The unbalanced spinning ring; an explanation

[ortwin]

Iacopo said:

Hmmm, actually I like to think that it is the same thing we are talking about. A regular throw top, a tippe top, an unbalanced ring, euler's disk ... should all be just special cases of a general top following the same laws.

..., which can produce a thrust in the case of the ring, can't do that in the case of the normal spinning top, ...

In a "normal" spinning top the tip moves in circles/spirals, is that not the produced thrust?


ta0 wrote:

....
Yes, but there is no clear relation with the case of a tippe top.

It is not originally my idea that that the rising of a regular top and the behavior of a tippe top are related.
For example in that book that goes with that lecture gyroscope the same  explanation is used. It does not go very deep though, I did not see much use in translating it. I could do if you want. Link to pages of that book.






I hope that link works for you, I can't upload images as I usually do, there seems to be a server problem.

[Iacopo]

In a "normal" spinning top the tip moves in circles/spirals, is that not the produced thrust?

It is not originally my idea that that the rising of a regular top and the behavior of a tippe top are related.

Yes, but what is the origin of that thrust ?
I know two explanations for the rise of the "normal" spinning tops:
1- The tip, rolling on the ground, working like a traction wheel, could push the top in the direction to accelerate the precession, in which case a torque in the top is produced, that, through a gyroscopic effect, makes the top to rise in sleeping position.
2- The rolling resistance at the tip too produces a torque, with the same orientation as above, which too helps the top to rise in sleeping position.

I don't believe that these two explanations are suitable for the tippe tops, the eggs, and the unbalanced rings and discs, because, even if certainly there are similitudes in their behaviour with that of the "normal" spinning tops, there are also significative differences.
On the other hand, the explanation which I am giving in this thread, which is different from these two ones, is not suitable for the normal spinning tops.

If you think that there is something interesting in the book you found, you could say it, even with your words, if you don't want to translate the text.  I can see the page, but I can't understand what there is written.

[Iacopo]

I added a tip and a stem to the unbalanced ring.
The ring, with the tip, is very unstable and it spins only for a few rounds before to topple, but this is sufficient for to observe something.
This is how the tipped ring spins:

https://youtu.be/G8V2C9TPDcA

The ring is unbalanced, so it wobbles.  It tends to wobble staying leaned towards the light side. The effect of the added weight is not that to pull the ring away from the spin axis, at the countrary, the side with the added weight becomes nearer to the spin axis.  This should be not surprising, many tops spin in this way, but I will add more about this later.  The explanation, again, could be that the tipped ring is spinning in counterphase, or that its center of mass wants to stay in the rotation axis.



The direction of the tilting is the same in both cases, with and without the tip, it is the direction that makes the heavy side to rise up, (pics below).
The difference between with and without the tip, is that, with the tip, the ring doesn't tip over completely, it only tilts a bit.

To understand why some unbalanced tops spin staying leaned towards their light side, is propaedeutic to the understanding of the unbalanced ring, because the causes of the initial tilting are the same. Then, in the ring, the shifting contact point makes possible the continuation of the tilting, (the rolling), until a complete overturning.

[ortwin]

Google translate did its thing with the two pages I uploaded before in German language.

[Iacopo]

Google translate did its thing with the two pages I uploaded before in German language.

I didn't know that Google could translate the text from pictures. Very nice!

The explanation for the rise of the "gaming top" seems basically like the first one I gave. The rolling tip that "forces the top to rotate around the vertical axis" was, in my words, the tip that, acting like a traction wheel, "accelerates" the precession.
The text mentions a "theorem of the parallelism of the axes of rotation in the same direction".  I never heard of something like that, do you know what is it ?

It says very little about the tippe top. To say that "the rising of a regular top and the behavior of a tippe top are related" is a bit vague, in any case it doesn't seem to me that the book makes to think in any way that the reason of the rise is the same, in the two cases. 

[ortwin]

"theorem of the parallelism of the axes of rotation in the same direction"

That is something I first read in that book I think. The Author gets to it in the first few pages and makes frequent use of it in the book.
In German that theorem looks like this:







Google translate gives this:







The translation could be better, I would say:



"Theorem of parallelism of the axes of rotation in the same direction:


A gyroscope behaves under the influence of a moment or a forced rotation in
such a way that the gyro axis tries to move via the  the shortest possible path to be parallel to the axis of the forced rotation."

