Differential equations applied to the pendulum with time-dependent mass: study of mass with exponential and polynomial variation
DOI:
https://doi.org/10.35819/remat2021v7i1id4164Keywords:
Damping, Bessel Equation, Substitution of VariablesAbstract
Differential equations are one of the contents that are applied in several areas. In Physics, one of the applications is the simple pendulum that has oscillation independent of the mass, when it is constant. However, when the mass is no-constant, the variation of linear momentum must be rewritten. In this work, two types of variable mass are proposed, as an exponential function and in terms of the time variable powers. In cases of gain of mass in the exponential variation, there is damping that is shown by the graphs of their solutions. When the mass is written in terms of powers, after substitution of variables, the problem is modeled by the Bessel Equation which has a dependent order of the power used in the mass function. At the end, the participation of the mass in the damping was verified and the analyzed problems are shown as applications that enrich the differential equations study field.
Downloads
References
AGUIAR, V.; GUEDES, I. Osciladores harmônicos amortecidos dependentes do tempo. Revista Brasileira de Ensino de Física, São Paulo, v. 35, n. 4, p. 1-9, out./dez. 2013. DOI: http://dx.doi.org/10.1590/S1806-11172013000400011.
BOYCE, William; DIPRIMA, Richard. Equações Diferenciais Elementares e Problemas de Valores de Contorno. 8. ed. Rio de Janeiro: LTC, 2006.
Caccamo, M. T.; Magazù, S. Variable mass pendulum behaviour processed by wavelet analysis. European Journal of Physics. v. 38, n. 1, p. 1-9, 30 nov. 2016. DOI: https://doi.org/10.1088/0143-0807/38/1/015804.
ÇENGEL, Yunus; PALM III, William. Equações Diferenciais. 1. ed. Nova Iorque: MCGRAW HILL, 2014.
CORREA, Eberth; ORTIZ, J. S. Espinoza; VALÉRIO, Mauro; DUTRA, Jomhara. Oscilador harmônico com massa variável e a segunda lei de Newton. Revista Brasileira de Ensino de Física, São Paulo, v. 33, n. 4, p. 1-9, out./dez. 2011. DOI: https://doi.org/10.1590/S1806-11172011000400007.
LENSKI, Richard. Dynamics of interactions between bacteria and virulent bacteriophage. In: Advances in microbial ecology. Boston: Springer, 1988.
LINDE, Andrei. Inflationary cosmology. Physics Reports, v. 333, p. 575-591, ago. 2000. DOI: https://doi.org/10.1007/978-3-540-74353-8_1.
NASCIMENTO, J. P. G.; GUEDES, I. Osciladores clássicos com massa dependente da posição. Revista Brasileira de Ensino de Física, São Paulo, v. 36, n. 4, p. 1-6, out./dez. 2014. DOI: http://dx.doi.org/10.1590/S1806-11172014000400009.
ROMEO, F. A simple model of energy expenditure in human locomotion. Revista Brasileira de Ensino de Física, São Paulo, v. 31, n. 4, p. 1-5, dez. 2009. DOI: https://doi.org/10.1590/S1806-11172009000400008.
SALAMANGA, Marcin. An Application of the Harmonic Oscillator Model to Verify Dunning’s Theory of the Economic Growth. Statistika: Statistics and Economy Journal, Prague, v. 93, n. 3, p. 56-69, set. 2013.
SILVA, Adilson Costa da; HELAYËL NETO, José Abdalla. Simulador de Oscilações Mecânicas. Revista Brasileira de Ensino de Física, São Paulo, v. 38, n. 3, p. 1-14, 7 jun. 2016. DOI: https://doi.org/10.1590/1806-9126-RBEF-2016-0042.
THORNTON, Stephen T.; MARION, Jerry N. Dinâmica Clássica de sistemas e partículas. Tradução da 5ª edição norte-americana. São Paulo: Cengage Learning, 2014.
ZILL, Dennis. Equações Diferenciais com aplicações em modelagem. 3. ed. São Paulo: Cengage Learning, 2016.
Downloads
Published
Issue
Section
License
Copyright (c) 2021 REMAT: Revista Eletrônica da Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
REMAT retains the copyright of published articles, having the right to first publication of the work, mention of first publication in the journal in other published media and distribution of parts or of the work as a whole in order to promote the magazine.
This is an open access journal, which means that all content is available free of charge, at no cost to the user or his institution. Users are permitted to read, download, copy, distribute, print, search or link the full texts of the articles, or use them for any other legal purpose, without requesting prior permission from the magazine or the author. This statement is in accordance with the BOAI definition of open access.
How to Cite
https://orcid.org/0000-0002-0893-7426
