Abstract
Since the times of Galileo, it is well-known that a simple pendulum oscillates harmonically for any sufficiently small angular amplitude. Beyond this regime and in absence of dissipative forces, the pendulum period increases with amplitude and then it becomes a nonlinear system. Here in this work, we make use of Fourier series to investigate the transition from linear to nonlinear oscillations, which is done by comparing the Fourier coefficient of the fundamental mode (i.e., that for the small-angle regime) to those corresponding to higher frequencies, for angular amplitudes up to 90∘. Contrarily to some previous works, our results reveal that the pendulum oscillations are not highly anharmonic for all angular amplitudes. This kind of analysis for the pendulum motion is of great pedagogical interest for both theoretical and experimental classes on this theme.
Highlights
The measurement of the period of a simple pendulum is a popular experiment for undergraduates
Its simplicity is subjected to a singular and important condition that the initial angular displacement θm rendered to the bob is small
This reduces the non-linear equation of motion [1]
Summary
The measurement of the period of a simple pendulum is a popular experiment for undergraduates. Its simplicity is subjected to a singular and important condition that the initial angular displacement θm rendered to the bob is small This reduces the non-linear equation of motion [1]. Where T0 is the period of the pendulum in the limit of small-angle oscillations, in which it exhibits a simple harmonic motion (SHM). It is only for small initial displacements, namely θm π/2 rad, that the pendulum. We address the distinction between small and large-angle oscillations based upon Fourier series analysis This method has not been explored in literature, except by the works by Gil and co-workers in Ref. A consolidated approach can be developed following our method for studying the nonlinear oscillations of a simple pendulum
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