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[Jeremy McCreary]

Your impression is correct.  My top Nr. 12 is a "conical pendulum with spin", and it was designed intentionally to be so.    Nr. 14 is the other one.  Probably you have seen the videos about them and you remember them.   About the Nr. 14, there was an error, because I didn't want it to be self-righting; its tip was near the CM but still above it;  but I used a ball tip in it, without thinking that a ball tip, near the CM, even if above it, makes the top self-righting in any case.
The rule in this case should be that the center (and not the contact point) of the ball tip must be below the CM, to avoid the self-righting effect.   

I have no fabrication skills whatsoever. If I made tops with anything other than LEGO, I'd spend all my time (and materials) throwing out botched jobs and starting over. How you make such gorgeous high-precision finished pieces is truly beyond me.

The good news: You can't beat LEGO tops for experimentation purposes.

[ta0]

Perhaps your tops have dense internal mass rings just inside their equators. That wasn't an option for me, but it would explain why your tops work about the same right-side up and upside down. Mine have no chance upside down.

The tops have holes for whistling, so I can look inside. They are just shells, with the axle going through. At the equator there is a little extra material where the two halves are clamped, but not much.
It is a mystery to me the difference between our tops.

[Jeremy McCreary]

It is a mystery to me the difference between our tops.

Me, too.

[Iacopo]

truly spherical mass distributions (including thin shells) are inherently unstable as tops at any speed.

You made me think to one of my first tops, which is quite unstable, (I can't make it spin for more than 15 seconds and topples down at 1250 RPM).  I thought that the reason was simply a too high CM but maybe it could be a case of spherical mass distribution.
I suppose that "spherical mass distribution" means that AMI and TMI are equal.
Do you know if the TMI in this case should be measured at the tip or at the CM ?

[Jeremy McCreary]

Couldn't resist making a pair of top-like spinners with CMs below their tips...



See video description for details. Needed a name for this kind of spinner there, so for lack of a better term, I settled on "rigid conical pendulum with spin", or "RCPS" for short. Ugly, I know.

As expected with a negative CM-tip distance, the 2nd RCPS exhibits clear retrograde precession (opposite the spin direction) after 3:42. The 1st RCPS appears to precess the wrong way after 0:50, but a prominent looping nutation obscures the slight precession in this case. In later off-camera runs, precession of this RCPS was always retrograde when present.

I've been looking for a good pedestal with a smooth, slightly concave low-friction upper surface for many moons. Finally found it in the unopened Snapple bottles I've been handling every day for years. (Opened caps lose their concavity permanently.) Among other things, a pedestal makes topple speed determinations more accurate in broad, low-slung finger tops, which in my experience can precess themselves into the ground long before actually falling over.

[Jeremy McCreary]

I suppose that "spherical mass distribution" means that AMI and TMI are equal.

Yes, AMI = TMI about the CM. In fact, all 3 principal moments of inertia about the CM are equal in balls (solid spheres), Platonic solids, and other regular polyhedra with isometric symmetry if density is uniform. Ditto for their hollow counterparts if wall thicknesses are also uniform.

Do you know if the TMI in this case should be measured at the tip or at the CM ?

In tops spinning on the ground, the only TMI that counts is the one taken at the tip. For tops in the air in string play, the TMIs about the CM and tip can both be important, though probably at different times.

For a solid sphere of mass M and radius R spinning on any point on its surface, AMI = 2 M R² / 5 and tip TMI = AMI + M R² = 7 M R² / 5. Hence, the effective AMI/TMI ratio at the tip is 2/7 = 0.29, not 1.

For a thin spherical shell of the same mass and radius, AMI = 2 M R² / 3 and tip TMI = AMI + M R² = 5 M R² / 3. Hence, the effective AMI/TMI at the tip is 2/5 = 0.40, a little closer to 1.

I read that bit about spherical tops being inherently unstable in a source that escapes me at the moment but was probably fairly reliable. In view of these formulas, I'm also beginning to wonder exactly what was meant by that. Nevertheless, it takes unusually high spin rates to keep my spherical tops up, and the same with yours, so there must be something to it.

