Foucault Pendulum without Driving

Abstract

Foucault pendulums are two-dimensional harmonic oscillators to which the Coriolis force is applied. The Foucault parameter ΩF is the ideal rate of rotation of the plane of oscillation due to the Coriolis force. Kamerlingh Onnes pointed out that the actual behavior of Foucault pendulums can be understood only by taking into account the unavoidable mechanical asymmetry, i.e., the difference between the frequency in the X-direction and that in the Y-direction, which is called the asymmetry parameter δ. Our equations of motion for Foucault pendulums are linear and their solutions are easily obtained. We are interested in slowly varying rotations of the plane of oscillation, where the angle of rotation is denoted by φ. The motion of the bob consists of rapid simple oscillations and the slow rotations mentioned above. Eliminating the rapid oscillations, we obtain an analytic expression for tan 2φ, which is a periodic function with period \(\pi /\sqrt{\Omega _{\text{F}}^{2} + \delta ^{2}} \). The graphs of the rotation φ versus time t have two distinct appearances. Under the condition ΩF ≥ δ|cos(2θ)| (θ is the initial value of φ), φ decreases monotonically, which is the behavior expected for Foucault pendulums. Otherwise φ repeatedly increases and decreases, which is the behavior of pendulums describing Lissajous figures. The vertical component of angular momentum is also calculated analytically.