A Short Vector Solution of the Foucault Pendulum Problem

I A Ciureanu,D Condurache

doi:10.4236/wjm.2015.52002

Abstract

The paper studies the motion of the Foucault Pendulum in a rotating non-inertial reference frame and provides a closed form vector solution determined by vector and matrix calculus. The solution is determined through vector and matrix calculus in both cases, for both forms of the law of motion (for the Foucault Pendulum Problem and its “Reduced Form”). A complex vector which transforms the motion equation in a first order differential equation with constant coefficients is used. Also, a novel kinematic interpretation of the Foucault Pendulum motion is given.

Highlights

Swinging with elegance across the meridian of Paris inside the grand hall of the observatory, the pendulum built by Bernard Léon Foucault (1819-1868) proved the rotation of the Earth for the first time by terrestrial methods
The present paper presents a closed form vector solution which exploits the benefits of the dualism of vector calculus and matrix calculus with extension to tensors
The relative derivation with respect to angular velocity ω, and like operator for symbolic representation the trivial derivation of vector functions of real variable with respect to the variable. This transformation “gives an algebraic form” to a class of vector differential equations that model the motion of mechanical systems in non-inertial frames, whom are in the motion of non-uniform rotation, on fixed direction, the motion with respect to the inertial frames in the fields of gyroscopic forces

Summary

Introduction

Swinging with elegance across the meridian of Paris inside the grand hall of the observatory, the pendulum built by Bernard Léon Foucault (1819-1868) proved the rotation of the Earth for the first time by terrestrial methods. In this case, the function f from Equation (1.1) has the particular expression of a constant real number and r is the position vector, ω is the angular velocity of the reference frame (an arbitrary differential vector map) and ω0 is the pulsation of the pendulum which depends on its length and the gravitational acceleration at the experiment place. The present paper presents a closed form vector solution which exploits the benefits of the dualism of vector calculus and matrix calculus with extension to tensors It is structured in five sections described below. The solution to the Cauchy problem (1.3) is given only for the planar case, using polar coordinates [6] or Cartesian coordinates [7]

Mathematical Preliminaries

Short Solution of Foucault Pendulum Problem

The Symbolic Tensor Representation of a Vector Functions

The Solution of the Reduced Form Problem

Conclusion