Video analysis of a massive coiled spring transverse oscillations described by Fourier series

Abstract

Abstract This work presents a low-cost and straightforward experiment to visualise Fourier synthesis using a coiled metallic spring whose weight is not negligible, oscillating in a vertical plane and fixed at both ends. A massive coiled spring still obeys the standard linear differential wave equation, but a new term accounting for the effect of gravity must be introduced in the equation. Hence, if the spring is plucked vertically at its centre so that its initial configuration is a triangle, the trapezoidal shape of the resulting oscillation, typical of a spring of negligible mass, is clearly distorted by the effect of gravity, when the amplitude of the oscillation moves away from its maximum value. The hypothesis that this distortion comes from some nonlinear effect produced by the larger pluck-length ratio used in the experiment was discarded when evaluating the magnitude of a mathematical criterion to estimate the nonlinear impact of oscillations with large amplitudes. As the spring’s tension is small (∼4 N, on average) and the frequencies of transverse waves are low (close to 1 Hz), oscillations could be filmed with a basic and cheap camera (60 fps), which makes it possible to carry out the experiment without the need to purchase equipment. In addition, as the frequencies involved are low, the oscillations could be reproduced and visualised in slow motion. The free software Tracker was used to collect the position data of the massive coiled spring at each instant and compare them with the related solution of the wave equation obtained by Fourier synthesis of the first six terms.