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[Iacopo]

Tops slip while spinning.
They slip backwards, a bit like a starting dragster that skids, so that the distance traveled by spinning tops is smaller than it would be without slipping.
The strange thing is that this doesn't happen only when the top is spun, at the start, but also during the spin.
I didn't believe this could be true, but Ta0 persuaded me to try and see by myself, and, with my big surprise, I saw it.

I started wondering, why does this happen ?
Is there some sort of force that pushes the top back, or that brakes the top, making it slip backwards ?
Maybe something related to the gyroscopic motion, the resistance of the flywheel to be tilted... ??

The first experiment is intended to reveal this force:

I put my concave glass mirror in the trifilar pendulum, so that the mirror rotates when a torque is applied to it.
The trifilar pendulum is very sensible and a very weak torque is sufficient to rotate it at least for a few degrees, because the three threads of the pendulum are long and very near each other where they start, overhead.
So certainly it has no difficulty to reveal a force strong enough to make a top to slip.
I added a vertical needle below the mirror, and a slot, with the aim to prevent the side oscillations of the pendulum when the top precesses on the mirror; the slot was larger than the diameter of the needle and there was very low friction because of it.

I spun a top, (with a 3/16" carbon steel ball), on the cleaned and degreased mirror.

If there is a force pushing the top backwards, or braking the precession, this will show as the mirror rotating in the opposite direction of the precession.
 
In other words, instead of the top slipping backwards, we should see the top not slipping and the mirror rotating instead, pushed by the spinning ball tip traction, and by the top, that, for some reason, refuses to reach its natural rate of precession.

This is what I saw, I will continue in the comments:
 


I can see the video in YouTube but not here, I don't know why. If you can't see the video here, you can see it in YouTube.
 

[Iacopo]

The torque needed for to rotate the mirror by 5 degrees with the top on it is about 0.00003 Nm.
The force needed for to make this top to slip while spinning is about 0.05 N, (I will show you this in the next experiment);
this supposed force, applied to the mirror by the spinning tip, at about 15 mm from the center of the mirror, and pivoting on the needle at the center, would produce a torque on the mirror of about 0.007 Nm.

This should produce a clear counterclockwise rotation of the mirror, (the top spins and precesses clockwise).
But you can see in the video that there isn't any tendency of the mirror to rotate in any direction;
there are some back and forth oscillations, but on average the mirror stays in the same position as when the top does not spin.

This means that there isn't any force braking the precession.
The reason of the slipping must be something else.

The result of this experiment is consistent with two other observations:

1 - If the backwards slipping of the top was due to a braking force coming from the top itself, it could be supposed that with a slippery tip, (a teflon one), the top should slip more than with a non slippery one, (carbon steel).  But in my experiments the amount of backwards slipping is about the same with the two tips.

2 - 0.05 Newton is a relatively strong force; such a force, applied to this 119 grams top, at the height of the flywheel, in backwards direction, should make it rise almost instantly. But this does not happen, and the top rises very slowly. 

Even theorically, I find difficult to imagine a force braking/pushing the top backwards. Where would this force come from ?

At this point I believe that there isn't any force braking the precession of tops, and that the reason of the backwards slipping is another one. The next experiment will show something about a possible alternative explanation.
   
 
 

[Iacopo]

The second experiment was started with the intention to cancel the precession in a normal spinning top, to see whether this would remove the backwards slipping.  In fact, the experiment with the ball top showed no backwards slipping, and the ball top walks along a straight trajectory, without precessing, so maybe the two things stay together.

For to eliminate the precession in a normal top I simply make the top spin on a slightly tilted glass pane;
spinning the top in a slightly tilted position, so to have its center of mass on the vertical line passing through the contact point, eliminates the torque from gravity which produces the precession, so the top walks straight, with no precession.



The glass pane was tilted by 4°. 
It is not too difficult to spin the top in that position, because the top itself tends to dispose in that way spontaneously, because of the rising effect, which is still present. The position in the drawing above is stable and it is the equivalent of the "sleeping position" for a tilted spinning surface.