[ta0]

Needed a name for this kind of spinner there, so for lack of a better term, I settled on "rigid conical pendulum with spin", or "RCPS" for short. Ugly, I know.

RCPS Hanging Top? Inverted Top?
In the tricking world we call tricks with the top hanging from the tip: upside-down (UD) tricks.
When a top is played UD from the tip, the direction of precession does not change because the spin direction with respect to the vertical is reversed. But with a double-tip top, when using the upper tip the direction of precession does reverse.

I suppose that "spherical mass distribution" means that AMI and TMI are equal.

I read that bit about spherical tops being inherently unstable in a source that escapes me at the moment but was probably fairly reliable. In view of these formulas, I'm also beginning to wonder exactly what was meant by that.

You may have read it on this forum  because that is a conclusion that we have arrived as a group. In particular, Neff noticed that if he made a top too round, it was unstable. The wobbling happens mainly when the top is in the air, but not so much in the hand. I suggested that this could be because, when there is not a clearly defined principal axis, any imperfection (e.g. wood density variation) will move the actual principal axis a large amount away from the spin axis. We believe this is why the Short Circuit top when fitted with the aluminum tip instead of the plastic tip becomes unstable: it equals the nominal moments of inertia.

[Jeremy McCreary]

Needed a name for this kind of spinner there, so for lack of a better term, I settled on "rigid conical pendulum with spin", or "RCPS" for short. Ugly, I know.

RCPS Hanging Top? Inverted Top?

Ah, but these spinners don't really qualify as tops because they can't fall over! I'd be much happier with a name like "[] spinner", where the "[]" is something catchy that also evokes the right mental image. "Hanging spinner" is as good as anything I've come up with, but there's surely something better.

Iacopo: Any suggestions? You made these spinners long before I got in on the act.

In the tricking world we call tricks with the top hanging from the tip: upside-down (UD) tricks.
When a top is played UD from the tip, the direction of precession does not change because the spin direction with respect to the vertical is reversed. But with a double-tip top, when using the upper tip the direction of precession does reverse.

Very interesting. I'll have to think that one through.

You may have read it on this forum  because that is a conclusion that we have arrived as a group. In particular, Neff noticed that if he made a top too round, it was unstable. The wobbling happens mainly when the top is in the air, but not so much in the hand. I suggested that this could be because, when there is not a clearly defined principal axis, any imperfection (e.g. wood density variation) will move the actual principal axis a large amount away from the spin axis. We believe this is why the Short Circuit top when fitted with the aluminum tip instead of the plastic tip becomes unstable: it equals the nominal moments of inertia.

I've no doubt seen that spherical top stability thing on the forum, too, but I remember coming across it in a textbook or journal article as well. Theory's nice, but observations are better, and these are some very valuable observations. Your theory about them sounds right to me.

Alas, I'd have a much better first-hand understanding of general top behavior if only I could move beyond twirling finger tops and do string tricks myself. Really need to get down to business with that Saturno Larry was kind enough to send me, but new LEGO tops keep popping into my head!

[Iacopo]

Iacopo: Any suggestions? You made these spinners long before I got in on the act.

I don't know..  maybe it could be called also "self-righting pedestal spinner"...?

When the tip is above the CM, I noticed that the precession direction becomes reversed relatively to that of spinning.
In regular tops instead precession and spinning have the same direction.

[Iacopo]

The wobbling happens mainly when the top is in the air, but not so much in the hand. I suggested that this could be because, when there is not a clearly defined principal axis, any imperfection (e.g. wood density variation) will move the actual principal axis a large amount away from the spin axis.

This seems quite plausible to me.

[Jeremy McCreary]

I don't know..  maybe it could be called also "self-righting pedestal spinner"...?

Accurate enough, but still too many words, I think.

When the tip is above the CM, I noticed that the precession direction becomes reversed relatively to that of spinning.
In regular tops instead precession and spinning have the same direction.

Good observation. Precession in the opposite direction of spin is called "retrograde", and it's to be expected when the CM is below the tip.

