also turns out to be equal to zero, since the displacement $\Delta s'$, corresponding to the expression of the point $B$ relative to the point $A$, is perpendicular to the force $\vec{F}_{BA}$, which is directed along the straight line connecting the points $A$ and $B$. In the same way, it can be shown that the virtual work (that is, the total work during virtual displacements) of the reaction forces for any other rigid body is equal to zero. Thus, by considering the example, we were convinced of the validity of the discussed property of ideal connections.

In order to finally formulate a new principle of equilibrium, it remains for us to consider the virtual work of external forces. Note that each external force applied to the system under equilibrium conditions must be in equilibrium with the reaction forces caused by

47

by her. Pozny, if we now calculate the total work done by external forces and reaction forces for any virtual displacement, then it turns out to be equal to zero. But regarding reaction forces, we already know that they do not do any complete work. Therefore, for any virtual displacement, the total work done by external forces must also be equal to zero.

So, the equilibrium conditions of a mechanical system can be formulated in the form of the so-called principle of virtual (possible) displacements:

For the equilibrium of a mechanical system with ideal connections, it is necessary and sufficient that the sum of the work of all external forces acting on the system for any virtual displacements of the system is equal to zero.

Thanks to this principle, formulated by Johann Bernouli in 1717, to find equilibrium conditions there is no need to consider a large number of reaction forces. It is enough to select convenient virtual displacements, calculate the corresponding total work of only external forces and equate it to zero.

Let us illustrate the application of this principle with a simple example: we will find the elastic force of a spring from which a load of mass $m$ is suspended. Let’s choose the load itself as the system. Two external forces are applied to it - the elastic force of the spring $T$ and the gravitational force of the load $mg$. Let’s imagine that we have shifted the load in equilibrium by a small amount $\Delta x$ down. Then the work done by the gravitational force will be equal to

$$\Delta A_1 = mg\Delta x,$$

and the work done by the elastic force of the spring is

$$\Delta A_2 = -T\Delta x.$$

According to the principle of virtual movements, the sum of the work of both should be equal to zero:

$$\Delta A_1 + \Delta A_2 = mg\Delta x - T\Delta x = 0,$$

where do we get it from

$$T = mg.$$

This problem, of course, could be solved in the usual way - using the conditions of equilibrium of bodies. Moreover, in this case, both methods are the same in terms of complexity. However, in a number of cases, the method of virtual displacements leads to a faster and simpler solution, and sometimes even allows one to solve problems that, in principle, cannot be solved on the basis of general equilibrium conditions.

Note also that the method of virtual displacements can be successfully used in solving not only mechanical problems, but also problems of electronics and molecular physics.

A. A. Varlamov

How many states does a substance have?

In this note we want to talk a little about the different states of matter - the most famous, somewhat less known and very little known.

Any body is a huge number of molecules moving and interacting with each other. Every perfection is natural, that when the interaction of molecules with each other is weak, the molecules must form a gas; otherwise, when the interaction is strong, it is a solid body; in the intermediate case - liquid. This is certainly true. Only in physics there is no understandable small or large quantity without comparison with something. In this case, the interaction energy of molecules must be compared with their kinetic energy.

From the molecular-kinetic theory it is known that the average kinetic energyxaotic energyxaotic energy of molecules $\bar{E}$ is directly related to the temperature $T$ of the system: $\bar{E} = \frac{3}{2}kT$. Since we are only interested in qualitative considerations, we can

48

we will not take into account the tel $^3/2$. Let us denote the average energy of interaction of molecules with their environment by $U$. Let us introduce the parameter $\varepsilon$, equal to the ratio of the average interaction energy to the average kinetic energy: $\varepsilon=U/kT$. Now we can write down the conditions for the existence of gas, liquid and solid. If $\varepsilon\ll 1$, we have a gas (the molecules move quickly, with almost no interaction with each other). When $\varepsilon\gg 1$, the system is a solid (the molecules are “sandwiched” in their places). The intermediate case, when $\varepsilon\approx 1$, corresponds to a liquid. But within each of these very large classes of condition there is quite a lot of diversity.

