How do scientist know how much the world really rotates

“The Smithsonian pendulum, like all pendulums, moved in accordance with Foucault’s sine law, which predicts how much a pendulum’s path will distort each day based on its latitude. Absent any exterior forces, a pendulum would swing back and forth in a single plane forever—there would be no gradual angular shift. But the Earth is rotating, so the story isn’t that simple.

Since all points on Earth’s surface rotate as a unit, it follows that those located on the wider portions of the planet—nearer to the equator—must cover more meters each second (i.e., go faster) to “keep up” with the points tracing smaller circles each day at the extreme northern and southern latitudes. Though they don’t feel it, a person standing in Quito, Ecuador, is moving with appreciably higher velocity than one in Reykjavik, Iceland.

Because each swing of a pendulum takes it from a point farther from the equator to a point nearer to the equator and vice versa, and the velocities at these points differ, the path of the pendulum is subtly distorted with every swing, gradually torqued away from its original orientation. The extent of this effect depends on where on Earth the pendulum is swinging.

At the North Pole—where small changes in latitude have big implications—the path traced by a pendulum would shift through a full 360 degrees in a mere 24 hours, explains Thompson. At the equator, meanwhile, a pendulum’s motion would not be seen to distort at all.” From the Smithsonian Magazine

Pe = 1/2k(x)^2  and .05n/m = .05x10^-9m = x