[Iacopo]

"Theorem of parallelism of the axes of rotation in the same direction:

A gyroscope behaves under the influence of a moment or a forced rotation in
such a way that the gyro axis tries to move via the  the shortest possible path to be parallel to the axis of the forced rotation."

Thank you, Ortwin.  So it is a rule about the gyroscopic motion. In fact, if I apply a torque to the spin axis of a spinning body, that spin axis tilts in a direction to become parallel to the axis of the applied torque. I had to think about it but it is true.

[ortwin]

Today I found this article: Can we understand the tippee top?


It definitely is not able to clarify everything, but I think it can help.

[Iacopo]

Today I found this article: Can we understand the tippee top?

I almost had an headache trying to understand it. Apparently it is written with clarity, but the direction of the forces in play is not always so clear. There is some inaccuracy also.

When a regular top rises, one explanation, the most known one I believe, is as follows;
the tip, spinning with the top in tilted position, acts like a traction wheel, and pushes the top along the walking trajectory on the ground. The friction at the tip is important because it makes possible for the tip to adhere to the ground, so that the traction becomes effective. The traction is a force directed forwards, at the contact point, in the walking direction. Then there can be an inertial resistance to this traction, for example like when a throw top lands and starts to walk on the ground but the tip is still slipping before the top to reach its full walking speed.  The forwards traction, and the backwards inertial resistance which we can imagine like located at the center of mass, produce a torque on the top, which is in the direction to make the top to tend to fall backwards.  Like when someone on a motorbike accelerates, and the motorbike rears up. This torque, in the top, becomes movement 90 degrees later, and the top rises. If you think to it, you see that it works.

In the article only "friction" was mentioned for the regular top, not traction, so it is not clear what the author thinks about it. It is not an insignificant detail, because this might lead the reader to think to a force with opposite direction to the real one. In the figures 7 and 8 I see that a regular top and a tippe top are drawn in the same position and with the same spinning direction, and in both the figures there is written "frictional force into paper", so, the same direction in both the cases, but this is misleading, because reality is that the friction produces forces with opposite directions in the two cases, that in the direction to accelerate the rotation about the vertical axis in the regular tops, and that to brake it in the tippe tops, (which is especially evident in the second half of the spin, when the direction of the tippe top spin about the stem axis reverses).

Furthermore, in the tippe top, the friction force direction spins together with the top, and the movements of the top relatively to the applied friction force are quite slow, so that the associated gyroscopic forces must be weak. Particularly, when the rotation about the stem axis ceases, at that moment the position and direction of the friction force relatively to the body of the top is steady, the gyroscopic effect is cancelled completely in those conditions, and still the tippe top continues rising, even energically. The gyroscopic effect cannot tilt a rotation axis if there is no rotation about that axis. For this reason too I believe that the explanation for the rise of the regular tops can't be suitable for the tippe tops.   

[ortwin]

One thing that leads to views that differ here, I believe, is the way one thinks of friction and traction.

...

When a regular top rises, one explanation, the most known one I believe, is as follows;
the tip, spinning with the top in tilted position, acts like a traction wheel, and pushes the top along the walking trajectory on the ground. The friction at the tip is important because it makes possible for the tip to adhere to the ground, so that the traction becomes effective. The traction is a force directed forwards, at the contact point, in the walking direction. Then there can be an inertial resistance to this traction, for example like when a throw top lands and starts to walk on the ground but the tip is still slipping before the top to reach its full walking speed.  ...

The "traction wheel" explanation where there is no slipping, only walking, can not be  a very general one. It would require a very exact match of tilting angle, tip curvature, spin speed, precession speed, ... . I think in general the tip is slipping, it seldom just walks on the ground. While the tip is slipping there is a frictional force directed in the opposite direction of the relative speed of the touching objects. That is why in figures 7 and 8 the frictional forces point into the same direction.

..."frictional force into paper", so, the same direction in both the cases, but this is misleading, because reality is that the friction produces forces with opposite directions in the two cases, that in the direction to accelerate the rotation about the vertical axis in the regular tops, and that to brake it in the tippe tops, (which is especially evident in the second half of the spin, when the direction of the tippe top spin about the stem axis reverses).

...

