Approximate expressions for the period of a simple pendulum using a Taylor series expansion

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Title: Approximate expressions for the period of a simple pendulum using a Taylor series expansion
Authors: Beléndez, Augusto | Arribas Garde, Enrique | Márquez, Andrés | Ortuño, Manuel | Gallego, Sergi
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 | Universidad de Alicante. Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías | Universidad de Castilla-La Mancha. Departamento de Física Aplicada
Keywords: Nonlinear pendulum | Period | Approximate formula | Taylor series expansion
Knowledge Area: Física Aplicada
Date Created: 22-Mar-2011
Issue Date: 27-Jul-2011
Publisher: Institute of Physics Publishing
Citation: BELÉNDEZ VÁZQUEZ, Augusto, et al. "Approximate expressions for the period of a simple pendulum using a Taylor series expansion". European Journal of Physics. Vol. 32, No. 5 (2011). ISSN 0143-0807, pp. 1303-1310
Abstract: An approximate scheme for obtaining the period of a simple pendulum for large-amplitude oscillations is analysed and discussed. When students express the exact frequency or the period of a simple pendulum as a function of the oscillation amplitude, and they are told to expand this function in a Taylor series, they always do so using the oscillation amplitude as the variable, without considering that if they change the variable (in this paper to the new variable m), a different Taylor series expansion may be performed which is in addition more accurate than previously published ones. Students tend to believe that there is one and only one way of performing a Taylor series expansion of a specific function. The approximate analytical formula for the period is obtained by means of a Taylor expansion of the exact frequency taking into account the Kidd–Fogg formula for the period. This approach based on the Taylor expansion of the frequency about a suitable value converges quickly even for large amplitudes. We believe that this method may be very useful for teaching undergraduate courses on classical mechanics and helping students understand nonlinear oscillations of a simple pendulum.
Sponsor: This work was supported by the ‘Vicerrectorado de Tecnología e Innovación Educativa’ of the University of Alicante, Spain (GITE-09006-UA), and by the Generalitat Valenciana, Spain (project PROMETEO/2011/021).
ISSN: 0143-0807 (Print) | 1361-6404 (Online)
DOI: 10.1088/0143-0807/32/5/018
Language: eng
Type: info:eu-repo/semantics/article
Peer Review: si
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