Tippe top; some observations, and an attempt to explain its behaviour

[Iacopo]

Recently I made an experiment with a tippe top, which I will describe later, and that led me to a possible explanation of its behaviour.

This is a slow motion video of a tippe top making its inversion:



I extrapolated some data from the video and put them in the following graph:



The highest rotational speed is about the vertical axis, (rotation axis), which remains always vertical, for all the duration of the spin.
The top tilts and inverts, but not the rotational axis.
The rotation speed decreases more rapidly when the top tilts faster.

There is some slow spin speed about the stem axis; in the beginning, the top spins against the ground. Then, the spin speed slows down until ceasing, when the top is tilted by about 90°, (but it can cease even sooner), then it starts spinning in the opposite direction, with the ground;
anyway, the spin motion, (about the stem axis), is always slow, and, whatever its direction, the top slides, always braking the fast rotational speed.

[Iacopo]

I made this animation: here, the point of view is spinning about the vertical axis at the same speed of the rotation, so the rotation about the vertical axis is cancelled in the video.
This can help showing the relationship of the movements between the top and the ground.

In the beginning, the top slides, spinning against the ground.
Then the direction of the spin about the stem axis reverses, but the spin speed is always slow, and the top continues to slide.
The direction of the friction force at the contact points is always in the same direction, into the screen, before and after the reversal of the spin motion about the stem axis.

[ta0]

Great work, Iacopo!

It seems difficult to measure by eye the spin around the stem.

I'm surprised that the spin is so much slower than the rotation at the beginning and the end of the movement: to me they look similar.

I look forward to your conclusions.

[Iacopo]

Great work, Iacopo!

Thanks !

I'm surprised that the spin is so much slower than the rotation at the beginning and the end of the movement: to me they look similar.

You are right, they look similar.
I explain.
What you see at the beginning and at the end is not simply the speed about the stem, but the sum of the speed about the stem and that about the vertical axis.
This is not evident in the beginning, when the stem is still directed upwards and near the rotation axis.
But, as the top tilts more and more, (if you try taking the measurements from the video, you will see what I mean), it becomes easier, and necessary, to distinguish the two speeds separately.

Maybe this can help; in this animation a ball is rotating about the vertical axis, there are no other movements, the stem is steady and not rotating. Anyway, when the stem is near the rotation axis, in the beginning, it looks like to rotate, (but that is the rotation about the vertical axis, it's not the stem):

[ta0]

Maybe this can help; in this animation a ball is rotating about the vertical axis, there are no other movements, the stem is steady and not rotating.

I don't see it like that. A pure rotation around the vertical with no rotation around the stem would be just like a wobble, with a mark on the stem always looking in the same direction. If the earth didn't spin around its axis and only rotated around the sun, a day would last 1 year.

[Iacopo]

I don't see it like that. A pure rotation around the vertical with no rotation around the stem would be just like a wobble, with a mark on the stem always looking in the same direction. If the earth didn't spin around its axis and only rotated around the sun, a day would last 1 year.

When I made the last animation, I set the speed of the rotation about the vertical axis to 12 RPM and that about the stem to 0 RPM. The stem is not spinning, but just following the movements of the ball about the vertical axis.
You know that because the side of the stem facing the "north pole" of the ball is always the same side, at all the angles of tilting.

For to have the effect you say, I should set the speed about the vertical axis to +12 RPM and that about the stem to -12 RPM. So the stem would appear not rotating, but just wobbling, when it is near the vertical axis.

[ortwin]

I explain.
What you see at the beginning and at the end is not simply the speed about the stem, but the sum of the speed about the stem and that about the vertical axis.
This is not evident in the beginning, when the stem is still directed upwards and near the rotation axis.
But, as the top tilts more and more, (if you try taking the measurements from the video, you will see what I mean), it becomes easier, and necessary, to distinguish the two speeds separately.

Iacopo, I also have trouble seeing how the spin speed and the rotation speed can be substantially different in the situations when the stem is almost perfectly aligned vertical axis. With larger tilt it is very clear, but with negligible tilt at the beginning and the end I can't see neither theoretical reasons nor observations that would justify the huge difference (factor ~7?) in the values.

[ta0]

Iacopo: for dynamic reasons, you want to measure the spin around the axis in reference to an inertial frame.
If you have a bar connecting the object to the center of rotation and fixed to the object, you are forcing the object to rotate one time around its axis for each revolution. The torque required will depend on the inertia to revolve the object around the center (due to radius and mass) plus the inertia to spin the object around its axis (due to its moment of inertia).

The moon faces the earth the same way during its orbit. The moon has a spin around its axis equal to the orbital revolution (27.3 days).

EDIT: You can also work on a rotating frame and then your definition of spin would be correct. But it gets more complicated (e.g. Coriolis force).

