6.6 Worked problems: Foucault pendulum
A Foucault pendulum is a mathematical pendulum set up on the Earth surface where we consider only small amplitude oscillations $\theta \ll 1$ with vanishing angular momentum, $C=0$. According to Equation 6.4.11 one expects that it follows the EOM \begin{align} \ddot \theta(t) &= -\frac{g}{\ell} \; \sin\theta(t) \simeq -\frac{g}{\ell} \: \theta(t) \, . \tag{6.6.1} \end{align}
Remih, CC BY-SA 3.0, via Wikimedia Commons}
Figure 6.20: Foucault pendulum in the Panthéon, Paris, France.
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This is a harmonic oscillator with frequency $\omega = \sqrt{g/\ell}$ and solution $\theta(t) = \theta_{\text{max}} \: \sin\bigl( \alpha + \omega \, t \bigr)$. The amplitude $\theta_{\text{max}} \ll 1$ and the offset phase $\alpha$ must be chosen to match boundary conditions. One expects that this motion will be confined to a fixed plane $\varphi=$const. In contrast the plane of motion is turning slowly, and the angular speed of the turning depends on the location where one sets up a Foucault pendulum. This is a striking consequence of the fact that a laboratory on the Earth surface is not an inertial system since it is turning around the Earth axis once every 24 hours.
In order to account for the Earth rotation we adopt spherical coordinates $\hat{\boldsymbol r}(\lambda,\phi)$ with origin at the Earth center and directions indicated by the longitude $\lambda$ and latitude $\phi$ (see sketch in the margin). Earth is rotation around the axis $\hat{\boldsymbol z}$ with a fixed angular frequency $\Omega = 2\pi/24 \, \text{h}$. Hence, $\phi=$const and $\lambda = \Omega\, t$ for a pendulum at a fixed position on the Earth surface. We denote the distance of the pendulum plane from the Earth center as $R$, and the deflection of the pendulum towards the East and North as $x(t)$ and $y(t)$, respectively. Consequently, the position $\mathbf q(t)$ of the pendulum weight amounts to \begin{align*} \mathbf q(t) = R \: \hat{\boldsymbol r} \bigl(\lambda(t), \phi(t) \bigr) + x(t) \: \hat{\boldsymbol \lambda} \bigl(\lambda(t) \bigr) + y(t) \: \hat{\boldsymbol \phi} \bigl(\lambda(t), \phi(t) \bigr) \end{align*}