Foucault pendulum revisited, the determination of precession angular velocity using Cartesian coordinates

Abstract

A review on the Foucault pendulum motion is presented using Cartesian coordinates for the ideal case and for small amplitude of oscillations. The choice of referential frames, the formulation and solution of Newton's differential equation for the non-inertial frame of the Earth and the validity of the approximations used to simplify the determination of the solution are given. Using the angular position of the trajectory cusps, a new method to determine the precession angular velocity of the Foucault pendulum is shown. The pendulum bob trajectories and velocities for the referential frame in rotation with the Earth as well as for the inertial frame are given and the Chevilliet theorem was demonstrated. In addition, the pendulum bob trajectories are shown when an initial velocity is impinged in the direction perpendicular to the pendulum oscillation plane.

On the Foucault pendulum

“…In general, the modulus of the tensile force T = T (r, ṙ) is a nonlinear function of r and ṙ. Under the condition of a very large wire length compared to vertical displacements of the pendulum bob, the nonlinear terms of T (r, ṙ) can be neglected and the modulus of the tensile force can be approximated to T = mg [1,3,5], leaving Eq. ( 1) in a simpler form…”

“…where both have already been implicitly adopted to justify T = mg [1,3,5]. By expanding φ(t) = arctan(y(t)/x(t)) in Taylor series to first order in both ω and θ 0 = θ(0) and considering that the bob starts from rest, the following result is reached:…”

“…In the literature, ∆φ day is given only by −2π sin(λ), independent on the initial conditions [1][2][3][4][5][6]. The additional term for the angle variation in (11) shows that the initial displacement of the plumb is decisive for the rotation of the oscillation plane [2], this difference being maximum if the bob is abandoned in such a way that the line that joins the bob and the equilibrium point of the pendulum lies exactly on a meridian (φ 0 = ±π/2).…”

“…The French physicist Jean Bernard Léon Foucault installed, in 1851 in the city of Paris, a simple pendulum whose string length was very large compared to the height reached by its bob [1]. The suspension point of this pendulum was fixed relative to a reference system located on the Earth's surface.…”

Algumas Percepções De Estudantes Do Ensino Médio Sobre Ciências, Pseudociência E Movimentos Anticientíficos

“…Atualmente é possível ver fotos de nosso Planeta retiradas do Espaço exterior a ele (Burt & Smith, 2012); realizar experimentos utilizando óptica, acelerômetros ou pêndulos de Foucault para medir efeitos da rotação da Terra (Foucault, 1851;Furukawa & Sano, 1991;Marion & Thornton, 1995;Giacometti, 2021); ou mesmo podemos vislumbrar a curvatura da Terra acompanhando o pôr-do-Sol.…”