Exact solution for the nonlinear pendulum
Autor(a) principal: | |
---|---|
Data de Publicação: | 2007 |
Outros Autores: | , , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Revista Brasileira de Ensino de Física (Online) |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172007000400024 |
Resumo: | 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 0.75pi (135º) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency w in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135º 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. |
id |
SBF-1_19a9d988860bf415a80578e2f4bf50a2 |
---|---|
oai_identifier_str |
oai:scielo:S1806-11172007000400024 |
network_acronym_str |
SBF-1 |
network_name_str |
Revista Brasileira de Ensino de Física (Online) |
repository_id_str |
|
spelling |
Exact solution for the nonlinear pendulumsimple pendulumlarge-angle periodangular displacementThis 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 0.75pi (135º) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency w in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135º 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.Sociedade Brasileira de Física2007-01-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172007000400024Revista Brasileira de Ensino de Física v.29 n.4 2007reponame:Revista Brasileira de Ensino de Física (Online)instname:Sociedade Brasileira de Física (SBF)instacron:SBF10.1590/S1806-11172007000400024info:eu-repo/semantics/openAccessBeléndez,A.Pascual,C.Méndez,D.I.Beléndez,T.Neipp,C.eng2008-03-18T00:00:00Zoai:scielo:S1806-11172007000400024Revistahttp://www.sbfisica.org.br/rbef/https://old.scielo.br/oai/scielo-oai.php||marcio@sbfisica.org.br1806-91261806-1117opendoar:2008-03-18T00:00Revista Brasileira de Ensino de Física (Online) - Sociedade Brasileira de Física (SBF)false |
dc.title.none.fl_str_mv |
Exact solution for the nonlinear pendulum |
title |
Exact solution for the nonlinear pendulum |
spellingShingle |
Exact solution for the nonlinear pendulum Beléndez,A. simple pendulum large-angle period angular displacement |
title_short |
Exact solution for the nonlinear pendulum |
title_full |
Exact solution for the nonlinear pendulum |
title_fullStr |
Exact solution for the nonlinear pendulum |
title_full_unstemmed |
Exact solution for the nonlinear pendulum |
title_sort |
Exact solution for the nonlinear pendulum |
author |
Beléndez,A. |
author_facet |
Beléndez,A. Pascual,C. Méndez,D.I. Beléndez,T. Neipp,C. |
author_role |
author |
author2 |
Pascual,C. Méndez,D.I. Beléndez,T. Neipp,C. |
author2_role |
author author author author |
dc.contributor.author.fl_str_mv |
Beléndez,A. Pascual,C. Méndez,D.I. Beléndez,T. Neipp,C. |
dc.subject.por.fl_str_mv |
simple pendulum large-angle period angular displacement |
topic |
simple pendulum large-angle period angular displacement |
description |
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 0.75pi (135º) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency w in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135º 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. |
publishDate |
2007 |
dc.date.none.fl_str_mv |
2007-01-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172007000400024 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172007000400024 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/S1806-11172007000400024 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
dc.source.none.fl_str_mv |
Revista Brasileira de Ensino de Física v.29 n.4 2007 reponame:Revista Brasileira de Ensino de Física (Online) instname:Sociedade Brasileira de Física (SBF) instacron:SBF |
instname_str |
Sociedade Brasileira de Física (SBF) |
instacron_str |
SBF |
institution |
SBF |
reponame_str |
Revista Brasileira de Ensino de Física (Online) |
collection |
Revista Brasileira de Ensino de Física (Online) |
repository.name.fl_str_mv |
Revista Brasileira de Ensino de Física (Online) - Sociedade Brasileira de Física (SBF) |
repository.mail.fl_str_mv |
||marcio@sbfisica.org.br |
_version_ |
1752122420310835200 |