Problems on Mechanics

This booklet is a sequel to a similar collection of problems on kinematics. Similarly to that collection the aim here is to present the most important ideas using which one can solve most (> 95%) of olympiad problems on mechanics. Usually a problem is stated first, and is followed by some relevant ideas and suggestions (letter ‘K’ in front of the number of an idea refers to the correspondingly numbered idea in the kinematics booklet). The answers to the problems are listed at the end of the booklet. They are preceded by quite detailed hints (no full solutions), but think carefully before reading the hints as a last resort! The guiding principle of this booklet argues that almost all olympiad problems are “variations” on a specific set of topics — the solutions follow from corresponding solution ideas. Usually it is not very hard to recognize the right idea for a given problem, having studied enough solution ideas. Discovering all the necessary ideas during the actual solving would certainly showmuchmore creativity and offer a greater joy, but the skill of conceiving ideas is unfortunately difficult (or even impracticable) to learn or teach. Moreover, it may take a long time to reach a new idea, and those relying on trying it during an olympiad would be in disadvantage in comparison to those who have mastered the ideas. In science as a whole, solution ideas play a similar role as in olympiads: most scientific papers apply and combine known ideas for solving new (or worse, old) problems, at best developing and generalising the ideas. Genuinely new good ideas occur extremely rarely and many of them are later known as masterpieces of science. However, as the whole repertoire of scientific ideas encompasses immensely more than mere mechanics, it is not so easy to remember and utilise them in right places. The respective skill is highly valued; an especial achievement would be employing a well-known idea in an unconventional (unexpected, novel) situation. In addition to ideas, the booklet also presents “facts” and “methods”. The distinction is largely arbitrary, some facts could have been called methods and vice versa. In principle, an “idea” should have wider and/or more creative applications than a “fact”; a “method” is a universal and conventionalized “idea”. Several sources have been used for the problems: Estonian olympiads’ regional and national rounds, journal “Kvant”, Russian and Soviet Union’s olympiads; some problems have been modified (either easier or tougher), some are “folklore” (origins unknown).