Clearly you have a different view on the forces, but I do not really understand what that is, how do you get to the statement that friction produces forces with different directions in the two cases?
What I could agree to would be a statement like: ... The frictional force accelerates the rotation about the vertical axis (~precession?) in regular tops  and in tippe tops. The rotation about the vertical axis in the tippe top in fact does not change that much, it is the orientation of the stem axis that reverses.

...

Furthermore, in the tippe top, the friction force direction spins together with the top, ... Particularly, when the rotation about the stem axis ceases, at that moment the position and direction of the friction force relatively to the body of the top is steady, the gyroscopic effect is cancelled completely in those conditions, and still the tippe top continues rising, even energically. The gyroscopic effect cannot tilt a rotation axis if there is no rotation about that axis. For this reason too I believe that the explanation for the rise of the regular tops can't be suitable for the tippe tops.   

I surely do not have an absolute precise understanding of all things happening there, especially during that phase when the tippe top does not rotate about the stem axis. But before that phase I have some picture now and after that it is the picture of the rise of the regular top.
Plus I can see that there is of course lots of rotation about another axis in that intermediate phase which is clearly not stable and small disturbances point towards the stable situation with the elevated center of mass.

[Iacopo]

The "traction wheel" explanation where there is no slipping, only walking, can not be  a very general one....  I think in general the tip is slipping, it seldom just walks on the ground.

This is completely unimportant. What does it matter if the tip slips, or it doesn't ? If the resistance to the traction is strong enough, it will make the tip to slip backwards a bit, while walking. If the resistance is not strong enough, the tip could roll against the resistance without slipping. But in both the cases the resistance is there, with the same orientation. Nothing changes in the explanation I gave. It is not necessary to have slipping and dynamic friction for to have that resistance, in any case.

It would require a very exact match of tilting angle, tip curvature, spin speed, precession speed, ... .

This seems a strange statement to me. Generally tops walk apparently without slipping. If you replace the ball tip of a top and use a twice larger ball tip, at parity of spin speed and angle of tilting, the top now will walk at a double speed, and nothing will prevent the top from doing so, in fact this is what it does. So I don't understand what it should be this match you are saying about.   

how do you get to the statement that friction produces forces with different directions in the two cases?
What I could agree to would be a statement like: ... The frictional force accelerates the rotation about the vertical axis (~precession?) in regular tops  and in tippe tops. The rotation about the vertical axis in the tippe top in fact does not change that much, it is the orientation of the stem axis that reverses.

No, it's different in the two cases.
We agree that, when there is an inertial resistance, the traction in regular tops is in the direction to ACCELERATE the rotation about the vertical axis.
The opposite happens in the tippe tops; when the spin about the stem axis reverses, it is evident that the friction is in the direction to BRAKE the rotation about the vertical axis.

You can see it well enough in this video, starting from 2:16, when there is no rotation on the stem axis.
The video is in slow motion but it is still a bit too fast, set the reproduction speed at 0.25, so you can see it better. 

https://youtu.be/8kAX9AIS3Q8

I surely do not have an absolute precise understanding of all things happening there, especially during that phase when the tippe top does not rotate about the stem axis. But before that phase I have some picture now and after that it is the picture of the rise of the regular top.

I too can't say to really understand the tippe top, it is too much complicated, I have some ideas but not a complete and fully coherent theory about it in my mind. But certainly it doesn't move like a regular top. It is easy to fall in wrong conclusions if the observations are not accurate, incoherences are neglected, and vague rules are used in the reasoning.

[ortwin]

Iacopo, I think we have to go back a few steps to gain some solid ground, some things we can completely agree on, otherwise we will continue to missunderstand each other on this rather complicated topic.


I cannot understand for example what you mean by "resistance to traction" in the last post. And "... slip backwards a bit, while walking..." makes me think what exactly what you mean when you say "walking". I thought you might mean simply rolling, no slip? No?
Just now I found this topic where you talk about related things.It is from a time before I joined the forum, but of course that does not matter in any way. Now I have to first get (again) comfortable myself with the concepts of  "inertial resistance", couple, torque, force and their mutual relations in this setting. Before that I should shut up on this.


Iacopo: "...This seems a strange statement to me. Generally tops walk apparently without slipping. If you replace the ball tip of a top and use a twice larger ball tip, at parity of spin speed and angle of tilting, the top now will walk at a double speed, and nothing will prevent the top from doing so,..."


