Differential equations applied to the pendulum with time-dependent mass: study of mass with exponential and polynomial variation

Authors

  • Otávio Paulino Lavor Universidade Federal Rural do Semi-árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil http://orcid.org/0000-0001-5237-3392
  • Antônio Nunes de Oliveira Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE), Campus Cedro, Cedro, CE, Brasil https://orcid.org/0000-0001-5697-8110

DOI:

https://doi.org/10.35819/remat2021v7i1id4164

Keywords:

Damping, Bessel Equation, Substitution of Variables

Abstract

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.

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Author Biographies

Otávio Paulino Lavor, Universidade Federal Rural do Semi-árido (UFERSA), Departamento de Ciências Exatas e Naturais (DECEN), Pau dos Ferros, RN, Brasil

http://lattes.cnpq.br/1857806253382088

Antônio Nunes de Oliveira, Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE), Campus Cedro, Cedro, CE, Brasil

http://lattes.cnpq.br/0413684696036057

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.

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Published

2021-01-05

How to Cite

LAVOR, O. P.; OLIVEIRA, A. N. de. Differential equations applied to the pendulum with time-dependent mass: study of mass with exponential and polynomial variation. REMAT: Revista Eletrônica da Matemática, Bento Gonçalves, RS, v. 7, n. 1, p. e3001, 2021. DOI: 10.35819/remat2021v7i1id4164. Disponível em: https://periodicos.ifrs.edu.br/index.php/REMAT/article/view/4164. Acesso em: 3 jul. 2024.

Issue

Vol. 7 No. 1 (2021)

Section

Mathematics

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