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[Jeremy McCreary]

For how much time your tops can stay slipping, after you launch them ?

With a low-friction ball tip on a fixed table top, I'd say up to several seconds in some cases. But I have tops I can launch at 5,500 RPM with a single twirl.

You mean that the top slips forward, along the temporary descents of the arena... ? (Since it is the arena that pushes and makes the top to spin).

With a ball tip, slip can be in any direction relative to the path taken by the contact point. I think the positive slip heard in the video closely paralleled the observed path most of the time. A sudden increase in friction coefficient would have accelerated the top mostly forward along the path.

The arena pushed on the top through its tip in a number of ways. In the rolling regeneration style shown in the video, the user-imposed pushes mainly served to turn the top at the bottom of each descent and then lift and turn it to the start of the next. Gravity and various tip processes dominated the descents, though not necessarily in straightforward ways.

[ta0]

My intuition is that most tops wtih ball tips will get to the rolling condition pretty fast. I'm very curious to see what results Iacopo gets for his tops.

I believe that if the precession rate driven by the rolling is faster than the free precession of the top, it will make the top rise.

[Iacopo]

For how much time your tops can stay slipping, after you launch them ?

With a low-friction ball tip on a fixed table top, I'd say up to several seconds in some cases. But I have tops I can launch at 5,500 RPM with a single twirl.

I agree that if the top is launched with very high speed, it could stay slipping for some seconds. 
The speed of the top should be observed, not the vibrating sound from the tip, this sound depends on the high spin speed and not on the acceleration.

[Iacopo]

My intuition is that most tops wtih ball tips will get to the rolling condition pretty fast. I'm very curious to see what results Iacopo gets for his tops.

My sensation is that my tops get to the rolling condition almost immediately after they are spun.
I sold my top Nr. 25 some time ago, at present I don't have other tops with an external ball tip, so it could take some time before I can test it.

I believe that if the precession rate driven by the rolling is faster than the free precession of the top, it will make the top rise.

I too believe the same.  But it seems to me that the positive slipping phase lasts generally for a very short time.
I don't think that the rolling could continue to accelerate the precession, after the acceleration phase with positive slipping is finished.  From that moment on, since the spin speed is decreasing, and the linear speed of the tip too is decreasing, I believe that the tip is decelerating, and not accelerating the precession.   

[ta0]

I don't think that the rolling could continue to accelerate the precession, after the acceleration phase with positive slipping is finished.  From that moment on, since the spin speed is decreasing, and the linear speed of the tip too is decreasing, I believe that the tip is decelerating, and not accelerating the precession.   

The tip is decelerating but still driving the top faster than what it wants to precess.

The top has a natural free rate of precession for each spin rate and inclination angle. The rolling condition can be higher or lower than this, because it also depends on the size of the ball tip. If the rolling condition sets a forced precession that is higher than the free precession, the top will rise. The rolling will slow down because of both the decrease of spin rate (rotational energy converted to potential energy) and the decrease of lean angle. But as long as the forced precession is faster than the natural precession, the top will continue to rise. This could last a long time.

[Iacopo]

The rolling condition ... also depends on the size of the ball tip. If the rolling condition sets a forced precession that is higher than the free precession, the top will rise... This could last a long time.

I can't figure out which forces exactly could cause a forced precession.
If it is through the rolling resistance of the ball tip, in this case we are talking about the same thing.

Anyway I suspect it is not rolling resistance but something else, because Perry doesn't mention it, and because he instead talks about the tip as a traction wheel, pushing the top forward;
It makes me think to a biker opening the throttle and doing a wheelie; in fact the motion of a biker tending to fall backward applied to a spinning top would make the top rise. It would be perfect. 
But the tip is not accelerating the top forward, so it doesn't work in this way.
A bike going at constant speed is not doing a wheelie.
So, if not transational inertia, what else could push the top backwards while the tip is pushing it forward ?
Where is the resistance ?

Could the precession itself pose, in some way, a resistance to the traction tip ?

I may be wrong, but I don't see how. I believe that, if I replace the ball tip in a top, and I use a double diameter ball tip, this larger ball will still let the top free to continue to precess at the angular speed it wants. Nothing prevents the top to walk at a double speed, and make circles of precession twice larger, because of the larger tip;
the angular speed of the precession could be a bit slower because the increased centrifugal force would lower the torque of the precession, then there could be faster rising associated to higher rolling resistance.
It's different from a gyroscope, where exactly the lack of this freedom of motions, (because of the gimbals), makes possible to force a precession by pushing on a gimbal. 

