How deep is the underground station

How to esti­mate the depth of a sub­way sta­tion, to which you are access­ing via an esca­la­tor? It hap­pens that even to answer this ques­tion the knowl­edge of math­e­mat­ics can be very use­ful. And in par­tic­u­lar, of trigonom­e­try.

The esca­la­tor of the met­ro­pol­i­tan rail­way... How many things are hid­den behind these words for a curi­ous observer. A huge machine in con­stant motion, a “liv­ing scale” ...

All began in the late nine­teenth cen­tury, when the Amer­i­can man­u­fac­turer Jesse W. Reno (1861-1947) patented the first “liv­ing scale.” In its con­struc­tion instead of the steps fixed to an “infi­nite” tape there were cylin­ders. But the first pub­lic esca­la­tor was made, accord­ing to its inven­tor, Charles D. See­berger (1857-1931), from “Otis” indus­try and was exhib­ited at the Paris Expo­si­tion of 1900. This esca­la­tor had hor­i­zon­tal steps, com­ing out from under a fence on a plat­form of entry and dis­ap­peared under the same fence on another plat­form. How­ever, this mech­a­nism gave a lot of prob­lems. In 1921 both ideas – that of hor­i­zon­tal steps and of cylin­ders — were put together to build a new model which, from that moment on, was always used.

When in the thir­ties the design of the Moscow sub­way started, the engi­neers tried to make use of for­eign expe­ri­ence. How­ever, both the cost and time required for its imple­men­ta­tion by for­eign com­pa­nies were so large that the idea was aban­doned. At that time the head of the Lon­don branch of the com­pany “Otis” wrote to the rep­re­sen­ta­tive of the Coun­cil of Moscow: “Your spe­cial­ists are capa­ble men. But the esca­la­tor is a very com­pli­cated affair, and you will not thrash out this prob­lem. Even for us, with our capa­bil­i­ties and decades of expe­ri­ence, to run the pro­ject with those dead­lines is impos­si­ble. As a friend of the Soviet Union, it is my duty to warn you that the date on which the sub­way will start can be not respected.” Yet, the Soviet engi­neers and sci­en­tists were able to solve this very spe­cial prob­lem, and in Feb­ru­ary 1935 an esca­la­tor began to carry pas­sen­gers to the sub­way sta­tion in Moscow.

One of the impor­tant ele­ments of the esca­la­tor is the step. It has four rollers, two large and two small. Both large and small rollers roll along their own tracks.

When the esca­la­tors were designed, even the choice of mate­ri­als for the rollers was a very impor­tant and dif­fi­cult prob­lem. The Moscow Metro is open from 6 am until about one o’clock at night. That is, more than 19 hours — that is, more than 68000 sec­onds per day. The min­i­mum speed of oper­a­tion of a slid­ing scale is 0.75 m / s, which means that each step runs 50 km in one day. And so, tire­lessly, day after day, over a year more than 18 thou­sand kilo­me­ters! Can you imag­ine of which kind of mate­r­ial the rollers should be made, to hold, with­out reg­u­lar repairs and replace­ments, a fairly high amount of pas­sen­gers that move on the steps. And this is just one detail and just one of all prob­lems that the Soviet engi­neers had to solve, and of such prob­lems and there were thou­sands.

Here is how the scheme of an esca­la­tor looks like. If we observe it in pro­file, we see that the mutual posi­tion of the rails of large rollers and of those of small rollers defines the fun­da­men­tal prop­erty of the esca­la­tor: in the upper part of the “liv­ing scale”, on which the pas­sen­gers stay, the steps are always hor­i­zon­tal. But at the bot­tom the steps are reversed and become par­al­lel to the rails, sav­ing space in the tun­nel where they slide.

But let us come back to our ques­tion about the depth to which we are car­ried by the esca­la­tor. The sur­pris­ing fact is that all Russ­ian esca­la­tors, from the first ones up to the pre­sent days, are inclined by 30 degrees with respect to the hor­i­zon­tal!

Let us men­tally con­struct an esca­la­tor on a right tri­an­gle. The length of its hypotenuse is the length of the esca­la­tor, while the length of the shorter cathetus is approx­i­mately equal to the depth of the sta­tion to which this stair­case leads.

But how to cal­cu­late the length of the esca­la­tor while you are going on it? You could mea­sure the time, but then for the cal­cu­la­tion of the course you should know exactly the speed of the motion. But the speed can range from .75 m/s to 1 m/s and the error — a fourth — is quite large.

You could cal­cu­late the mea­sures of a step, but in this case to know how many there are on the hypotenuse, while mov­ing on the esca­la­tor, is com­pli­cated ...

What we can still use? Going down or going up an esca­la­tor, we con­tin­u­ally meet lamp­posts! The dis­tance between them is not fixed, but accord­ing to gov­ern­ment rules, the tun­nel light­ing must have a cer­tain inten­sity. And in total we obtain that the lamp­posts are located at about five meters one from the other.

Going down the esca­la­tor, we can count the num­ber of lamp­posts. What should we do then, to cal­cu­late the length of the hypotenuse?

Do not rush to mul­ti­ply by 5. For the cal­cu­la­tion of the length we do not need the num­ber of lamps, but the amount of dis­tances between them! From the num­ber of lamps we must sub­tract 1, then we can mul­ti­ply by 5 and by the sine of 30°.

The great thing at this point is that the sine of 30 degrees is equal to ½, and this num­ber is easy to count in your head! So the for­mula obtained for the depth of a sta­tion is very easy both to remem­ber and to cal­cu­late.