The Foucault pendulum and the additivity of infinitesimal rotations

Abstract

The purpose of this work is to present the Foucault pendulum precession without making recourse to the Coriolis acceleration or to geometrical methods. The exposition makes use of the fact that, contrary to finite rotations, infinitesimal rotations about different axes in three dimensions are both additive and commutative and that these important properties are readily extended to their time-derivatives, i.e. angular velocities. The angular velocity of the Earth about the polar axis is then decomposed into a pair of orthogonal angular velocity vectors having a common start point at the center of the Earth; the first vector is parallel to the axis that, in the local horizontal plane, points toward the north, while the second vector is directed upward, pointing toward the site where the pendulum is positioned. For the validity of the procedure it is essential to show that an infinitesimal rotation about the horizontal axis can be shown to have a negligible effect on the orientation of the plane of oscillation of the pendulum. At this point, only an infinitesimal rotation about the vertical axis is enough for calculating the change of orientation of the pendulum. Finally, owing to the constancy of the Earth’s spin, the obtained result can be readily extended to any finite rotation of the Earth, for example through the angle 2π that corresponds to a whole sidereal day.