= Gyroscope direction reference
For a rotor spinning about a <principal axis> with its support at the <centre of mass>, uniform <gravity> supplies no <torque> about that centre. With other external <torques> negligible, <conservation of angular momentum> preserves its direction in an <inertial frame>. A mount that permits free axis motion lets the housing rotate relative to that persistent direction, giving an orientation reference when external landmarks are unavailable. This is the useful limit opposite to the deliberately off-centre support in the <spinning-wheel precession experiment>. It supplies a direction, not a position or a full three-axis orientation from one axis alone.
For a small accumulated transverse <torque> about the <centre of mass>, the change in the direction <unit vector> is, to first order,
$$
\Delta\widehat{\mathbf L}\simeq\frac{1}{L}\int\boldsymbol\tau_\perp\,dt,\qquad
\boldsymbol\tau_\perp=\boldsymbol\tau-(\boldsymbol\tau\cdot\widehat{\mathbf L})\widehat{\mathbf L},\qquad L=|\mathbf L|.
$$
Here $L$ is approximately constant over the short interval. Large <angular momentum> reduces drift for the same integrated transverse <torque>; parallel <torque> instead changes its magnitude. For $L=8.0\,\mathrm{kg\,m^2/s}$, a constant transverse <torque> of $0.010\,\mathrm{N\,m}$ over $1.0\,\mathrm s$ produces approximately $0.00125\,\mathrm{rad}$, or $0.0716^\circ$, of direction change. This is another synthetic calculation. The persistent direction belongs to an <inertial frame>; its components can change in a rotating <reference frame> even with zero external <torque>.
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