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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.

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. However, a pendulum with an elastic suspension string is described by nonlinear differential equations, which couple horizontal and vertical oscillations [

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 rate [

Instead of directly solving the equations of motion in the noninertial laboratory system by including the centrifugal and the Coriolis inertial forces, we have calculated the pendulum trajectories using a slightly different procedure. Such a procedure was followed also in deriving analytical solutions for the ideal elastic pendulum [

By neglecting the tidal gravitational forces of external bodies, such as the sun and moon, we have a quasi-inertial system

Views along the

Neglecting dissipative forces, we have Newton's law for the motion of the pendulum bob

We further assume that

The force of the string is proportional to its dilatation and points towards the suspension point, that is, towards the origin of

Let us observe that the above nonlinear differential equation (

From the differential equation (

The motion of the bob has been studied for the usual initial conditions with which a Foucault pendulum is started in the system

The oscillation plane is defined with the

For various values of the string elasticity, we show in Figure

The dependence of the azimuth angle of the oscillation plane on time for latitude 55°. Different curves represent different string elasticities (see text).

In the limit of a very stiff string, that is, for

In the limit of an ideally elastic string (

In order to learn more about the transition from the 24-hour precession time to the precession time of the Foucault pendulum, we plot in Figure

The dependence of pendulum precession time on the suspension string elasticity, that is, on the length

Finally, we should point out that in addition to the discontinuity in precession time at

We have performed calculations of the precession of the elastic pendulum and have found a most interesting behaviour of the precession time as a function of suspension string elasticity. In the limit of an ideally elastic string with zero unstressed string length (

Surely, the reader is wondering why should there be a discontinuity of the pendulum precession time and why should it appear at that particular value of string elasticity. The authors do not have answers to these questions, but they do believe that answers will be found by further investigation of the long-term behaviour of the elastic pendulum. Maybe, this simple system can teach us something about more complex systems.