Prevent Foucault's Pendulum from an Ellipticity

Question

I am using a 15kg dumbbell as the bob of a Foucault's pendulum attached to a string of 4 meters. However, using the common method of burning a wire to set the bob in motion, it will begin to oscillate in an elliptical path. I attribute this to the rotation of the dumbbell itself. No matter how long I wait before and after tying it to a stationary pole at the greatest displacement to set the dumbbell in equilibrium (no motion at all), the bob would start to swing in ellipticity right away, disrupting all data. Any suggestions to improve this experiment so I can determine the period of Earth's rotation using ${\displaystyle \omega =360^{\circ }\sin \varphi \ /\mathrm {day} ,}$ where $\omega$ is the angular speed of the bob and $\varphi$ is the attitude (this translate to about 10 degrees per hour)?

Answer 1

The classic method is to add a ring somewhere near the top, through which you pass the wire or string that suspends your dumbbell. Make sure the amplitude is so large that with every swing, the wire clips on the ring. That way, every swing adds a little force that keeps the elliptical motion from building up. @Cleonis pointed out that this ring is called a "Charron Ring". The original publication is here but perhaps more of historical interest. @Farcher provided a description (pdf) of a Focault pendulum in the UN building, figure 3 shows the Charron ring. Here is another example from the University in Magdeburg:

BTW you would also get this elliptic motion from a spherical bob. It's a possible degree of freedom of a real pendulum's motion. Any torque that isn't perfectly symmetric in azimuth will pump energy into that degree of freedom. That can be a twist in the wire, finite size of the bob, local disturbances in the gravitational field, etc. Thus, every Foucault pendulum I have ever seen has such a ring to prevent this precession.

Answer 2

About the cause of the plane of swing opening up.

(I'm not sure what you mean by 'rotation of the dumbbell itself', I will return to that at the end of this answer.)

The upper attachment point of the suspension cable is critical. For every orientation of the plane of swing the effective length of the suspension wire must be the same.

A worst case scenario would be where the pendulum wire runs through a slit (for example a slit between two ceiling boards) and secured at the top of the slit.

In that worst case scenario: when the plane of swing is parallel to that slit the effective length of the pendulum wire is longest, and when the plane of swing is perpendicular to the slit the effective length of the pendulum wire is shortest.

Another factor for the effective length is whether the wire is equally flexible in all directions. For a Foucault pendulum setup the wire must bend equally easily for all directions of swing.

The effective length of the pendulum wire affects the period of swing.

The jargon name for any effect that causes the period of the swing to be not the same in all orientations is anisotropy (an-isotropy; not isotropic)

Let's say there is an anisotropy. The simplest instance of a period bias is where there is a single bias as you go through a full circle. Let's call the orientation of largest period of swing the 0 degrees plane. With a single bias: the orientiation with the shortest period of swing will be at 90 degrees to the orientation of largest period.

Let's say you happen to release the pendulum bob along a plane that is at a 45 degrees angle with respect to the 0 degrees plane. When released from that angle the ensuing swing is a superposition of two swings with unequal period of swing. This unequality then causes the plane of swing to open up.

Generally it takes a while for the plane of swing to open up. You describe: "the bob would start to swing in ellipticity right away" That is actually worse than what you would expect with an anisotropy of the pendulum wire.

With any Foucault pendulum setup the aim is to make the plane of swing opening up happen slower.

A Foucault pendulum setup is very sensitive. The rotation-of-Earth-effect is very, very small. The Foucault pendulum setup accumulates that very, very small effect, making the effect visible to the naked eye. The thing is: the Foucault pendulum setup accumulates everything. The effect of anisotropy of the pendulum wire accumulates too!

If as you describe the bob starts to swing in ellipticity right away then you have a major, major problem.

About rotation of the dumbbell.
It could be that you are referring to the fact that the dumbbell is co-rotating with the earth.

Let me tell the story of how the idea of the Foucault pendulum setup occurred to Léon Foucault.

Being an experimental physicist Foucault had a workshop with a lathe, and for some job he had clamped a rod in the jaws of the lathe. He twanged the rod, and then he was curious what would happen to the plane of swing upon changing the orientation of the chuck of the lathe. So he turned the chuck, and the plane of swing remained in the same orientation.

Foucault recognized that the momentum of the swing of the twanged rod is not bound to the physical orientation of the rod.

If occurred to Foucault that if a pendulum would be suspended over one of the poles then the fact that the pendulum bob is co-rotating with the Earth is not a factor; the plane of swing will remain in the same orientation anyway.

The first Foucault pendulum setup was in Foucault's basement workshop. The second one had a 21 meter wire, in the 'Musée des Arts et Metieres', the third one had a 67 meter wire, in the Pantheon in Paris.

The longer the wire, the less sensitive the Foucault pendulum setup is to imperfections in the wire.

(A discussion of why on latitudes away from the poles the rate of change of pendulum swing orientation is slower is discussed in an article on my own website. Link to my website is on my stackexchange profile.)