O caos numa máquina de catástrofe

Abstract

This work has objective to investigate and analyze the points 𝐵𝑛 that corresponds to the collapse points produced in the sudden change of position of a point on the disk of a catastrophe machine, through the variation of the horizontal trajectory of the pencil on paper. To this, a theoretical foundation was sought about chaos and the catastrophe machine, the circular movement, considering the rotational movement and the movement in space. In this sense, we sought to understand the vector functions and solve line integrals, considering the elastic force and torque acting on the catastrophe machine. The methodology adopts assumptions of a bibliographic and experimental research, initially developed by analyzing the data collected in the machine and, later, studied 4 (four) cases involving the line integral. In the first case, the application of Newton's second law was prioritized, using the acceleration obtained through the experiment. In the 2nd case, a vector product module (torque) was applied between 𝑭⃗ (𝑡) = 𝑚 ∙ 𝑎 (𝑡) and 𝒖⃗ (𝑡) = Δ𝒓⃗ in the line integral. In relation to the 3rd case, we considered the product integral of the resultant force modulus in Newton's second law, taking the function 𝑥(𝑦(𝑡)), in the same interval of 𝜃 of the disk. And, in the 4th case, torque was again applied to the line integral, however, using the vector product module between 𝑭⃗ (𝑡) = 𝑚 ∙ 𝑭⃗ 𝑟𝑒𝑠(𝑡) and 𝒖⃗ (𝑡) = Δ𝒓⃗ . Data collection took place through notes of the researcher's observations during the experiment and the points were plotted in the GeoGebra software. As for the results, it was verified that the torque determines an accentuated catastrophe and that the material point will always have a single equilibrium point, in addition to a stable and unstable equilibrium point in the 1st and 4th quadrants. It was verified that the restoring strength of the elastics 𝑒1 and 𝑒2 contributed to the existence of the collapse point. It was concluded that 𝑠 (𝜃𝜔) = 10 ⋅ 𝜃 ⋅𝑠𝑒𝑛 (𝜃𝜔) is the equation that best corresponds to the data collected in the catastrophe machine. Also, the importance of applying this experiment in high school was found to work on concepts related to vectors, modeling equations on the GeoGebra software, working in an interdisciplinary way. Regarding graduation, one can understand the resolution and applicability of the line integral, the relationships between analytical geometry and vector calculus, being both motivating and challenging.