Hint 9

The rate of precession of a symmetric top rotating about an axis close to the vertical has one of two possible values

$$\Omega={I_3\omega\pm\sqrt{I_3^2\omega^2-4I_1 mgr}\over 2I_1},$$

where $r$ is the distance of its centre of mass from the point of support. This motion is stable only if $\Omega$ is real. This holds for

$$\omega^2>{4I_1mgr\over I_3^2}.$$

For the pencil, the minimum $\omega$ for stability is about 2900 rad s$^{-1}$!