Non-trivial solution to a simple problem

TL;DR

This paper examines the complex solution to a seemingly simple physics problem involving oscillations of a spring with finite mass and stiffness, revealing that common solutions are often incorrect and providing a correct analysis.

Contribution

It clarifies the correct approach to solving the oscillation frequencies of a finite-mass spring, correcting widespread misconceptions in educational contexts.

Findings

Common solutions to the problem are incorrect

The problem's solution is non-trivial and requires careful analysis

Provides a correct method for calculating oscillation frequencies

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.