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Very Large Angular Oscillations (Up to 3π/4) of the Physical Pendulum—A Simple Trigonometric Analytical Solution

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  • Joao C. Fernandes

    (Physics Department, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal)

Abstract

The oscillatory properties of pendular motion, along with the associated energetic conditions, are used to induce analytical functions capable of simultaneously describing the angular position and velocity. To describe the angular position of a generic pendulum, for very large amplitudes of oscillation, we used the numerical solutions obtained from the numerical resolution of the differential equation of motion. The solver software needed was built using the LabView 2019 platform, but any other ODE solver containing peak and valley detectors can be used. The fitting software and plots were performed with the ORIGIN 7.0 program, but also other equivalent programs can be used. For a non-damped pendulum, an analytical model is proposed, built from simple trigonometric functions, but containing the important physical information of the dependence between the period and amplitude of oscillation. The application of the proposed model, using the numerical solutions of the non-approximated differential equation of motion, shows very good agreement, less than 0.01%, for large amplitudes, up to 3π/4.

Suggested Citation

  • Joao C. Fernandes, 2025. "Very Large Angular Oscillations (Up to 3π/4) of the Physical Pendulum—A Simple Trigonometric Analytical Solution," Mathematics, MDPI, vol. 13(17), pages 1-21, September.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:17:p:2836-:d:1740970
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