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[Iacopo]

THE EFFECT OF THE SLOWING DOWN OF THE SPIN SPEED, (DUE TO AIR DRAG AND TIP FRICTIONS),
ON THE RISING OF THE TOP

When a top spins and precesses, like in the drawing below, the linear velocity of the top axis, (green line), along the precession trajectory, at the level of the tip, is higher than the linear velocity of the center of mass, (it traces a larger circle with the same angular speed).

In absence of frictions, the two linear velocities would be constant.
In this situation there would be no torque and the top would not rise nor fall.
The tip going faster than the center of mass, (as for linear speed), does not create a torque on the top axis;
an acceleration is needed for to create a torque.



Which is the case of real tops, their velocities are not constant.
Because of air drag and tip frictions, tops slow down while spinning.
The diminishing spin speed of the top causes a braking action at the contact point, so that the linear velocity of the tip along the precession trajectory is coinstrained to diminish, (if the tips doesn't slip, which is the most common case).

While the tip slows down along the precession trajectory, the center of mass would want to continue to go on with its constant linear speed, (inertial speed).

So a torque comes out from this situation.



This torque is in the direction to make the top to tilt more, not to rise.

So, the top slowing down works against the top to rise.
Air drag works against the top to rise.
Rotational sliding friction works against the top to rise too, by both the mechanisms explained, the one of this comment and the one explained in the video.
Rolling resistance is the only mechanism that makes the top rise, as explained in the video, but one effect of rolling resistance is to slow the spin speed of the top, and, in this sense, it fights against the top to rise.

Since real tops do rise if their spin speed is high enough, it can be supposed that the rising torque of the rolling resistance alone is stronger than all the other opposing torques together.

[ta0]

A lot of work and thinking went into that video explanation, Iacopo! Congratulations!

I have not commented yet, because I need to think more about it (and also read that paper that was mentioned above).

Jeremy, do you know if those who disagree about rolling resistance as the cause of the rising top, think the same as Perry, that the cause is the peg that, rolling on the table, " wants to roll the top faster than the precession lets it roll, so that it hurries on the precession, and therefore the top rises" ?  Is this the common given explanation by them ?

Yes, this is the standard explanation (Crabtree, etc.)

Your experience is with your low and heavy rimmed tops, while mine is mainly with throw tops. I will say that in mine the top rises and sleeps before reaching a stable rolling precession. The tip will be accelerating the precession during the rise (I think both sliding and rolling). So, I think Perry's explanation is correct for them, at least.

[Mermouy]

As usual your video is awesome and soooo interresting!
I will create a special "technical & theorical" page on spintricks as sonn I have time, (this will be an almost lacopo videos only page btw )

[Iacopo]

Yes, this is the standard explanation (Crabtree, etc.)

Thank you, Ta0.  Yes, I remember the explanation given by Crabtree is similar to that of Perry, but I didn't know about the others.

Your experience is with your low and heavy rimmed tops, while mine is mainly with throw tops. I will say that in mine the top rises and sleeps before reaching a stable rolling precession.

If the tip is slipping while rolling, it could mean that the tip is still accelerating the top along the precession trajectory;
in this condition, there is indeed a resistance, (inertial), of the center of mass to the push of the tip,  and actually this would cause a rising torque.

In the sample I showed in my video the acceleration phase is very brief because I used a wax spinning surface which provides high grip for the tip, but on more slippery surfaces, and especially when the top is started at higher RPM, like in the case of throw tops, the slipping/acceleration phase could last for more time. It is more than plausible to me that during this phase the top would rise above all for the reasons explained by Perry.

But I don't think very much that this is exactly what Perry intended;
he didn't talk about a slipping/acceleration phase, and it seems more like he intended the rules he explained to be valid in all cases, even with a stable rolling precession, without any slipping.

I believe Perry understood correctly great part of the process, he understood correctly the direction of the torque needed for the top to rise, he understood correctly that the processes between the contact points had to be responsible for this torque.
Then, maybe, at that time, the direction of the forces involved in rolling resistance was not known, (it is not very intuitive), so he couldn't find a better explanation than that he gave.
   

[Iacopo]

As usual your video is awesome and soooo interresting!
I will create a special "technical & theorical" page on spintricks as sonn I have time, (this will be an almost lacopo videos only page btw )

Thank you, Mermouy, I doubt I deserve an almost Iacopo videos only page as for "technical and theorical", but I'm glad you appreciated the video.

[ta0]

I read the paper by Cross. Very interesting. It's a perspective I had not seen and along the lines of the explanation you provide on the video.
He mentions two papers about rolling friction which I have not read (and I'm not very eager to read  ), so I guess his explanation of rolling resistance is correct. But my intuitive understanding of rolling resistance was different: I imagined that it came from the "sticking" of the surfaces, so it produced a downwards force behind the center of the wheel, not an upwards force in front. Anyway, either would produce the same torque. However, given the very small lever arm distance to the center, it's surprising to me that such force would cause the rising of the top.
On the other hand, it seems to me that a rolling "reaction" could explain the rise. Because the center of the top is rising, in principle there is an inertia and acceleration the rolling can act against. Because of the rotational inertia of the top, it pushes forward the tip by rolling on the surface, so the total rolling force is forward, even if there is rolling friction in the opposite direction. In which case the classical explanation of "speeding up precession" would still be valid for rolling tips.
The above was my intuitive representation, but it may well wrong.

[Iacopo]

I read the paper by Cross...
He mentions two papers about rolling friction which I have not read (and I'm not very eager to read  ).

Neither did I. Often these papers are too much complicated for an uneducated person like me. I learned the correct direction of the forces involved in rolling resistance from Wikipedia, it was eye opening to me because I realized soon that this could cause a rising torque.  The only other rising torque I found is that of air drag opposing the precession movement of the top, but this seems usually very week, and also distorted by the Magnus effect, perhaps it could have some influence only in tops walking rapidly along the spinning surface, because of a large radius of curvature of the tip.     

But my intuitive understanding of rolling resistance was different: I imagined that it came from the "sticking" of the surfaces, so it produced a downwards force behind the center of the wheel, not an upwards force in front. Anyway, either would produce the same torque.

I have read that there can be "sticking" too in the rolling resistance dynamics. But the most common cause seems the release phase not so complete and/or not so fast as the compression phase.

However, given the very small lever arm distance to the center, it's surprising to me that such force would cause the rising of the top.

I too was a bit surprised, for the same reason. Especially for more rigid materials like glass, ruby, steel.
The ball top rolling uphill, with the contact point shifted forward by about 1 mm, rises in about half a second.
So it seems like a microscopic shift could be enough for a top to rise in minutes.
I will think if I can make an experiment to understand better this aspect. 

On the other hand, it seems to me that a rolling "reaction" could explain the rise. Because the center of the top is rising, in principle there is an inertia and acceleration the rolling can act against. Because of the rotational inertia of the top, it pushes forward the tip by rolling on the surface, so the total rolling force is forward, even if there is rolling friction in the opposite direction. In which case the classical explanation of "speeding up precession" would still be valid for rolling tips.

I am still reasoning about it..