= Counterbalanced gyroscope precession
{title2=$\Omega_z=g(ml-m_cl_c)/L_x$}
Let a horizontal <gyroscope> have signed spin <angular momentum> $L_x$ along the axle. The disk's <centre of mass> is at $+l\widehat{\mathbf x}$ and a counterweight is at $-l_c\widehat{\mathbf x}$. Gravity supplies <torque> $g(ml-m_cl_c)\widehat{\mathbf y}$. Since $\boldsymbol\tau=\boldsymbol\Omega\times\mathbf L$, steady horizontal <precession> gives the displayed signed vertical <angular velocity>. Neglect the axle's <mass> and the counterweight's axial spin <moment of inertia>. The zero-torque choice $m_cl_c=ml$ allows the axle to remain fixed. Moving the counterweight through this position reverses the gravitational <torque> and the associated slow <precession>.
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