Yeah, I found a really good pdf article on the Foucault pendulum which went through the math, and then I saw how the Coriolis force was just given by twice the cross product between omega and the velocity of an object. And as you say, this means there is a Coriolis force when something moves east-west. It's just that I thought that there would be no force in this case because all I really knew of Coriolis was the demonstration with a canon ball getting shot due north from the equator, and how that resulted in a deflection. I imagined this wouldn't happen if something got shot east-west.

@FelisSuper About the cannon ball idea. Let's make that cannon ever larger. Set up a supercannon on, say, 45 degrees latitude. Fire to the west, at such velocity that the cannonball stays in flight for hours. During the flight the cannonball is effectively in a low orbit. Orbital motion is along a great circle. This shows that a westward travelling cannonball will proceed towards the Equator. That is: the cannonball idea predicts that the plane of swing turning will be different when swinging east-west. Observation: plane of swing turns uniformly.

@Cleonis Right, I did know that, but I thought that the amount of deflection would be too small as the bob on the pendulum moved just a tiny amount east-west. But as Matteo points out, the Coriolis force is indeed small (that's why we usually don't have to think about it), but during a long time span, that does lead to some deflection.

Just to note for those reading Cleonis' comment... his visual of a giant supercannon doesn't speak to Coriolis (indeed a westward shot in the NH would mean a deflection away from the equator (towards the north))

@JeopardyTempest Your comment was submitted 18 months ago, but I notice it only now. Unfortunately, your assertion violates the laws of motion. Let's review another example: satellite orbit. Let a satellite be orbiting in Low Earth Orbit, orbiting from east to west. Let the inclination be 30 degrees. The ground track of any satellite orbit is a great circle. Let the ground track of the satellite be tangent to the 30 degrees latitude line at $t=0$. One quarter of the orbital period later the ground track of the satellite crosses the equator.

@Cleonis: (Sorry, my 1st was just as late, 1 yr!) I certainly had to think this over (I'm not too heavily astronomy, only a meteorologist working on long-ago understanding)... and can always be wrong! But after a lot of pondering, I believe the changes in direction you are talking about due to the great circle path are centrifugal forces (they happen regardless of whether the Earth rotated). (If there were no gravitational force, the object would continue off to space rather than change direction centripetally/fugally). The Coriolis causes the shifts in each satellite pass.

At the "high" point (no idea the technical term!) on the orbit (whether + or - inclination (I don't know which is which!)), which would be the point where it's moving due west in this retrograde orbit... it will be hard to tell whether Coriolis shifted left or right (because it subsequently reshows up directly behind you)... but at a point "below" that "high point" in the NH... the next pass it winds up further north of you than the velocity suggested it would (or further south/left in the SH).

So certainly up to being wrong, and had a lot of head scratching, but I believe Coriolis due to the Earth's rotation still deflects right (in NH) from the Earth-based observer's perspective (where it's valid).

(If you had no gravity [and no friction, on equinox, without additional relative motions like Earth's orbit around the sun]... if you released an object from you hand, it would shoot initially west at the opposite of the local rotational speed because its linear momentum (it had from solid body rotation) would stop changing direction, and it would appear to move down into the Earth (Coriolis is 3D, with a vertical component caused by horizontal motion too, sometimes aliased as Eotvos)... but would begin to seem to be moving north immediately because of Earth's rotation...

... you can see this by the sun's path at (the sadly somewhat defunct, but still workable) Suncalc website (disregard error). The orange arc is sun's path across the sky... if you continue that for after the sun sets behind Earth, the ellipse starts to go N of you after being due W at sunset. Then emerges the next day from the E seemingly heading S (and up into the sky); right turning curved motion. The path idea would be the same with any extra W velocity added by a cannon.) Coriolis deflects right in the NH, which is N for a W initial velocity.

@JeopardyTempest I propose to migrate this discussion to a threaded forum, such as physicsforums. The stackexchange policy is that the comment section should not be used for extended discussion, and I endorse that policy. It appears to me you are giving yourself a very hard time, for no benefit. Low Earth Orbit is among the simplest of motions: along a circle around the Earth center of mass, uniform angular velocity, optionally inclined wrt Earth Equator, Earth rotating underneath the orbit. Other than the attraction to the Earth center of mass there is no interaction with the Earth.

(Sorry, before reading your comment, I realized an error in what I said on a release from the gravityless Earth... it won't shoot westward and set immediately if released, it'll rise first (because it has the linear momentum it had at the release point). Same elliptical arc, just at a different point in it. Poor error!)

That said, I am fine with not continuing in chat, will add the next comment to shorten the discussion and request mods to move inbetween comments to chat, but I believe I properly proved my point that Coriolis is a rightward deflection in NH, and Coriolis[/Eotvos] applies to any object moving with respect to the rotating body from the rotating bodies reference frame unless along the rotation axis (up/down at the Pole), and not sure there's much more to say about it. I'm still quite convinced still that Coriolis deflects right in the NH.

@Cleonis: I believe you've mixed Coriolis with Centrifugal Forces, but we'll move the subsequent conversation to chat where I try to show why, and we debated the point further...

@JeopardyTempest The stackexchange chat functionality is very awkward to use. The entries have the same length limit as the comments (600 characters), no possibility to upload pics. It's a pain. That is why I proposed migrating to physicsforums. Incidentally, the wikipedia article about the Eötvós effect; I am the author of that article. (The current article is still close to what I uploaded.)

@JeopardyTempest To give you an idea why I am confident. Some years ago I created a set of Java simulations, available on my website. The display shows a side-by-side view. On the left the inertial point of view, on the right a co-rotating point of view. The back-end computation is for the motion with respect to the inertial coordinate system. That motion is then transformed to the rotating coordinate system, and displayed in the co-rotating view panel. It's one thing to try and visualize mentally. Doing the work of programming a simulation, that's when the rubber meets the road.

@Cleonis feel free to start a thread wherever and copy in the discussion. Then add a comment with the link, and I'd think the mods can delete most of the remaining comments and leave our initial points and the link. It sounds like you're well versed in the topic, and we're probably saying the same thing in different ways... but I don't see the great circle path being evidence of Coriolis, which is what the OP seemed to be trying to understand, but of centrifugal force (again, it would cross the equator for a non-rotating Earth too).

I'd expect if you're calculating the transformed path, you'd have both "forces" resulting in the calculation to do so. But please let's stop discussing here, and create your link if you want to go elsewhere so mods can clean up comments once you do :-)

(Ops, please don't clean comments until ample time has passed for the thread to be transitioned to his alternative location)

@JeopardyTempest I created a thread on physicsforums. title: On whether the motion of a Foucault pendulum bob is comparable to ballistics