= Latitude law of Foucault precession
{c}
{title2=$\dot\psi=-\Omega\sin\varphi,\quad T_{\mathrm p}=T_{\mathrm{sid}}/|\sin\varphi|$}
Use local east, north, and up <unit vectors>, with $x$ eastward, $y$ northward, and positive orientation $\psi$ anticlockwise as viewed from above. Let $\varphi$ be signed <geographical latitude>, positive north, and $\Omega>0$ the magnitude of <Earth>'s <angular velocity>. In the spherical-<Earth> approximation,
$$
\boldsymbol\Omega=\Omega\cos\varphi\,\mathbf e_N+\Omega\sin\varphi\,\mathbf e_U,\qquad
\alpha=\Omega\sin\varphi,\qquad \omega_0^2=\frac g\ell.
$$
Here $g$ is the local <effective gravity in a rotating frame>, including the steady terrestrial <centrifugal acceleration>. For small horizontal displacements compared with $\ell$, the horizontal restoring <acceleration> from <tension> is $-\omega_0^2(x,y)$. Vertical displacement is second order in the swing <amplitude>, so vertical <velocity> can be omitted at this order. The horizontal <Coriolis acceleration in local Earth coordinates> then gives the coupled <linear differential equations>
$$
\ddot x+\omega_0^2x=2\alpha\dot y,\qquad
\ddot y+\omega_0^2y=-2\alpha\dot x.
$$
These equations neglect <damping>, finite-amplitude corrections, support <anisotropy>, and spatial changes of the <centrifugal force> of order $\Omega^2$; they retain the leading terrestrial <Coriolis force>. A related derivation uses south/east axes instead.<Foucault's pendulum - the physics (and maths) explained by Joe Wolfe>[Foucault's pendulum - the physics (and maths) explained, equations (9.1\*)–(11): horizontal small-oscillation equations]{ref}
Introduce the <complex number> $z=x+iy$. Multiplying the second equation by $i$ and adding gives
$$
\ddot z+2i\alpha\dot z+\omega_0^2z=0.
$$
Set $z=e^{-i\alpha t}u$. Direct <differentiation> cancels the first <derivative> of $u$ and yields
$$
\ddot u+(\omega_0^2+\alpha^2)u=0,\qquad
\beta=\sqrt{\omega_0^2+\alpha^2},\qquad
z=e^{-i\alpha t}(C\cos\beta t+D\sin\beta t).
$$
The $\alpha^2$ in $\beta$ comes from solving these reduced equations exactly; it is not a complete prediction of terrestrial corrections of order $\Omega^2$. Thus the horizontal <trajectory> is an <ellipse> in coordinates rotating at <angular velocity> $-\alpha$ relative to the building. When $|\alpha|\ll\omega_0$, each rapid swing samples an almost fixed long axis; its slow <precession> is
$$
\dot\psi=-\Omega\sin\varphi,\qquad
T_{\mathrm p}=\frac{2\pi}{\Omega|\sin\varphi|}=\frac{T_{\mathrm{sid}}}{|\sin\varphi|}.
$$
The <sidereal day> $T_{\mathrm{sid}}=2\pi/\Omega$ is a rotation relative to distant directions. $T_{\mathrm p}$ denotes a full $2\pi$ turn of a continuously followed orientation; an unlabelled swing line repeats after $T_{\mathrm p}/2$. At the equator there is no finite <precession period> in this model. The <Coriolis parameter> is $f=2\alpha$, so the plane's rate is $-f/2$, not $-f$.
The limiting cases check the sign and scale. At the north pole $\alpha=\Omega$: the swing direction turns clockwise once per <sidereal day>. At the equator $\alpha=0$: both horizontal equations reduce to independent <simple harmonic motions> and a line launch stays on its original line. At the south pole $\alpha=-\Omega$: the direction turns anticlockwise once per <sidereal day>. Vanishing horizontal <precession> at the equator does not mean that every component of <Coriolis acceleration> vanishes there.
For a stipulated $\Omega=7.292115\times10^{-5}\,\mathrm{s^{-1}}$, the <sidereal day> is approximately $23.9345\,\mathrm h$. At $45^\circ$ north the predicted rate is $-10.6356^\circ/\mathrm h$ and $T_{\mathrm p}\simeq33.8485\,\mathrm h$; at $45^\circ$ south the rate changes sign. At the illustrative <geographical latitude> $34^\circ$ south, the prediction is $+8.4109^\circ/\mathrm h$. These calculations are model predictions, not new observations of the UNSW apparatus.
A swing <plane> fixed among the stars is generally only a polar picture. A north–south swing at the equator turns with <Earth> in an <inertial frame>, while having no <precession> against the building.<The Foucault pendulum by Joe Wolfe>[The Foucault pendulum, The 'plane' of the pendulum's swing is not fixed in space: equatorial counterexample]{ref}
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