**Some more realistic critical speeds for Diaper Genie balloons (DGBs)...**Rough DGB specs estimated from video...

*M* = mass (unknown but eventually cancels out of critical speed calculations; ditto for density)

*R* = outer radius = 0.125 m

*L* = total length = 7.5 m

*a* = balloon aspect ratio =

*L* /

*R* = 60

*G* = Ground clearance (depends on DGB orientation)

*b* = relative ground clearance =

*G* /

*R**H* = CM-tip distance (depends on DGB orientation)

*g* = acceleration of gravity = 9.8065 m s

^{-2}**Subscripts: **"3" points to the spin axis and "1" to any transverse axis through the tip-floor contact; "v" points to the 1st case of a DGB with its long axis vertical and "h" to the horizontal case.

**Vertical DGB: **Consider a single

*vertical* DGB spinning on one end about its long axis with negligible ground clearance. Ignoring the closed ends and modeling the DGB as a thin-walled open tube,

*G*_{v} = 0

*H*_{v} =

*G*_{v} +

*L* / 2 =

*L* / 2

*I*_{3v} =

*M* *R*^{2}*I*_{1v} =

*M* *R*^{2} (3 + 2

*a*^{2}) / 6 =

*I*_{3v} (½ +

*a*^{2} / 3)

Critical speed for a sleeping vertical DGB...

*w*_{Cv} = (30 / pi) sqrt [(g / 3 R) (3 + 2

*a*^{2})

*a*] ~ 32,000 RPM

So not as fast as we thought. But if the spin-up method didn't destroy the balloon directly, centrifugal force probably would, and long before reaching this speed.

**Horizontal DGBs: **Now form a symmetrical 4-spoke rotor from 2

*horizontal* DGBs and mount it on a long rod-like tip of negligible mass so that

*G*_{h} = 2 m

*b*_{h} =

*G*_{h} /

*R* = 16

*H*_{h} =

*G*_{h} +

*R* / 2 = 2.06 m

Since these DGBs are spinning about their short axes now,

*I*_{3h} = 2

*I*_{1v,cm} =

*M* *R*^{2} (6 +

*a*^{2}) / 6

*I*_{1h} = 2 (

*I*_{3v} +

*M* *H*_{h}^{2})

Critical speed for a sleeping top made of 2 crossed

*horizontal* DGBs...

*w*_{Ch} = (30 / pi) [ 6 / (6 +

*a*^{2})] sqrt [(2

*g* /

*R*) (5 + 4

*b*_{h}^{2} + 4

*b*_{h}) (

*b*_{h} + ½)] ~ 27 RPM

So this spinthing would totally work as a top -- provided you could spin it up without damaging it. Since there would be some centrifugal straightening and stiffening of the flexible spokes, the rotor might even hold some of its shape against gravity and air resistance.

**Safety tip: **Replacing the rod-like tip with Mike on his unicycle would increase

*w*_{Ch} a good bit.

Note that the critical speed formulas depend only on the proportions

*a* and

*b*, the scaling factor

*R*, and

*g*. Mass and density don't enter the picture at all. So you could replace the DGB plastic with stronger plastic or aluminum, and the critical speeds would stay the same. The new wall could even be thicker as long as it remained very thin compared to

*R*.