Abstract
The elastic pendulum is a simple physical system represented by nonlinear differential equations. Analytical solutions for the bob trajectories on the rotating earth may be obtained in two limiting cases: for the ideally elastic pendulum with zero unstressed string length and for the Foucault pendulum with an inextensible string. The precession period of the oscillation plane, as seen by the local observer on the rotating earth, is 24 hours in the first case and has a well-known latitude dependence in the second case. In the present work, we have obtained numerical solutions of the nonlinear equations for different string elasticities in order to study the transition from one precession period to the other. It is found that the transition is abrupt and that it occurs for a quite small perturbation of the ideally elastic pendulum, that is, for the unstressed string length equal to about 10−4 of the equilibrium length due to the weight of the bob.
Highlights
The pendulum is such a fundamental physical system that all the details and various aspects of its motion are of interest
This coupling has been mainly investigated in the vicinity of the so-called autoparametric resonance, where the string elasticity is such that the frequency of vertical, that is, spring-mode oscillations is double the frequency of horizontal, that is, pendulum-mode oscillations 1, 2
As the string elasticity parameter l0 is reduced towards zero, the precession time tends towards 24 hours
Summary
The pendulum is such a fundamental physical system that all the details and various aspects of its motion are of interest. In the limit of inextensible suspension strings and small amplitudes, the behaviour of the system is described by simple, linear equations. A pendulum with an elastic suspension string is described by nonlinear differential equations, which couple horizontal and vertical oscillations 1. This coupling has been mainly investigated in the vicinity of the so-called autoparametric resonance, where the string elasticity is such that the frequency of vertical, that is, spring-mode oscillations is double the frequency of horizontal, that is, pendulum-mode oscillations 1, 2. It is perhaps surprising that no published work about the influence of earth’s rotation on the behaviour of the nonlinear elastic pendulum could be found in the available literature. In the limit of an inextensible suspension string, one obtains the well-known and popular Foucault pendulum, for which the oscillation plane rotates, that is, precesses at a constant
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