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The science
Why the plane turns
In an inertial frame the bob swings in a fixed plane. The floor, however, turns with the Earth. Only the component of the Earth's angular velocity along the local vertical matters for a horizontal swing: Ωz = ΩE sin φ, with ΩE = 7.292×10-5 rad/s. In the floor frame the small-amplitude equations are x'' = -(g/L) x + 2 Ωz y' and y'' = -(g/L) y - 2 Ωz x'. The plane therefore rotates at -Ωz, which is 15.04 sin φ degrees per hour, once round in 23.93 h / sin φ. This page integrates those two equations with a fourth-order Runge-Kutta scheme and reads the plane angle from the successive swing tips.
Limits
At the equator sin φ = 0 and the plane does not turn; at the poles it turns once per sidereal day; in the southern hemisphere the sense reverses.
Honest simplifications: small-angle swing (no amplitude-dependent period), no air drag or support imperfections, a flat floor, and a uniformly rotating Earth. A real pendulum needs a carefully symmetric suspension or the plane drifts for other reasons.
What to try
- Choose Paris and raise the time warp: the dotted swing tips walk clockwise round the floor in about 32 hours.
- Choose Equator: the plane never turns. Then North Pole: one turn per sidereal day.
- Pick Sydney and notice the sense of rotation reverses.
- Change the length: the swing period changes, but the precession rate does not.