A Dinâmica clássica de uma partícula massiva em circuitos fechados nos formatos circular e elíptico sujeita à força de atrito

Abstract

This work investigates the dynamics of a massive particle moving along closed circuits (Loops) in circular and elliptical geometries, considering both the conservative regime and the dissipative regime that models the presence of kinetic friction forces. The central objective is to understand how the geometry of the track and dissipative forces influence the conditions required for the particle to complete the trajectory without losing contact with the surface. In addition, we aim to explore the didactic potential of this physical system in the teaching of Mechanics, through the development of a didactic sequence at the undergraduate level, grounded in the educational framework proposed by Jerome Bruner. The methodology is based on the systematic application of Newton’s laws and generalized energy balance equations. In fact, we show that this approach leads naturally to the formulation of Volterra integral equations of the second kind for the particle’s velocity along the track. In the case of a circular circuit with a constant coefficient of friction, the formalism allows us to obtain closed-form analytical solutions for the velocity as a function of the angle ϕ. This, in turn, enables the explicit determination of the minimum initial velocity conditions, as well as the domains of validity associated with the maintenance of contact and the continuity of the particle’s motion along the track. For elliptical circuits, in both horizontal and vertical orientations, we adopt a parametrization that preserves the total length of the trajectory. In this case, we employ Ramanujan’s approximation for the perimeter of the ellipse, thereby ensuring a physically consistent comparison among the different geometries under consideration. Within this context, the resulting integral equation does not admit a simple analytical solution, requiring a numerical approach to analyze the dynamical behavior of the particle. In summary, the problem of a particle’s dynamics in closed circuits, when extended to non-circular geometries and the presence of friction, reveals a rich physical and mathematical structure. It provides a class of physical systems in which fundamental dynamical concepts—such as energy, velocity, and dissipative forces—can be studied while simultaneously employing more advanced tools, such as integral equations and elements of differential geometry. Moreover, it constitutes an effective pedagogical instrument for the progressive teaching of Mechanics at introductory and intermediate undergraduate levels.