Let us first focus on the solid body. From the condition $\varepsilon\gg 1$ it is clear that the state of a solid is determined mainly by the energetic interactions of molecules. As is known, any system presented to itself tends to occupy a position such that its potential energy is minimal (by potential energy here we must remember precisely the energy of interaction of molecules with each other). So, it turns out that the minimum energy corresponds to the state when the molecules are arranged strictly periodically. In other words, stable equilibrium corresponds not just to a solid body, but specifically to a crystal. This is the first, most studied type of solid. The properties of crystals are entirely determined by the type of crystal lattice. There are lattices composed of cups, hexagonal prisms, parallelepids, etc. When crystals are heated (for example, at atmospheric pressure), there is a temperature (attention! a very specific temperature) at which the crystal lattice becomes unstable. Swimming begins (it is clear that the swimming temperature must be determined from condition $\varepsilon\approx 1$, i.e. $kT_{\pi l}\approx U$).

Another type of solid will arise when, upon cooling, liquids lose their mobility before they have time to protrude into the crystal lattice. Now they would “want” to get organized, but they can’t. More precisely, they can, but for this they need a lot of time. We get a solid, but not a crystalline, but an amorphous body. A typical example of such bodies is glass. When heated, glass gradually softens and eventually turns into a liquid, but there is no specific melting point.

Whether cooling a given liquid produces a crystal or an amorphous body strongly depends on the cooling rate. For example, to obtain amorphous metals, the cooling rate must be enormous (molten metal is sprayed onto a surface cooled with liquid nitrogen). But this is not the only condition. For example, no matter how hard it melts, glycerin will not form a crystal (the reason for this is not entirely clear today).

If we are talking about bodies consisting of molecules of simple shape, then, apparently, there are no other possibilities. But fortunately, the world is not that simple. You know very well that there are limited (and not only limited) molecular properties of complex shapes. Substances built from such molecules can be in non-forming states that cannot be classified as either liquid or solid. Here are some examples.

A typical property of a liquid is its isotropy, i.e., the same properties in all directions. However, now it is now possible to use mechanical properties, the speed of propagation of various waves (elastic or electromagnetic), etc. So, about a hundred years ago, liquids that were not isotropic were discovered - the so-called anisotropic liquids. Since then, huge amounts of liquid have been discovered (and artificially created). Their main feature is that in some directions they have

49

give the properties of the crystal (for example, the periodicity of the internal structure), while in others they do not. These are liquid crystals. For the combination of such seemingly incompatible properties as fluidity and orderliness, they received the name mezofaz (mezo means intermediate; i.e., intermediate phases).

Long polymer molecules can form another class of states, which include, for example, aspic or rubber. In these states, long molecules are combined into branched chains or networks. As a result, a peculiar gland-like body will appear, which is called a “gel”. Conditions of this type are also extremely common in nature.

Finally, let us dwell very briefly on some sense of extreme states of matter.

When a gas is heated, the kinetic energy of its molecules increases and may be on the order of the atomic infusion energy. Then, when molecules collide with each other, the atoms can become intertwined, and we get a mixture of neutral and loaded (positively and negatively) particles. It is very important that the amount of positive and negative contaminants is always the same, so that in general the gas is electrically neutral. This is plasma, a completely special state of matter with a unique property.

And in conclusion, let’s turn to…
to the stars. A star is a giant cloud of gas and dust that tends to shrink under the influence of gravitational attraction. As a result of this decline, the temperature in the core of the star increases, and at some point a thermonuclear reaction is ignited: hydrogen nuclei merge and turn into helium. The energy released in this case further transforms the star into scatia, and the star stabilizes (it is at this stage of stellar evolution that the Sun is located). But gradually the hydrogen burns out, and tell us the possibility. The colossal pressure that arises in this case crushes the atoms. A state arises in which electrons seem to float freely in the field of bare nuclei. If the mass of the star is not too large (less than 1.25 solar masses), then the specific scaling that exists between the electrons transforms it further into compression (this repulsion is not associated with electrical forces, but is purely quantum in nature). As a result, a completely special state arises with enormous density (about 60 t/cm³). Stars arranged in this way are called white dwarfs (due to their light blue glow and small size). If the mass of the star is greater (more than 1.5-2 solar masses), then electrons cannot resist gravitational shrinkage. As a result, they (electrons) move into nuclei and, merging with protons, form neutrons. A substance appears, consisting not of atomic nuclei, but of neutrons, with a very fantastic high density (≈2·10⁹ t/cm³). These are neutron stars.

