MODELING OF DYNAMIC PHENOMENA THROUGH DIFFERENTIAL CALCULUS: CONTRIBUTIONS TO THE TEACHING OF MATHEMATICS AND PHYSICS
DOI:
https://doi.org/10.63330/aurumpub.050-064Keywords:
Differential Calculus, Derivative, Wave Mechanics, Classical Physics, KinematicsAbstract
This article examines the foundations of differential calculus and its applications to central problems in Classical Physics, with emphasis on kinematics and wave mechanics. Beginning from the rigorous limit-based definition of the derivative, as formalised by Cauchy and Weierstrass in the nineteenth century, the work develops the operational rules of differentiation, discusses the geometric and physical interpretation of the derivative, and analytically demonstrates the equations of uniformly accelerated motion. Subsequently, the differential operator is applied to the sinusoidal progressive wave equation, yielding explicit expressions for the instantaneous velocity and acceleration of a point in the vibrating medium. The mathematical treatment shows that the phase velocity of a transverse mechanical wave on a tensioned string is governed jointly by the elastic property (tension T) and the inertial property (linear mass density µ), producing the expression v = √(T/µ), which is confirmed by dimensional analysis. The methodology is analytical-deductive, employing the formal apparatus of differential calculus and Leibniz notation. Results demonstrate that differentiation constitutes an indispensable tool for the precise modelling of dynamic physical phenomena, transcending the limitations of elementary algebraic approaches.
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References
ÁVILA, G. S. S. Cálculo das Funções de Uma Variável. v. 1. 7. ed. Rio de Janeiro: LTC, 2003.
BRIDGMAN, P. W. Dimensional Analysis. New Haven: Yale University Press, 1922.
CRESWELL, J. W.; CRESWELL, J. D. Research Design: Qualitative, Quantitative, and Mixed Methods Approaches. 6. ed. Thousand Oaks: SAGE Publications, 2022.
FLEMMING, D. M.; GONÇALVES, M. B. Cálculo A: Funções, Limite, Derivação, Integração. 6. ed. rev. ampl. São Paulo: Pearson Prentice Hall, 2007.
GIL, A. C. Métodos e Técnicas de Pesquisa Social. 7. ed. São Paulo: Atlas, 2019.
HALLIDAY, D.; RESNICK, R.; WALKER, J. Fundamentals of Physics. 11. ed. New York: Wiley, 2021.
HALLIDAY, D.; RESNICK, R.; WALKER, J. Fundamentals of Physics. 12. ed. Hoboken: Wiley, 2022.
KREYSZIG, E. Advanced Engineering Mathematics. 10. ed. New York: Wiley, 2011.
LEITHOLD, L. O Cálculo com Geometria Analítica. v. 1. 3. ed. São Paulo: Harbra, 1994.
LIMA, E. L. Análise Real. v. 1: Funções de Uma Variável. 12. ed. Rio de Janeiro: IMPA, 2016. (Coleção Matemática Universitária).
MORIN, D. Introduction to Classical Mechanics: With Problems and Solutions. 2. ed. Cambridge: Cambridge University Press, 2022.
NUSSENZVEIG, H. M. Curso de Física Básica. v. 2: Fluidos, Oscilações e Ondas, Calor. 5. ed. São Paulo: Blucher, 2014.
PIETROCOLA, M. et al. Física em Contextos: Pessoal, Social e Histórico. v. 2. São Paulo: FTD, 2016.
REZENDE, F. As novas tecnologias na prática pedagógica sob a perspectiva construtivista. Ensaio - Pesquisa em Educação em Ciências, Belo Horizonte, v. 5, n. 1, p. 1–17, 2003. Disponível em: https://doi.org/10.1590/1983-21172003050102. Acesso em: 10 jan. 2025.
SOUZA, M. A.; PATARO, P. R. M. Física: Contexto e Aplicações. v. 2. 3. ed. São Paulo: Scipione, 2020.
STEWART, J. Calculus: Early Transcendentals. 8. ed. Boston: Cengage Learning, 2016.
STEWART, J.; CLEGG, D. K.; WATSON, S. H. Calculus: Early Transcendentals. 9. ed. Belmont: Cengage Learning, 2020.
THOMAS, G. B. et al. Cálculo. v. 1. 12. ed. São Paulo: Pearson, 2012.
TIPLER, P. A.; MOSCA, G. Física para Cientistas e Engenheiros. v. 1: Mecânica, Oscilações e Ondas, Termodinâmica. 6. ed. Rio de Janeiro: LTC, 2014.
THOMAS, G. B. et al. Thomas' Calculus: Early Transcendentals. 15. ed. London: Pearson, 2023.
THORNTON, S. T.; MARION, J. B. Classical Dynamics of Particles and Systems. 6. ed. Boston: Cengage Learning, 2021.
YOUNG, H. D.; FREEDMAN, R. A. Sears and Zemansky's University Physics with Modern Physics. 16. ed. Harlow: Pearson Education, 2023.
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