= Solution
One application is the slow steady <precession> of a rapidly spinning <gyroscope> supported at one point. Let its <centre of mass> be a distance $\ell$ from the pivot along its spin axis, at angle $\alpha$ to the upward vertical. Write $J\simeq I_1\omega$ for its dominant spin <angular momentum>. In a <reference frame> precessing steadily about the vertical with <angular velocity> $\boldsymbol\Omega_p$, the spin <angular momentum> is approximately stationary, so
$$
\mathbf G\simeq\boldsymbol\Omega_p\times\mathbf J.
$$
The gravitational <torque> has magnitude $mg\ell\sin\alpha$. The <cross product> on the right has magnitude $\Omega_pJ\sin\alpha$, giving, for a nonvertical axis,
$$
\boxed{\Omega_p\simeq\frac{mg\ell}{I_1\omega}.}
$$
Its direction follows from the <torque> equation. This is the slow <precession> approximation: the additional <angular momentum> associated with <precession> is small compared with the spin contribution.
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