= Gyrocompass oscillation
{title2=$\omega_0^2=(I_1/I_2)\omega\Omega_E\cos\lambda$}
For a rapidly spinning <gyrocompass> with axle constrained horizontally, take azimuth $\theta$ positive from north towards west and positive spin <angular momentum> $J_s=I_1\omega$. In the terrestrial <rotating reference frame>, the vertical component of the <torque> equation, with zero external vertical <torque>, is $I_2\ddot\theta+J_s\Omega_E\cos\lambda\sin\theta\simeq0$. Here $I_2$ is the vertical <moment of inertia> and terms of order $\Omega_E^2$ have been neglected. The <small-angle approximation> about $\theta=0$ gives <simple harmonic motion> with the displayed squared <angular frequency>. Linearizing about $\theta=\pi$ reverses its sign and gives an unstable <equilibrium>. At the poles this restoring effect vanishes. For a thin rotationally symmetric flywheel, the <perpendicular axis theorem> gives $I_1=2I_2$.
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