Geometric progression decrease

It is common to speak of “expo­nen­tial growth” when the aim is to empha­size the signif­i­cant speed of change (growth or decline) of some value. If we consider time to be discrete and changing in leaps, then the concept of “expo­nen­tial growth” is easily described using geometric progres­sions.

The most famous example, illus­trating the speed of growth of a geometric progres­sion, is the legend about the origin of chess. If you put one grain of wheat on the first field of the chess board, two — on the second, four — on the third and so on, then it turns out that filling the last 64-th field of the chess board requires to harvest crops from the entire Earth for more than a thou­sand years.

You can also get an idea of the speed of growth of a geometric progres­sion just by folding a piece of paper. After the first fold the thick­ness of the paper would increase twice, after the second — 4 times, and if one could manage to fold the paper 42 times, the thick­ness of the paper would be greater than the maximal distance from the Earth to the Moon.

Exam­ples demon­strating the decrease of the geometric progres­sion are no less fasci­nating.

Imagine a sequence of gears connected succes­sively in such a way that each gear is rotating 10 times less frequently than the previous one. If the first gear completed 10 full rota­tions, the second would complete a single one, the next one — 1/10 of a full rota­tion etc. There­fore, the angular veloc­i­ties of the gears form a geomet­rical progres­sion with common ratio 1/10.

In the movie the first gear turns with a rate of 3 full rota­tions per second. It turns out that even a few steps of the geometric progres­sion are suffi­cient for the last gear nearly not to move in any “rea­son­able” time and it is even possible to completely fix it in a wall.

Let us watch how many rota­tions will the gears complete after a long time. For example, what will happen 10 years later...

After 10 years of the drill's constant work the first gear will have completed more than 877 million rota­tions, and the last, 13th, gear will not have budged even one thou­sandth of a rota­tion.

What would happen if we applied the drill to one of the inter­me­diate gears? Despite the intu­ition suggesting that the gears to the right of the driven gear will move very fast, the system will not move at all. It turns out that the fric­tion force emerging on the right side of the driven gear grows in a geomet­rical progres­sion as well. Rotating such a system while applying force not to the right­most gear is impos­sible.

Other etudes in “Formula geometry”