The equation of motion for a pendulum in uniform gravity is

The equation is non-linear in theta. However, for small angles, the equation can be linearized to the simple harmonic oscillator equation, giving sinusoidal solutions. When a pendulum swings on the surface of the Earth, it is in a rotating reference frame. This introduces a Coriolis Force term into the equation of motion.

In Cartesian coordinates, the rotation of the Earth can be expressed as

where lambda is latitude, and z points north along the Earth's axis. This problem is most easily solved if it is transformed into a coordinate system that rotates with respect to the lab coordinate system with a speed omega prime. This transformation is

The equation of motion reduces to

This equation is simply a swinging small-angle pendulum that steadily precesses with an angular speed of omega prime. This rotation is the Foucault precession. Its observation in 1851 by Leon Foucault was a direct experimental confirmation of the rotation of the Earth. The path it traces out is shown in the picture below.

Last Updated: 04/18/2005