[lurquer]

I found it very difficult to understand WHY Foucaults pendulum behaves the way it does.

I could picture what was happening with the pendulum at the Norh Pole... earth rotates under the pivot point every 24 hours. And, I understand what happened st the Equator... no effect on the pendulum as nothing is rotating beneath it.

But, at points in between, it was hard to understand why the pendulum might only rotate 80% or so depending on how far north it was. (Easy to derive mathematically, but hard to truly intuit what was going on.)

As I doubt I'm the only one, allow me to provide the thought expirement that make it click for me; specifically, why the pendulum turns some fraction of a complete rotation while the Earth -- at any point -- makes a full circuit every 24 hours.

Put a pendulum in your car. If you drive straight, the pendulum won't be affected. If you turn, of course, the pendulum will turn a bit. Easy to visualize so far. Now, let's stop the Earths rotation for a moment. Let's get in the car and with the Pendulum and drive due East from L.A. All the way to N. Carolina (or wherever due east would end up.) Across the Atlantic. Across Southern Europe, Asia, Pacific, and back to L.A.

Now I've been driving straight this entire time, so would the pendulum have moved?

Well, as it runs out, I HAVEN'T been driving 'straight'. The entire time, I would necessarily had to have been veering a little bit to the left to keep me in my due East path. If I was truly driving straight the entire time, I would have made a Great Circle and dipped down into Africa at some point.

In any case, as my desktop pendulum moves around the globe every 24 hours, it isn't traveling in a straight line... just as a car transporting it along its path would have to be curving a bit to stay on course.

This was the 'ah-ha' moment that allowed to understand the gradual increase in the pendulums rotation as you move north (or south) from the Great Circle of the equator.

Maybe that is common-sense for everyone else. If so, disregard. ;)

[gregfjohnson]

Here's the intuition that clicked for me, similar to the above but with a different twist at the end.

Imagine a large flat parking lot, and drive around the parking lot in a circle. Say you have a small pendulum swinging freely, suspended in your car. If you drive the full 360 degrees, the pendulum will appear from the reference frame of the inside of the car to have rotated a full 360 degrees. (Let's say that you are driving fast enough to render the rotation of the earth, the latitude of the parking lot, etc. negligible.)

Whatever fraction of the circle you drive, the pendulum will appear to rotate that same fraction as viewed from inside the car.

Now, here is the fun part. Imagine that you are driving at some fixed latitude on a large sphere. Create a big cone that is just tangent to the sphere on your path. What happens if you drive the entire distance and return to your starting point?

Cut the cone in a vertical straight line starting at its apex, and then flatten out the cone. It will form a flat circle, but with a pie slice removed.

If you had driven the perimeter of this partial circle in the parking lot, you would see that the pendulum only rotated part-way around. The same thing will happen if you drive the full distance around the sphere at the corresponding latitude.

[lurquer]

I had heard it explained explained that way as well. Made sense then and now.

Not to digress, but IF the earth was stationary and everything rotated around it, can you think of a mechanism to explain the pendulums behavior? (Could be ether, or some medieval conception of ptolemic shells, you name it... but it has to be consistent and sort of reasonable sounding.)

[kortex]

This would make for a great 3blue1brown visualization.

[leni536]

This is a very good intuitive explanation.

For a quantitative explanation one can make use of the fact that angular velocities or infinitesimal rotations compose like vectors. Earth has an angular velocity that is parallel to its radius to the North pole. When you look at the movent of the surface of the Earth around an arbitrary point, then you can decompose this angular momentum to parallel-to-radius and parallel-to-surface components. The former describes how fast Earth's surface "twists", the latter describes how fast it tilts under your feet. The former is the relevant component for describing the pendulum.

Interestingly this angular-momentum approach has a deeper connection to your car analogy. It can be used to solve the following problem: The Little Prince buys a little car that can turn around on radius r on Earths surface. He brings this car to his home planet, which has radius R. On what radius can this same car turn around on this planet?

[fatnoah]

Thank you! I live in Boston and am a frequent visitor to the Museum of Science, which has a Foucault pendulum. I never truly grasped the explanation until reading your post.

[lurquer]

Thanks. I found that explanation a long time ago in some remote corner of the internet... some Japanese website about Transformers and Decipticons as I recall.

Ages ago -- when the Internet was young -- I tried to put the explanation in the Foucaults article on Wikipedia... it lasted about 30 minutes before getting deleted and replaced with the usual trig stuff.

[fatnoah]

> lasted about 30 minutes before getting deleted and replaced

My one and only attempt to edit Wikipedia was to redact a statement on the article about the Semantic Web that had recently been changed to note that someone named Dr. Foch was attempting to weaponize the semantic web in support of a Chinese takeover of the world. My edit was immediately removed for not having a valid reference...nevermind that the original note didn't either.

[Ididntdothis]

That's very good. There are a lot of things that sound simple but when you take a closer look suddenly a lot of questions come up. Examples would be this pendulum, aircraft wings or gravity assists of interplanetary probes.