I believe the formula Jeremy posted and a different one I once posted give an estimate on the transition speed between precession at a more or less uniform angle and the start of large oscillations (nutation). If I recall correctly, it starts when the energy of spin is of the magnitude of the gravitational energy.

I have seen a few tops rise from nearly touching the floor to standing up and sleeping.

The formula I gave really does estimate the minimum spin rate

*s*_{c} for stable sleeping, but some of the underlying assumptions might be questioned. A more general formula is

*s*_{c}^{2} = 4 (

*I*_{1} -

*I*_{3})

*M* *g* *h* /

*I*_{3}^{2},

where

*I*_{1} is the transverse moment of inertia

*about the contact point, not the center of mass*;

*I*_{3} is the axial moment of inertia;

*M* is the top's mass;

*g* is the acceleration of gravity; and

*h* is the axial distance from the contact point to the top's center of mass.

A similar formula gives the minimum spin rate

*s*_{cp} for stable steady precession at constant inclination angle

*a*:

*s*_{cp}^{2} = 4 (

*I*_{1} -

*I*_{3})

*M* *g* *h* cos(

*a*) /

*I*_{3}^{2},

These formulas skirt a thorny issue having to do with the fact that the precession rate

*p* about the vertical isn't necessarily zero in a sleeping top. They work only when

*I*_{1} >

*I*_{3}, but that's usually the case for realistic values of

*h*.

The spin rate at which axial kinetic energy equals gravitational potential energy (GPE) is different from both of these.

Assume steady precession at constant

*a*. Relative to the contact point, the GPE is

*V* =

*M* *g* *h* cos(

*a*).

The kinetic energy about the top's symmetry axis (subscript 3) is

*T*_{3} = ½

*I*_{3} *w*_{3}^{2},

where

*w*_{3} is the total angular speed about the symmetry axis given by

*w*_{3} =

*s* +

*p* cos(

*a*).

The total angular speed

*w*_{3eq} when

*T*_{3} =

*V* is then

*w*_{3eq}^{2} = 2

*M* *g* *h* cos(

*a*) /

*I*_{3}Unfortunately, one must specify the precession rate

*p* in order to extract the corresponding spin rate from

*w*_{3eq}. The usual formulas for estimating

*p* won't do here, as they all assume that

*w*_{3} >>

*w*_{3eq}.

Nor is it kosher to assume that

*s* >>

*p*in this low-speed regime, where the usual formulas surrounding nutation also break down.