Siasiev A. V., Melnik A. L., Rusakova A.V.

Oles Honchar Dnipropetrovsk National University

THE TASK OF SEVEN TWIGS

Everyone knows the old parable of the seven twigs. The father wanted his sons to live in peace and friendship and he offered them to break a bunch of seven twigs. Sons tried to do it, but it was unsuccessfully. Then the father took this bunch, untied it and then he broke each twig separately without any problems. That is why this parable can be quite clear to us only when we think about it from the mechanical point of view.

The amount of rod bending is measured, in mechanics, by so-called «bending deflection»   (Fig. 1). The greater this bending deflection in this timber is, the closer its fracture moment is. The amount of bending deflection is expressed by the following formula

, 

here   ‒ is the force applied to the rod;   ‒ is the length of the rod,  ;   ‒ is a number characterizing the elastic properties of the rod;   ‒ is the radius of the circular rod.

Fig. 1. Bending ( ) deflection

We apply this formula to the bunch of twigs. His seven twigs were likely located as it is shown on the Fig. 2. But we can only approximately consider this bunch as a solid rod (for that it must be tied hard). However, here we are not looking for an accurate solution. As it is clearly seen from the Fig. 2, the diameter of tied bunch is about three times bigger than the diameter of each twig separately. We can show that to bend a separate twig (that means ‒ to break it) is much easier than to break a whole bunch. If in these both cases, the same bending deflection must be obtained, then the force   must be applied to one twig and to the whole bunch – the power  . And the ratio between   and   arises from the following equation

,

whence

.

Fig. 2. Location scheme seven twigs

Thus, it is clear that the father had to apply force though 7 times, but he did it with less force that is 81 times less than his sons.

The list of references:

1. Перельман Я. И. Занимательная механика / Я. И. Перельман. – М.: Книговек, 2013. – 240 с.