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.

Other etudes in “Mathematics and Technology”