Could you show me the formula for the critical speed? If I am not too scared by the first look of it, I just might try to understand some of it and draw conclusions.

Not missed, just deferred, as the formula deserves some explanation.

For a top in steady precession (no wobble) at tilt angle

*θ*, the formula you usually see is for the square of the critical angular speed

*ω*_{C} in rad/s:

*ω*_{C}² = 4

*M* *g* *H* *I*_{1tip} cos(

*θ*) /

*I*_{3}²

where

*M* is the mass in kg,

*g* = 9.81 m/s² is the acceleration of gravity,

*H* is the CM-contact distance in m,

*I*_{1tip} is the transverse moment of inertia (TMI)

*about the tip (contact point)* in kg m², and

*I*_{3} is the axial moment of inertia (AMI) in kg m². When you have a sleeper with

*θ* = 0, the cos(

*θ*) factor goes away.

Technically,

*ω*_{C} includes both the pure spin rate and a component of the precession rate. But for design purposes, you can interpret

*ω*_{C} as the critical spin rate to a good approximation, as the precession rate involved is usually much smaller.

**Making that which is hidden seen**The problem with using

*I*_{1tip} is that it buries a hugely important dependency on

*H*. To get the latter out in the open where it belongs, I prefer the "central TMI"

*I*_{1cm} -- i.e., the TMI about the CM. An added advantage is that

*I*_{1cm} is the TMI you see in tables of formulas like

the one on Wikipedia. The parallel axis theorem shows how the two TMIs are related:

*I*_{1tip} =

*I*_{1cm} +

*M* *H*²

With

*H* exposed in all its glory, the critical speed is then

*ω*_{C}² = 4

*M* *g* *H* (

*I*_{1cm} +

*M* *H*²) cos(

*θ*) /

*I*_{3}²

So your plan to minimize

*H* and maximize

*I*_{3} for a given scrape angle is well founded, as

*H* is the most influential parameter here, with

*I*_{3} not far behind. You can also see why we sometimes talk about the TMI/AMI ratio.

**Forget the mass**But the mass

*M* turns out to be just confusing clutter, as both moments are also proportional to

*M*. To give

*M* the boot it deserves, I prefer to use the "specific moments"

*J*_{i} (aka moments per unit mass), like so...

*J*_{3} =

*I*_{3} /

*M* and

*J*_{1} =

*I*_{1} /

*M*For example, in the formula

*I*_{3} = 1/2

*M* *R*² for the AMI of a thin disk of radius

*R* , the specific AMI is just the 1/2

*R*² part.

The critical spin rate then reduces to

*ω*_{C}² = 4

*g* *H* (

*J*_{1cm} +

*H*²) cos(

*θ*) /

*J*_{3}²

In words, critical speed has nothing to do with mass. To minimize it, you need to focus on maximizing

*J*_{3} and minimizing both

*J*_{1cm} and

*H*. As an added bonus, you can then get all the AMI you need with the least weight on the tip.

**Practical applications**Unlike the regular moments, the specific moments are purely geometric in nature. For example, the specific AMI

*J*_{3} measures the efficiency of a top's mass distribution about the spin axis without regard to how much mass there actually is.

Suppose you have two lumps of clay, A and B, of the same mass

*M*. You mold A into a solid disk and B into a hollow toroid of the same maximum radius

*R*. Since B will get more AMI out of the same mass, it will have the greater specific AMI.

I find it useful to think in terms of specific moments when I'm optimizing top designs -- especially

*J*_{3}. But you also have to play a delicate trade-off with the absolute AMI

*I*_{3}, as AMI is the inertia that opposes

any change in spin rate -- during both spin-up and spin-down. Too much AMI, and you'll limit the release speed your fingers can attain in the brief time available during a single twirl. Too little, and air and tip resistances will chew up your hard-won release speed that much faster. Test, test test!