All the sources I see for the equations of motion for the Foucault pendulum start with the small angle approximation. Does anyone know a source or textbook that does the full derivation? Specifically, I want to determine the way that increases of the amplitude of the pendulum's oscillation affect the rate of precession. I think the Foucault experiment is designed to be carried out with very small vertical oscillations and I want to compute the effect on the rate of precession as one increases the vertical component of the oscillation by increasing the angle at release.
1 Answer
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Years ago I created a Java simulation titled 'Foucault rod'. The simulation allows a large amplitude of swing. What the simulation displays suggests that the Foucault effect is very robust.
The simulation is for the case of a flexible rod with a weight at the end, the rod points from outside towards the center of rotation.
(I haven't tested wether that simulation, created in 2009, is still played by the latest version of the Java VM.)
Since the simulation is numerical anyway the calculations are done for the motion with respect to the inertial coordinate system. The calculated motion is subsequently transformed to a rotating coordinate system.
The simulation displays the resulting motion in two side-by-side panels, left showing the motion with respect to the inertial coordinate system, right showing the motion with respect to the coordinate system that is co-rotating with the base of the rod.
Among the initial conditions that can be altered by the user is the amplitude of swing. The default value is that the rod is released at an angle of 5 degrees with respect to the midline of the vibration. The simulation accepts any value for that angle.
As I wrote at the beginning of this post, it appears the Foucault effect is remarkably robust. It appears that the effects of larger amplitude tend to average out. (The simulation doesn't provide the means to quantify this.)
The reasons for doing the calculations for the motion with respect to the inertial coordinate system:
Since the simulation is numerical anyway the cost of transforming the resulting motion to a co-rotating coordinate system is negligable.
Using the equation of motion for motion with respect to a rotating coordinate system comes with hidden assumptions. For instance the assumption that for a Foucault pendulum setup the Eötvös effect is negligably small.
(Eötvös effect: when the pendulum is on its swing from west to east the bob is circumnavigating the Earth faster than the Earth's rotation. So during the swing from west to east the bob has a smaller effective weight, and conversely a larger effective weight during swing from east to west. For north-south swing the effective weight is the same in both directions.)
When doing the calculation for motion with respect to the inertial coordinate system the calculation is automatically exhaustive.