Let me try to say why I see it differently: I envision a throw top (high CM) spinning fast and landing at about 45 degrees on the ground. The center of mass does not move much, but the top precesses while the tip goes in circles on the ground around a line that connects Cm and the ground. (Does this make sense up to now?)
The speed of  precession is dependent only on the general geometry of the top, its mass distribution and its speed - not on the exact radius of the tip at the contact point, correct?


Now on the other hand  if the tip would not slip, its speed going around the center of mass is fully determined by the speed of the top, the radius of the tip and the diameter of the circle it goes around. ( It is independent of things like the magnitude of the gravitational force for example)


That means if we want the two speeds that are determined very differently to coincide, we would need to adjust the indepent parameters in the two equations very carefully. But even than it would hold only for a moment because as the top slows down the precession asks for higher circular speeds of the top while the "no slip condition" will decrease its velocity.




About the friction forces in the figures 7 and 8 you say:


"No, it's different in the two cases.
We agree that, when there is an inertial resistance, the traction in regular tops is in the direction to ACCELERATE the rotation about the vertical axis.
The opposite happens in the tippe tops; when the spin about the stem axis reverses, it is evident that the friction is in the direction to BRAKE the rotation about the vertical axis."


Your argument seems to be, that because you observe certain effects, the cause can not be the one as indicated in the figures 7 and 8.
For me it seems very consistent and logical how he arrived at those figures 7 and 8 and now we need explain with them the different effects observed. 

I don't think I can see in that video you posted what you mean. A little slowing down of the speed around the vertical? For energy conservation reasons we know it must slow down a bit: the center of mass is rising and the rotational energy is the only reservoir this energy can come from. In some sense this corresponds to the regular top that has to go down a bit for the precession to speed up.

[Iacopo]

I cannot understand for example what you mean by "resistance to traction" in the last post. And "... slip backwards a bit, while walking..." makes me think what exactly what you mean when you say "walking". I thought you might mean simply rolling, no slip? No?
Just now I found this topic where you talk about related things.It is from a time before I joined the forum, but of course that does not matter in any way. Now I have to first get (again) comfortable myself with the concepts of  "inertial resistance", couple, torque, force and their mutual relations in this setting. Before that I should shut up on this.

I make a sample with the motorbike, which is simpler:
By "traction" I mean a force at the contact point of the traction wheel.
If the motorbike is accelerating, an "inertial resistance", which we can think about like located at the center of mass, opposes that acceleration. The traction and the inertial resistance together provide the torque which can make the motorbike to rear up during the acceleration, (the motorbike tilts backwards, clockwise in the photo). The motorbike can't rear up if it goes at constant speed because there is not inertial resistance at constant speed, the "traction" itself can't exist without a resistance to it. 
The torque in the direction to make a regular top to tilt backwards is the one needed for its rise. A torque is a couple of forces, so it is not sufficient the "traction", or the friction at the contact point. A second force, or resistance, should be defined.
By "walking" I mean the forwards movement of the top on the ground. It doesn't matter if there is a bit of slipping or not, I don't understand why you are so much focused on the issue slipping/no slipping.     

The speed of  precession is dependent only on the general geometry of the top, its mass distribution and its speed - not on the exact radius of the tip at the contact point, correct?

The angular speed of the precession, yes. But I was talking about the linear speed, not the angular speed, (the "walk"), and that is dependent on the radius of the tip too.

Now on the other hand  if the tip would not slip, its speed going around the center of mass is fully determined by the speed of the top, the radius of the tip and the diameter of the circle it goes around. ( It is independent of things like the magnitude of the gravitational force for example)

That means if we want the two speeds that are determined very differently to coincide, we would need to adjust the indepent parameters in the two equations very carefully. But even than it would hold only for a moment because as the top slows down the precession asks for higher circular speeds of the top while the "no slip condition" will decrease its velocity.

You seem to reason like if there were two separate and independent sets of parameters, which would determine two different speeds of the precession, which in turn, in some way, would constrain the tip to slip constantly while precessing.
Where does this strange idea come from ?
The parameters are all interdependent, and some of them are variable, so that they can adjust continuously and spontaneously to each other. For example, the diameter of the circle of precession is a variable parameter, not a fixed one.  The top is free to precess at the angular speed it wants, with the "walking" speed it wants, and this will determine the diameter of the circle of precession.  Not the opposite.

