The spring‐mass system revisited

A vertically oscillating spring of mass m and spring constant k suspended from its upper end and with a mass M attached to its lower end is a system often used for demonstrations and experiments in introductory physics courses. We discuss the motion of this system for arbitrary values of ε=m/M, 0≤ε<∞ and show explicitly why theory predicts that the amplitude of the lowest normal‐mode frequency makes the major contribution to the motion of M (or of the lower end of the spring) for all values of ε. Although a complete understanding of this fact involves detailed mathematical analysis, the results themselves are simply stated and readily verified even by students in an introductory calculus‐based physics course. The various predictions of the theory are easily demonstrated with simple equipment and lend themselves nicely to an introductory physics laboratory. These applications are discussed in some detail, and an analog electric circuit is given which exhibits similar behavior.