ta0 nicely tackled the offset top with a circular rotor. Here's a general solution for the not-so-simple square rotor case. (Sorry,
ortwin, have to shoot Miss Skimpy in Chains and her fluorescent friends at night and just haven't had the energy for all the takes that will involve.)
Setup: Consider a square top rotor of side
S. In map view, a straight "boundary" a distance
B < ½
S from the closer parallel side divides the rotor into 2 parallel rectangles — a smaller "Rectangle 1" with short side
B (white below) and a larger "Rectangle 2" with short side
S -
B (dark gray).
Each rectangle
i of mass
Mi, area
Ai, mass density
ρi, and thickness
Li has its own
areal density
αi =
Mi /
Ai =
ρi LiAreal density ratio: Rectangle 1 gets the greater areal density in this problem — whether by thickening it, densifying it, or both. Hence, the
areal density ratio given by
δ =
α2 /
α1 =
ρ2 L2 /
ρ1 L1 < 1
In a rotor of uniform thickness where
L1 =
L2, this density ratio reduces to
δ =
ρ2 /
ρ1. Likewise, the ratio reduces to
δ =
L2 /
L1 when
ρ1 =
ρ2. The wobble-free LEGO offset top above and below models the latter scenario with
δ =
L2 /
L1 = 1/3. The stem and tip accurately mark a rotor CM well off the boundary.
Rotor CM: The respective rectangle CMs are located at the following distances, not from the boundary, but from the parallel
rotor edge on Rectangle 1:
yCM,1 = ½
ByCM,2 = ½ (
B +
S)
The distance to the rotor's overall CM is then
yCM = (
M1 yCM,1 +
M2 yCM,2) /
M = ½
S (
β +
M2 /
M) = ½
S (
β² +
q) / (
β +
q),
where the proportion
β =
B /
S,
M =
M1 +
M2 is the rotor's total mass, and
q =
δ / (1 -
δ) =
α2 / (
α1 -
α2). The LEGO model above confirms this expression for
β = 1/4 and
δ = 1/3, which makes
q = 1/2 and
yCM = 3/8
S.
CM on the boundary: Putting the rotor's CM smack on the boundary means that
B =
yCM, or equivalently, that
β =
yCM /
S. To generalize the solutions forced by this restriction to a square rotor of
any size, we'll focus now on
β as a function of
δ via
q. The problem then reduces to the quadratic equation below. With this 1:1 relationship between
β and
q when the CM's on the boundary, you can set either one and easily solve for the other.
β² + 2
q β -
q = 0
Problem 1: For a desired areal density ratio
δ, find the
β needed to put the CM on the boundary using
q =
δ / (1 -
δ)
β =
q [√(1+ 1/
q) - 1]
Problem 2: For a desired
β, find the
δ needed to put the CM on the boundary using
q =
β² / (1 - 2
β)
δ =
q / (1 +
q)
Does it work? You bet! In the uniform density model below,
δ = 1/5 and
q = 1/4. The calculations give
B = 4.94 on this rotor with
S = 16, but couldn't get the boundary, stem, and tip any closer than
B = 5.00. Still, the wobble's not bad.
Golden ratio: Can you put the CM on the boundary with
δ =
β? Yes! Just set them both to 2 -
φ ~ 0.382, where
φ = ½ (1 + √5) ~ 1.618033 is the
golden ratio. Unfortunately, can't (yet) see a way to make a smoothly spinning golden LEGO offset top with
δ =
β = 0.382.