Your argument seems to be, that because you observe certain effects...
I don't think I can see in that video you posted what you mean. A little slowing down of the speed around the vertical? .

We were discussing about the direction of the friction force at the contact point of the tippe top.
It seemed obvious to me, (but maybe it wasn't), that for to see it, you have to look at the direction of the movement of the surface of the contact point of the top, relatively to the ground. Do you see in which direction it is slipping ? 

[ortwin]

Iacopo, today I don't think I will get to all the points that I think I need to mention, but over the next few days I hope can do it. Anyways:


You say: " It doesn't matter if there is a bit of slipping or not, I don't understand why you are so much focused on the issue slipping/no slipping.   "


I think slipping is the general, the "normal" case. That is when the friction force also has a quite simple form.


You say: "The angular speed of the precession, yes. But I was talking about the linear speed, not the angular speed, (the "walk"), and that is dependent on the radius of the tip too."


I wanted to keep it simple and considered only the precession, no linear speed. I hope that was not an oversimplification, but I saw enough examples to justify this I think.

You say: You seem to reason like if there were two separate and independent sets of parameters, which would determine two different speeds of the precession, which in turn, in some way, would constrain the tip to slip constantly while precessing.Where does this strange idea come from ?
The parameters are all interdependent, and some of them are variable, so that they can adjust continuously and spontaneously to each other. For example, the diameter of the circle of precession is a variable parameter, not a fixed one.  The top is free to precess at the angular speed it wants, with the "walking" speed it wants, and this will determine the diameter of the circle of precession.  Not the opposite.


I think there is only one set of parameters that determines the speed of the dynamic precession, the effect that keeps the top from falling down. But my point is that when using those paremeters to arrive at that precession speed , you cannot usually fulfill at the same time the requirements for the parameters that would lead to pure rolling of the tip without slipping.
And no, not all parameters are interdependent, the radius of the tip at the contact point plays hardly any role in determining the dynamical precession speed to keep the top at a given angle (lets say 45 degrees). But that tip radius plays a crucial role in determining the speed of the top that is asked to roll in circles around its CM  without slipping.
And also no, the diameter of the circle of precession is, at least in my picture, not a variable parameter. If I want the top to start at an angle of 45 degrees and the CM should not move sideways, then that diameter is very much fixed to a certain value. The top is not free to precess at the angular speed it wants, it needs to be a very specific speed to keep it from falling on the one hand and from rising on the other hand. But of course you know that very well.

[Iacopo]

I think slipping is the general, the "normal" case.

Why do you believe so ? You read it somewhere, or you saw it with your eyes, or it's your reasoning...?
Static friction exists too, and it works as well.

I think there is only one set of parameters that determines the speed of the dynamic precession, the effect that keeps the top from falling down. But my point is that when using those paremeters to arrive at that precession speed , you cannot usually fulfill at the same time the requirements for the parameters that would lead to pure rolling of the tip without slipping.
And no, not all parameters are interdependent, the radius of the tip at the contact point plays hardly any role in determining the dynamical precession speed to keep the top at a given angle (lets say 45 degrees). But that tip radius plays a crucial role in determining the speed of the top that is asked to roll in circles around its CM  without slipping.
And also no, the diameter of the circle of precession is, at least in my picture, not a variable parameter. If I want the top to start at an angle of 45 degrees and the CM should not move sideways, then that diameter is very much fixed to a certain value. The top is not free to precess at the angular speed it wants, it needs to be a very specific speed to keep it from falling on the one hand and from rising on the other hand. But of course you know that very well.

I don't understand where do you see a constraint that should force the tip to slip.
And I am not understanding very well what you are saying.
When you say speed, could you specify if you mean "angular" or "linear" speed ?
Also, what do you mean with "dynamic" precession ? 

I have to insist here, why do you think that "the CM should not move sideways", or that the diamater of the precession circle should be considered a fixed one ? It's not fixed. The fixed parameters are about the design of the top and its distribution of weights. Then we can decide arbitrarily a transient spin speed and a transient angle of tilting. These data will determine the other ones. Spin speed, angle of tilting and tip radius will determine the linear speed of the precession. Spin speed, weight, CM-tip distance and axial moment of inertia will determine the angular speed of the precession. In the end, the linear speed of precession and the angular speed of precession will determine the diameter of the circle of precession. It's that simple, what is your cut-off point, I don't understand.