Non-trivial solution to a simple problem

Abstract

The problem of finding the frequencies of small longitudinal oscillations of a spring having a finite mass and stiffness, attached at one end to a wall and at the other end to a body of finite mass, is discussed. This problem was repeatedly proposed at Olympiads for schoolchildren, in various lessons on the Internet, and even on tests in mechanics for students of universities. In all the cases known to me, the implied solution was actually wrong. I discuss two cases: (A) a spring lies on a smooth table, (B) a spring is attached to the ceiling. It is shown that the solution to this simply formulated problem is non-trivial.