Foucault pendulum

Example

A Foucault pendulum is a long pendulum that is free to swing in every direction. It is set up at a place of the Earth with latitude $\phi$. Let $\Omega$ be the angular velocity of the Earth, and let $\Omega_z = \Omega \sin \phi$ be its vertical component. In the rotating frame of the Earth, the Coriolis force acts on the swinging mass. For small swings, let $x$ and $y$ be the horizontal displacements to the east and to the north. Let $\omega_0 = \sqrt{g/\ell}$, where $\ell$ is the length of the pendulum and $g$ is the acceleration of gravity. Keeping only the vertical component of the rotation, the equations of motion are $$x'' - 2 \Omega_z \, y' + \omega_0^2 x = 0, \qquad y'' + 2 \Omega_z \, x' + \omega_0^2 y = 0.$$ With $z = x + i y$ they become $z'' + 2 i \Omega_z \, z' + \omega_0^2 z = 0$. Check: substituting $z(t) = e^{-i \Omega_z t} w(t)$ gives $w'' + (\omega_0^2 + \Omega_z^2) w = 0$. So $w$ is a planar harmonic swing that does not turn. In the northern hemisphere the factor $e^{-i \Omega_z t}$ turns it clockwise, seen from above, at the angular rate $\Omega_z = \Omega \sin \phi$.