But I am not totally sure, because the measurements of the rolling resistance seem too low for rolling resistance to be the only one cause of the rise.
I will perform the test to see whether there is some slipping of the tip during the rise. 

[ta0]

I believe that, if I replace the ball tip in a top, and I use a double diameter ball tip, this larger ball will still let the top free to continue to precess at the angular speed it wants. Nothing prevents the top to walk at a double speed, and make circles of precession twice larger, because of the larger tip;

Perhaps it never reaches the larger circle.  We agree that at the start the tip will accelerate the precession. Higher precession means a smaller radius to match the tip velocity. So before it reaches the free precession period with a matching large circle period it could get stuck in the faster forced precession with a matching smaller traveling circle. It's not clear to me if this condition is stable or not.

[Iacopo]

Perhaps it never reaches the larger circle. 

I believe that this would imply that something particular is happening, something that at present I don't understand.
Before to get an headache, I first will see whether there is really slipping in a precessing top.
I am modifying one of my tops, so it can spin with a ball tip; I will make the test today, or the next week.

[ta0]

Here is a hypothesis that perhaps can be tested experimentally.

We know that a top on a surface with zero friction will precess around its center of mass. So the tip will describe a circle determined by the height of the center of mass and the leaning angle.

If that surface is swapped with one with a lot of friction but the radius of the ball tip is such that the rolling condition is fulfilled for that circle, nothing would change.

My hypothesis is that if the tip radius is larger that this, the top will rise (and spiral in). If the tip radius is smaller, it will fall (and spiral out)

[Iacopo]

I am surprised, I didn't believe it but it's true.
The top slips !
In this first test the top, spinning at 2208 RPM (average), inclined by 10.25° (average), with the 3/16" carbon steel ball,
made one turn of precession in 11.1 seconds, by 408.5 spin revolutions.
The calculated diameter of the track should have been mm 346   
The observed "diameter" of the spiral was between mm 304 and 316

This was not caused by the initial acceleration, which was not included in the examined time lapse, and lasted about 0.5 seconds. 
Since the balance of the top was decent but not perfect, next I will try to see whether all of this has something to do with unbalance.

My hypothesis is that if the tip radius is larger that this, the top will rise (and spiral in). If the tip radius is smaller, it will fall (and spiral out)

As for my experience, you might be right, or, at least, not far from the truth.
Still the causes are not so clear to me.

[ta0]

Experimental data is always great!

What's the diameter of the ball tip?

[Iacopo]

What's the diameter of the ball tip?

It's 4.76 mm.

[Jeremy McCreary]

In this first test the top, spinning at 2208 RPM (average), inclined by 10.25° (average), with the 3/16" carbon steel ball,
made one turn of precession in 11.1 seconds, by 408.5 spin revolutions.
The calculated diameter of the track should have been mm 346   
The observed "diameter" of the spiral was between mm 304 and 316

Not sure it's meaningful to speak of an "average speed" over an 11 s segment of a nonlinear spin decay curve -- especially if you're just taking the arithmetic mean of the endpoint speeds. Might be close enough near critical speed, as the curve's probably roughly linear there, but might not be safe early on.

[Iacopo]

Not sure it's meaningful to speak of an "average speed" over an 11 s segment of a nonlinear spin decay curve -- especially if you're just taking the arithmetic mean of the endpoint speeds. Might be close enough near critical speed, as the curve's probably roughly linear there, but might not be safe early on.

In this case I counted the spin revolutions from the video, (it is accurate but it takes some time to do so), then I calculated the average speed, knowing the duration of the precession.  But even taking the arithmetic mean of the endpoint speeds the result would be very similar, because the change of the deceleration is very little in my tops, for a 11 seconds segment.

[Iacopo]

The same top, with some added weight for making it unbalanced, didn't behave very differently:
At 2126 RPM (average), tilted by 10.35° (average), made one turn of precession in 10.72 seconds, by 379.8 spin revolutions.
Calculated diameter of the precession track, mm 325
Observed "diameter" of the spiral, mm 296 - 301

The top produced a vibrating noise and the track tended to be dotted, because of the unbalance.
I expected more slipping in these conditions but there is even less slipping than in the first test instead.
So it seems that it is not unbalance with the vibrating tip that causes the slip.

I have a few ideas so I will continue making some other tests in the following days.
I used the top with the carbon steel ball in the first two tests;
next I will try with the teflon tip ball, (same diameter).