Exact solution for the nonlinear pendulum

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Title: Exact solution for the nonlinear pendulum
Authors: Beléndez, Augusto | Pascual, Carolina | Méndez Alcaraz, David Israel | Beléndez, Tarsicio | Neipp, Cristian
Research Group/s: Holografía y Procesado Óptico | GITE - Física, Óptica y Telecomunicaciones
Center, Department or Service: Universidad de Alicante. Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal
Keywords: Simple pendulum | Large-angle period | Angular displacement
Knowledge Area: Física Aplicada
Date Created: Jul-2007
Issue Date: Dec-2007
Publisher: Sociedade Brasileira de Física
Citation: BELÉNDEZ VÁZQUEZ, Augusto, et al. "Exact solution for the nonlinear pendulum". Revista Brasileira de Ensino de Física. Vol. 29, No. 4 (Dez. 2007). ISSN 1806-1117, pp. 645-648
Abstract: This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is different from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75π (135◦) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ω in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135◦ the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics.
Sponsor: This work was supported by the “Ministerio de Educación y Ciencia”, Spain, under project FIS2005-05881-C02-02, and by the “Generalitat Valenciana”, Spain, under project ACOMP/2007/020.
URI: http://hdl.handle.net/10045/9174
ISSN: 1806-1117 (Print) | 1806-9126 (Online)
DOI: 10.1590/S1806-11172007000400024
Language: eng
Type: info:eu-repo/semantics/article
Peer Review: si
Publisher version: http://dx.doi.org/10.1590/S1806-11172007000400024
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