Why doesn't Foucault's pendulum show Earth's spin at the equator but works at the North/South poles

Question

If you where directly above the North/South pole and set up Foucault's pendulum, why does it show the Earth's spin even though the Earth isn't spinning there? This is also where it takes the shortest amount of time.

And now you set up Foucault's pendulum exactly on the equator it doesn't show you that the Earth is spinning even though it spins the fastest here, it will just keep going in the same path.

My question is why at the North/South pole where the earth does not spin you can see Foucault's pendulum demonstrate spin? And at the equator where the Earth spins the fastest it doesn't demonstrate spin?

Thanks for any replies, I spent a few hours looking for an answer but, I couldn't find any answer that explained it well.

Answer 1

First, the surface of the Earth is (kind of) a solid mass and the whole thing spins at a (kind of) single rate.

At the North and South Poles, you are perfectly spinning. (I will drop all the 'kind of's now). At all other points on the Earth's surface, you are traveling in a circle around the axis through the poles. You are still spinning but you are also traveling in a circle.

What makes a difference in the rate of Foucault's pendulum is the angle between the axis of the rest position of the pendulum and the axis though the poles. At the poles, they line up and Foucault's pendulum rotates at the same rate as the Earth's spin.

Anywhere on the equator, these axis are perpendicular to each other so the Earth's spin does not affect the pendulum at all!

Answer 2

Why doesn't Foucault's pendulum show Earth's spin at the equator but works at the North/South poles?

Because the Coriolis effect is purely vertical at the equator.

The Coriolis acceleration is $-2\vec \Omega \times \vec v$, where $\vec \Omega$ is the Earth's angular velocity vector and $\vec v$ is the velocity of some object relative to the surface of the Earth. At the equator, $\vec \Omega$ points due north, i.e., parallel to the surface. This means that the Coriolis acceleration is either zero or vertical at the equator. There is no torque on a pendulum at the equator that can make the pendulum turn.

This, by the way, is one reason that all those videos showing water draining from a pan one way just north of the equator but the other way just south of the equator are hoaxes. (Another reason is this effect is very small, even at the poles.)