You can see this in the usual approximate expression for (slow) precession...

p = M g H / I₃ w₃,

where p is the precession rate in rad/sec, M is the mass, g is the acceleration of gravity, H is the distance from tip to CM, I₃ is the AMI, and w₃ is the total angular speed about the spin axis.

Strictly speaking,

w₃ = s + p cos(a),

where s is the pure spin rate, and a is the angle between the spin axis and the vertical. But when s >> p or a ~ 90°, w₃ ~ s, and the precession rate formula takes a more useful form...

p ~ M g H / I₃ s

Now, when the CM is above the tip, H is positive, but when the CM is below the tip, H is negative. In the latter case, p and s must be of opposite signs, and the precession must be retrograde, because M, g, and I₃ are always positive.

While we have the precession rate formula out, note that the AMI can be written in terms of the axial radius of gyration K₃ like so...

I₃ = M K₃²

The formula then becomes

p ~ g H / K₃² s

Hence, precession rate depends on the mass distribution as measured by K₃ but not on total mass.

[ta0]

You can see this in the usual approximate expression for (slow) precession...

For me, starting from an equation does not do it: I don't see it as an explanation. I know that is not necessary the case for other people.

An explanation can be given from the conservation of the rotational momentum. Say you start with at top spinning close to vertically and you let it go. Gravity will make it fall a bit and precess. Because of the now larger angle with the vertical, the vertical momentum due to the spin of the top will have decreased. For the total momentum to remain the same, the momentum added by the precession has to add up: spin and precession have to be in the same direction.
Now imagine the top is hanging and you let it go. Again the top will fall a bit before precession takes over. But because the final angle of the axis of the top with respect to the vertical is smaller, the contribution of the top spin to the vertical momentum would increase. To keep the total momentum constant the momentum due to the precession has to substract: spin and precession are in opposite directions.

One of my favorite spherical tops (even if it's a tin plunge top):



It is called Sun Loved and is from Chein Playthings. Looks like an ad for oranges or juice.
It has some metal balls inside that make a really loud rattle while you are spinning it up.
At Lassanske's I saw three different sizes. Mine is 5.25 inches in diameter and either the medium or the small size.

[Iacopo]

Hence, precession rate depends on the mass distribution as measured by K₃ but not on total mass.

This is consistent with what I have observed;
in fact generally my light wooden tops don't seem to precess slower or faster than my heavy metal tops.
But larger tops do precess slower, whatever their weight.
And lower CM tops precess slower too. 
You didn't mention the TMI but I am not too surprised if it doesn't influence the precession speed.
 

[Jeremy McCreary]

You can see this in the usual approximate expression for (slow) precession...

For me, starting from an equation does not do it: I don't see it as an explanation. I know that is not necessary the case for other people.

An explanation can be given from the conservation of the rotational momentum....

Excellent physical explanation, ta0! It builds on the one given by Feynman, the great explainer of physics, when his famous lecture series in undergrad physics turned to the behavior of gyroscopes and tops.

It's no coincidence that he took that opportunity to warn his students that it's often easier to come to an accurate mathematical description of a phenomenon than to understand it on a deep physical level. He clearly valued both, but he urged his students to work hard for the latter whenever possible.

I happen to agree with you and Feynman on that score, but simple formulas still have their place as concise summaries of important relationships among key parameters -- even if they're never used to calculate anything practical. That's certainly the case with the precession rate formula here.

So, it's good to have both perspectives -- the gut-level physical understanding and the summary of relationships.

One of my favorite spherical tops (even if it's a tin plunge top)...

Very cool top. Your collection just seems to go on forever. If His Orangeness ever decides to dump Milania, this lady would make a perfect 4th wife. 

[Jeremy McCreary]

Couldn't resist making a pair of top-like spinners with CMs below their tips. ... Needed a name for this kind of spinner there, so for lack of a better term, I settled on "rigid conical pendulum with spin", or "RCPS" for short. Ugly, I know.

Writing a blog about these CM-below-tip spinners and needed a catchy name for them. Since we couldn't agree on a name, catchy or otherwise, I just went with "spindulum".