We’ll probably stop there. Although, frankly speaking, the list of amazing states of matter that exist in nature is far from exhausted.

E. E. Gorodetsky

A few changes regarding the photo effect

In the school physics course, you became acquainted with the phenomenon of the photoelectric effect, i.e. the emission of electrons by a substance under the influence of light, and its laws (“Physics 10”, chapter 10).

What is the main thing in the theory of photoelectric effect? Of course, the hypothesis of light quanta - photons. The photoelectric effect can be represented as the result of two successive processes: 1) absorption of a light quantum by an electron, 2) emission of an electron outside the substance. If it happens

50

Both processes and the phenomenon are more correctly called external photoeffect. If the absorption of photons does not lead to the emission of electrons from the substance, but changes its electrical conductivity, then they speak of an internal photoelectric effect (usually observed in semiconductors). The actual photoelectric effect is often called the act of absorption of a photon by an electron.

The question arises: can the photoelectric effect occur on a single free electron? At first glance - why not? After all, we say: the photon is absorbed by the electron. What does the substance have to do with it?
Let’s take an electron, shine a flashlight on it, and it begins to “swallow” photons and accelerate them! It turns out that nothing will work out. A free electron cannot absorb a single photon. True, it will move from its place, but the reason will not be absorption, but scattering of photons. This is also a very interesting process - the well-known Compton effect, in which the quantum properties of light are clearly manifested, but… not the photo effect.

Why can’t a photon be absorbed by a free electron? Let us carry out, as mathematicians say, a proof “by contradiction”. Let an electron (at rest or moving) absorb a photon incident on it, and at the same time its speed changes. It turns out that such a process is prohibited by the laws of conservation of energy and momentum. This becomes very serious if you choose an inertial reporting system in which the electron ends up dead after the photoelectric effect. See for yourself. What do we have in the final state? An electron at rest and nothing else. And in the initial state? Moving an electron and a photon to boot. Real energy is not conserved.

This means that the photo effect “according to all laws” is possible only in the presence of a third participant. In half-shaped metals, this role is played by ions of the crystal lattice. But when calculating energy (for example, in Einstein’s equations for the photoelectric effect), we do not take them into account because, due to their large mass, they quickly take up a very small part of the energy (playing an important role in the law of conservation of momentum).

Is a photo effect possible on a single atom (or molecules) - for example, in a gas? It turns out, yes. The photon is absorbed by one of the electrons of the atoms, and only the impulse is carried away by the atomic nucleus. It is interesting to note that the photoelectric effect was first discovered by G. Herme precisely in experiments with gases (1887). He studied the electrical breakdown of the air gap between the electrodes and discovered that when this gap is irradiated with light, the breakdown occurs at a lower voltage.

Why is it that only the external photo effect is discussed in detail in a school physics course? After all, with any photoelectric effect, the main quantum phenomenon occurs - the absorption of a photon by an electron. The thing is that the laws of the external photoeffect are quite simple and are relatively easy to study experimentally. In this case, the quantitative characteristics of the photoeffect can be found both from the experiments on the external photoeffect and by independent methods. Let’s talk about this in a little more detail.

Einstein’s equation

$$hv = A + \frac{mv^2}{2}$$

contains two directly variable quantities: the frequency of light $v$ and the maximum kinetic energy of knocked out electrons $mv^2/2$. Optical methods can create beams of light with well-extracted frequencies $v$. For information