[Iacopo]

with negligible tilt at the beginning and the end I can't see neither theoretical reasons nor observations that would justify the huge difference (factor ~7?) in the values.

You can also work on a rotating frame and then your definition of spin would be correct. But it gets more complicated (e.g. Coriolis force).

I took the measurements from two different frames of reference, which is necessary if I want to know the speed of the stem indipendently from the speed about the rotation axis.

If I took all the measurements from the inertial frame of reference, the stem would have a sum of the two speeds, in different percentages, depending on the angle of tilting, maybe perfect for computer calculations, also depending on the aims of these calculations, but unnecessarily complicated in my case.

I took the measurements of the speeds for to calculate the speeds of the surfaces at the two contact points, for to know if the top was sliding, and for to know the direction of the friction force, at the various angles of tilting.

[Iacopo]

Friction causes the tippe top inversion. But, in which way ?

The tippe top rotates rapidly about the center of mass.
In the past I made a test for to see if this was true, and I found that the top spins exactly about the center of mass, or really very close to it.
The contact point is dragged in circular motion about the rotation axis, so the top slides.
In the image, the top rotates about the vertical axis clockwise, so the direction of the friction force is into the screen.
The difference of speed between the two surfaces of the contact points diminishes by the time, but it never ceases, nor inverts, so the direction of the friction force is always the same, for all the duration of the spin.

We know that the friction at the contact points is the cause of the tippe top inversion; we know it because the tippe top spinning on slippery surfaces does not invert.

Anyway it is difficult to understand in which way exactly the inversion happens;

The direction of the friction force, (into the screen, in the image), seems completely unrelated to the direction of the inversion, a rotation of the top with the stem moving downwards: this rotation happens on a plane which is orthogonal to the direction of the friction force, so the friction force cannot cause it directly.

[Iacopo]

What gyroscopic effect ?

It would be very tempting to believe that there must be some gyroscopic effect in play, making it possible.
In fact, in spinning bodies, the gyroscopic effect makes movements to happen along planes which are orthogonal to the direction of the applied force.
The movement of the inversion would be a vertical precession, (like someone said).

Anyway, in this case, this is impossible, for the following reasons:

1. In a gyroscopic motion, the body must spin relatively to the direction of the applied force. But in the tippe top the direction of the friction force spins together with the top, at its same speed, so that the top is steady relatively to it. The gyroscopic motion cannot take place in these conditions.
2. The gyroscopic effect would tilt the rotation axis, not the stem alone.
3. Even ignoring the two first points, if we consider the directions of the rotation and of the friction force, and we apply the rules of the gyroscopic motion, we would find out that the stem would move upwards, not downwards, and this is wrong.
4. If then we consider the spin about the stem axis, instead of the rotation about the vertical axis, we can see that the friction force is in the direction to slow down the spin speed, (which in fact it is what it happens), and not that to cause a gyroscopic effect.

Dropping the idea of the gyroscopic effect makes the reasoning a bit simpler, but the mystery thickens:
what causes the tippe top inversion ?

[Iacopo]

Tippe top precession

By the way, I don't want to mean that a tippe top cannot precess;
it can, if it is spun like a normal spinning top, in which case the top walks, without slipping, in a dynamics totally different from that of the inversion.

Interestingly, the tippe top precesses in a direction opposite to that of the spin.
If the top spins clockwise, it precesses counterclockwise.
This depends on the fact that the tippe top is a self-righting top, so the torque on the top due to gravity has an opposite direction compared to that of normal spinning tops, (which spin and precess in the same direction).

[ta0]

I'm reading this on my phone in a hotel. I cannot check everything you wrote but it sounds reasonable. I look forward to your explanation of the Tipe Top.

[ortwin]

Before I go into really looking in any depth at the analysis of Iacopo in this topic, I feel I need to state shortly what I personally was left with to believe to be true after one of our last rounds discussing the topic:
I believe the four following cases show in essence the same effect:

• the inversion of the Tippe top
• the rising of a top with a rounded tip
• the behaviour of a spinning unbalanced ring
• the rising of a spinning egg

Looking quickly through our discussions from last time, I see that we did not agree on much, especially not on my statement above. So I would need to go in more detail into the last thread and especially into your analysis in this topic to try to understand again the differences of our views. Lets see how far we get this time ........

[Iacopo]

I believe the four following cases show in essence the same effect:

• the inversion of the Tippe top
• the rising of a top with a rounded tip
• the behaviour of a spinning unbalanced ring
• the rising of a spinning egg

The same effect in the sense that all their centers of mass rise.
The dynamics of the tippe top and of the egg seem similar to me, but especially that of the rising top with a rounded tip is very different from the other three.