An asymmetric isochronous pendulum

Abstract

In this work, we propose a novel asymmetric isochronous pendulum, where half of the trajectory corresponds to that of the (non-isochronous) simple pendulum, while the remaining part corresponds to a specific trajectory. The pendulum's complete swing is isochronous—i.e., its period is not dependent on the oscillation amplitude. This new design is inspired by the symmetric isochronous Huygens pendulum, in which the trajectory is modified by cycloidal guides. In our case, only one guide is needed, for which analytical expressions are given for the first time. The objective of the paper is to add a new pedagogical tool in the undergraduate-level physics courses for understanding the isochronism concept, specifically the time delay for large amplitudes in the simple pendulum must be compensated by accelerating it along the new trajectory. We also introduce a novel numerical method to find the trajectory, based on the convergent superposition of straight trajectories (inclined-plane type) to approximate the curve. The procedure is not only accurate, but it is also appropriate for introducing the concepts of differential calculus and inclined-plane mechanics.