The top in this position, anyway, does not want to go downhill.
It wants to go sideways, at 90° from the downhill direction, (this is the no slip direction):  in that direction the top walks along a horizontal line.

But the top is unable to maintain this no slip trajectory and it slips downhill a bit, while it spins, so the resultant trajectory is a mix of the no slip trajectory with a bit of downhill slipping.

I checked backwards slipping and I found no backwards slipping for the top spinning in this way;
anyway the downhill slipping makes the picture a bit confused, so this result doesn't seem very interesting.

Then I spun the top on different tilted surfaces, (glass, with and without grease, paper, silicon), with two different tips, (teflon and carbon steel), and I discovered something interesting, which I am going to explain.

 

[ta0]

All these experiments are very original 

Does the top on the torque balance right itself? I don't see how that could happen without applying torque to the balance. A longer video would be helpful to convince myself what is going on.

Then I spun the top on different tilted surfaces, (glass, with and without grease, paper, silicon), with two different tips, (teflon and carbon steel), and I discovered something interesting, which I am going to explain.

You got me intrigued. I'm eager to hear your results. 

[Jeremy McCreary]

Ingenious experiments! Still digesting your results, but I have a cautionary comment...

From my own experiments, I think it's a mistake to equate travel and precession in the general case -- even when the contact trace is a circle.

Textbook treatments of top precession assume a tip contact fixed in space and a constant total angular momentum. In other words, they enter the movie after the acceleration phase leading up to that total. But that's where all the slip comes in.

Once the tip contact starts to translate, with or without slip, other tendencies start competing with the top's tendency to precess. The balance of tendencies controls both the slip and the contact's trace across the table.

The common real-world case of a top with its CM stuck on the vertical orbited by its ball tip is already a substantial departure from the textbook case and must be analyzed as such. Having seen no such quantitative analysis, I conclude that it's too scary even for the pros. Which should give us pause.

Further complications arise when the CM's path deviates from the fixed circular path associated with textbook precession.

I'm still trying to understand the dynamics of a traveling top, but we mustn't neglect the linear inertia and momentum associated non-circular CM motion. For straight-line CM travel, the resisting inertia is measured solely by the top's mass.

[Iacopo]

This is the top spinning on the tilted glass mirror.
When the top spins in this way, there is not backwards slipping.
The lines in the background are vertical, and in the two mirrors there is the side view of the top, (which is the exact side view when the top passes in correspondence of the vertical red lines).

[Iacopo]

Does the top on the torque balance right itself? I don't see how that could happen without applying torque to the balance. A longer video would be helpful to convince myself what is going on.

Yes, the top rises normally. 
In the video I posted, the top rised by about 2.5° during the six precession revolutions.
I didn't make calculations but I suppose that the torque for this rise could be very approximately 0.00001 Nm.
The resistance of the trifilar pendulum to rotation is about 0.00003 Nm with a shift of 5° and maybe 0.00030 Nm with a shift of 60°, so it would withstand the torque for the top to rise without difficulty, (if this resistance is necessary, because I don't understand the dynamics involved yet).

At least part of this rising torque, should be due to rolling resistance, which works through vertical forces, so I suppose that the trifilar pendulum would not reveal the rolling resistance in any case, nor would prevent the top from rising in any case, because rolling resistance needs a vertical resistance, not horizontal.

But I made the test thinking to the slipping issue, and not very much to rising.
If the backwards slipping is considered as a proof of the existence of a force braking the precession, (which would have the correct direction for causing the rise of the top), it makes sense that the magnitude of this force should be at least sufficient to cause this backwards slipping. 
But this is not the case, because the force necessary for the slip in the tested top is about 0.05 N, the pendulum should see this force very easily, instead it doesn't see it at all.  Something is not right here.

[Iacopo]

... enter the movie after the acceleration phase leading up to that total. But that's where all the slip comes in.

I too believed the same.
But it seems instead that tops continue to slip backwards even after the acceleration phase.
Tops decelerate, and, at the same time, they slip backwards.

If you don't believe it, it is understandable, because I too didn't believe it.
But I observed it in my tops and I know that it is true.

