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

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.

本文针对理想工况与小幅振幅振荡工况下的傅科摆（Foucault pendulum）运动，采用笛卡尔坐标系（Cartesian coordinates）展开综述研究。文中系统阐述了参考系的选取规则、地球非惯性系下牛顿微分方程的建立与求解过程，以及为简化求解流程所采用近似方法的合理性。本文利用运动轨迹尖点的角位置，提出了一种求解傅科摆进动角速度的全新方法。给出了随地球旋转的参考系与惯性参考系下的摆锤运动轨迹与速度，并证明了谢维列定理（Chevilliet theorem）。此外，本文还展示了当沿垂直于摆振荡平面的方向施加初始速度时的摆锤运动轨迹。