Simple ‘log formulae’ for pendulum motion valid for any amplitude

Abstract

By combining a logarithmic approximate formula for the pendulum period derived recently (valid for amplitudes below π/2 rad) with the Cromer asymptotic approximation (valid for amplitudes near to π rad), a new approximate formula accurate for all amplitudes between 0 and π rad is derived here. It is shown that this formula yields an error that tends to zero in both the small and large amplitude limits, a feature not found in any previous approximate formula. Some ways of refining this formula are also presented. Interestingly, when one of the improved expressions is taken for building a sinusoidal (harmonic) approximation to the solution of the pendulum equation of motion very good agreement is found. The simple log formulae derived here require only a few elementary function calls in a pocket calculator for accurate evaluations, being useful for analyzing pendulum experiments in introductory physics labs. They may also be of interest for those specialists working with nonlinear phenomena governed by pendulum-like differential equations, which arise in many fields of science and technology (e.g., analysis of acoustic vibrations, oscillations in small molecules, optically torqued nanorods, Josephson junctions, electronic filters, gravitational lensing in general relativity, advanced models in field theory, oscillations of buildings during earthquakes, and others).