setting I1 = I3 gives h = r/2
I find this an interesting formula, which gives an immediate idea of the proportions of a top with the two moments of inertia equal.
I deduce from it that h is 0.707 times the radius of gyration in tops with I1 = I3.
True, but keep in mind that h = r/2 assumes the mass of the top is in the shape of a thin disk. Your Nr 30 top is more of a torus shape (beautiful top by the way) Using your mass of 0.119 Kg and radius of 0.02875 m and assuming a thin disk:
I
3 = 1/2(.119)(.02875)
2 = 4.9x10
-5 Kg m
2you give I
3 = 6.7x10
-5 Kg m
2which makes since because more mass is on the outside.
By the way, how did you get this value of I
3?
Using your value of 140 rpm for the toppling down speed and using the critical speed formula I get
I
1 = 3.1x10
-5 Kg m
2 for your Nr 30 top.
Not sure how accurate this is but it's probably in the ball park.
Edit:
For a symmetric mass distribution, its impossible to have I
1 < I
3/2. If h > 0 and or thickness along the symmetry axis then I
1 > I
3/2. So this calculation is not accurate.
Did you measure or calulate I
1 for your Nr 30 top?