A low-cost experiment to visualise the Fourier series: video analysis of a real plucked coiled spring

In the present work, we develop a low-cost and simple experiment to visualise Fourier's synthesis using a short, soft, and light plastic coiled spring oscillating in a horizontal plane, and a basic camera (120 fps). It is shown that the spring obeys a linear wave differential equation, as gravitational influence is neglected. A nonlinear criterion is evaluated to determine if magnitudes of the parameters in the initial conditions satisfy the linear wave equation. Our setup promotes some desirable characteristics that make Fourier's synthesis experiments feasible, visual, and enlightening: (i) it requires few, common, and cheap resources, and the experiment can be carried out even in a high-school laboratory; (ii) since the spring's tension is small (∼1 N, on average), the frequencies of normal modes are low (close to 2 Hz), and therefore, it is possible to record the oscillations just with the camera and extract a considerable number of position and time data in just one cycle; (iii) when the video is loaded in the Tracker free software, it can be reproduced in slow motion. Since the frequencies involved are low, an interesting and instructive temporal sequence of images of the spring displaying the typical trapezoidal shape appears clearly; (iv) the tools associated with the Tracker software tools can yield the relevant oscillation parameters, such as the damping constant, amplitudes, frequencies, and phases; and (v) it is possible to carry out superposition of a snapshot of the spring in Tracker at any time, and to draw the related Fourier synthesis graphs. The visual match between the shape of the spring and the theoretical graph is remarkable, and can be enhanced by adding the damping term. (paper)