Dagnabbit, wrong again!

But air resistance ... may well favor smaller tops. Many tops have disk-like rotors. Consider, then, the swirling air flow induced by a thin smooth spinning disk of radius *R*.

The resulting aerodynamic braking torque generated by the disk faces grows with *R*^{4}. Empirically, the braking torque from the disk's edge seems to be negligible up to thicknesses approaching *R*/10.

Made 2 big mistakes here:

1. Lost sight of the fact that what a player experiences during spin-down is the top's

*deceleration rate* *α*, not the underlying braking torque.

2. And when you expose all the contributing factors involving

*R* in a disk this thin, you find that

*α* actually

improves with absolute size

*as scaled by* *R*.

Good to know, as keeping

*α* to a minimum at operating speeds is a powerful way to boost spin time.

**von Karman's disk**As an oversimplified but useful model of a common top genre, consider the "von Karman disk" — a smooth "thin" disk of radius

*R* and axial length

*L* spinning at constant speed

*ω* about a fixed axis in otherwise still air. Per von Karman, the steady-state air resistance the disk encounters is then

*Q*_{A} =

*k*_{A} *ω*^{n} *R*^{4},

where

*Q*_{A} is the

*aerodynamic* braking torque about the spin axis, and

*k*_{A} is a constant depending only on ambient air properties. For the speed exponent, von Karman got

*n* = 1.5.

But just how "thin" does this disk have to be? Per many, many studies, von Karman nailed it as long as the disk's lateral aspect ratio

*λ* =

*L* /

*R* is under 10% or so. Meaning that aerodynamic edge effects are negligible for disks this thin. Only the faces count.

**Deceleration rate**Graphically, a top's deceleration rate

*α* is just the slope of its spin decay curve — a plot of

*ω* over time. Mathematically,

*α* = d

*ω*/dt =

*Q* /

*I*_{3},

where

*Q* < 0 is the net braking torque about the spin axis, and

*I*_{3} is the disk's axial moment of inertia (AMI). For a von Karman disk subject only to air resistance,

*Q* =

*Q*_{A}.

**Absolute size and AMI**The AMI of a von Karman disk is

*I*_{3} = ½

*M* *R*² = ½ (

*ρ* *V*)

*R*² = ½ π

*ρ* *R*^{5} *λ*,

where

*M* is the disk's mass,

*ρ* is its uniform mass density, and

*V* = π

*R*²

*L* = π

*R*³

*λ* is its volume. We can now expose all the ways that absolute size, as scaled by

*R*, affects the disk's deceleration rate:

*α* =

*k*_{A} *ω*^{1.5} *R*^{4} / (½ π

*ρ* *R*^{5} *λ*) =

*k* *ω*^{1.5} /

*R*,

where the new constant

*k* = 2

*k*_{A} / (π

*ρ* *λ*) is effectively fixed at design time.

So while air resistance grows very rapidly with increasing absolute disk size, the more practically important deceleration rate actually improves!

**Absolute size and critical speed**If a von Karman disk were made into a top with a stem and tip of negligible mass, the square of its critical speed for stable sleep would be

*ω*_{C}² = 4

*M* *g* *H* (

*I*_{1} +

*M* *H*²) /

*I*_{3}²,

where

*g* is the acceleration of gravity,

*H* is the top's CM-contact distance, and

*I*_{1} is the disk's

*central* transverse moment of inertia. To a very good approximation,

*I*_{1} = ½

*I*_{3} in a von Karman disk.

After exposing all the ways that

*R* affects critical speed and simplifying, we're left with

*ω*_{C}² =

~~4 ~~*g* *η* / *R* 4

*g* (4

*η*³ +

*η*) /

*R*where

*η* =

*H* /

*R* is relative CM height -- another fixed top proportion. Note the cubic depence on

*η*, and the complete lack of dependence on

*ρ* and

*M*.

*This result can only be good for spin time, as increasing absolute size now improves both deceleration rate and critical speed — at least in a von Karman-like top.***Absolute size and spin time in real tops**So far, we've been scaling absolute size by max radius

*R*. This turns out to be useful for real tops with other simple shapes as well, as the moments of inertia then scale as

*R*^{5} *G*, where

*G* is a purely geometric factor depending only on fixed top proportions like

*λ* and

*η* above.

On hard, smooth surfaces, these tops will generally lose most of their speed early on, when

*Q* ~

*Q*_{A}. Of course, they'll also be subject to tip resistance, and aerodynamic edge effects generally won't be negligible.

But in light of the von Karman model above, seems likely that spin time would increase with absolute size in most tops — at least to a point. That's certainly the trend I've seen in my own tops, and Iacopo's reported the same trend in his.