Now, question is, what causes this odd behaviour ?
Is there some force pushing the top backwards, braking the precession, while the top spins ?
The trifilar pendulum says that there isn't any significative force in this direction.
So the cause of the slipping should be something else. 

[Iacopo]

I applied a dynamometer to the tested top, (the top was not spinning), to see how much force is needed for making it starting to slip.
Depending on the kind of the tip and of the spinning surface, the readings were approximately between 0.06 and 0.30 Newton.
The top weighs 119 grams.

When the top spins, less force should be enough for causing the slipping, since dynamic friction is lower than static friction.

When the top spins on a tilted surface, there is a force pushing the top downhill;
this force can be calculated;
In this test, the glass mirror was tilted by 4°.  The top weighs 119 grams.

119 x (sin 4°) = 8.3 grams force = 0.08 Newton.

The drawing below represents the glass mirror seen from above.
The arrows represent the trajectories of the top on it.  The top, spinning clockwise, would want to go towards the left, but, while it spins, it slips downhill;  depending on the amount of the slipping, the direction of the top changes.

The top, with the teflon tip, slips more than with the carbon steel ball, (in fact teflon is more slippery).
If there is a thin layer of grease on the mirror, the top slips almost directly towards the downhill direction, especially with the teflon tip.

The amount of slipping for the top with the carbon steel ball tip spinning on the clean, degreased mirror, is dependent on speed:
At slow speed there is little slipping, (direction of spinning at 5° - 8° from the no slip direction).
At high speed the slipping increases and the direction falls to 13° - 17° from the no slip direction.

The top, with the teflon tip, walks at 20° - 25° from the no slip direction.
I didn't notice changes of direction related to speed.  This fact by itself is curious. I don't imagine the reason of this.


 
 

[ta0]

The first time I became aware that a top travels approximately perpendicular to the slope was while trying to level the mirror base of a Quark top. It's something that at first sight is counter-intuitive. I have found it mentioned in some papers (e.g. about the traces left by a top), but I think your study of the actual angle depending on the tip friction is new and original. Great work!

[Iacopo]

The first time I became aware that a top travels approximately perpendicular to the slope was while trying to level the mirror base of a Quark top. It's something that at first sight is counter-intuitive.

It's a bit counterintuitive but not illogical; given the position of the contact point relatively to the "traction wheel" of the ball tip, (the red circle), when the top spins the traction is directed sideways, (view from above).
If the top appears exactly vertical when seen from the side view from the downhill direction, (not the side view in the drawing), the no slip direction of the walking top will be exactly perpendicular to the slope.

[ta0]

It's a bit counterintuitive but not illogical; given the position of the contact point relatively to the "traction wheel" of the ball tip, (the red circle), when the top spins the traction is directed sideways, (view from above).

It was logical once I thought about it. But if you have a top that is going towards the edge of the platform, there is a strong urge to lift that edge.

[Iacopo]

if you have a top that is going towards the edge of the platform, there is a strong urge to lift that edge.

In fact I never saw a top walking uphill.  Downhill, and less or more sideways, but never uphill.
I suppose that the pure no slip direction at 90° from the slope is not reachable neither, because I never saw a top going in that direction, they always slip downhill, at least a bit, while they spin.

[Jeremy McCreary]

The first time I became aware that a top travels approximately perpendicular to the slope was while trying to level the mirror base of a Quark top. It's something that at first sight is counter-intuitive. I have found it mentioned in some papers (e.g. about the traces left by a top), but I think your study of the actual angle depending on the tip friction is new and original. Great work!

Agree, great work.

I'd love to see those articles. Citations?

[ta0]

In fact I never saw a top walking uphill.  Downhill, and less or more sideways, but never uphill.

Neither have I. But if you tilt it up the top will not go back to the center but travel parallel to the edge. If you are trying to level a platform (specially a circular one like the one for the Quark) that is a problem.

I'd love to see those articles. Citations?

Lourens mentioned this paper in the Gyrograph thread:
C. Barus: 'Science', 1896, part IV, page 444-446 'A curve-tracing top. Available here: www.archive.org/stream/science41896mich/science
Barus is concerned with a top precessing in an inclined plane and the explanation he gives is interesting but when I originally read it was not too convincing.