# Motion of a Spinning Coin

hapefish
I have several questions about the motion of spinning coins. I won't get to them all in this post, but hopefully some of you can help get me started. (Note: I am a mathematician, not a physicist - that might help color your responses.)

First I want to think about spinning an idealized penny - a perfect disk with radius, R, width W, and constant density, ρ.

1. Would you expect such a coin to tend to spin on the flat of it's edge or on one of the corners? I would expect the corners for two reasons: First, friction is minimized when the coin is on a corner and second the center of mass is at its highest point when the coin spins on a corner:

1'. I have noticed that objects spinning at very high speeds seem to move their center of mass to the highest point possible. Two prime examples are the Tippe Top and the hard-boiled egg. Is this a general phenomenon, or are these two examples special cases? I would expect it to be a general phenomenon due to the very large angular velocity causing an upward force on the object, but I really can't say why...

2. I would expect the coin to ideally spin in a perfectly balanced state; however, as the coin spins I expect friction to cause small perturbations in this state making the coin wobble from a perfectly balanced state. When such a wobble occurs, I would expect the coin to stand itself back up into an upright state if the angular velocity is large enough and start the slow descent downwards if the angular velocity is not large enough. But, the question is: What is "large enough?" If given a perturbation that causes the coin to tilt to an angle of θ with the z-axis, is it possible to determine the angular velocity needed to "stand the coin back up?"

Thanks for any insight you can give me!

1) Your assumption that friction is the smallest on the edge is not necessarily true. In the most simple model the area does not make a difference.

1') Objects that are spinning do not have a tendency to bring the centre of mass to the highest point. Your mentioned examples are special cases that took the experts a lot of time to understand. They depend on an interaction of the rotation with the friction on the surface leading to something called precession. In special cases this can cause the object to find a configuration that is more stable even though it has a higher centre of mass.

2) The coin starts its descend immediately, but it doesn't tip because the torque induced precession causes the axis to rotate instead of the coin to tip. http://en.wikipedia.org/wiki/Precession

hapefish
1) Your assumption that friction is the smallest on the edge is not necessarily true. In the most simple model the area does not make a difference.

1') Objects that are spinning do not have a tendency to bring the centre of mass to the highest point. Your mentioned examples are special cases that took the experts a lot of time to understand. They depend on an interaction of the rotation with the friction on the surface leading to something called precession. In special cases this can cause the object to find a configuration that is more stable even though it has a higher centre of mass.

2) The coin starts its descend immediately, but it doesn't tip because the torque induced precession causes the axis to rotate instead of the coin to tip. http://en.wikipedia.org/wiki/Precession

Thanks for the response - I really appreciate it.

let me see if I can clarify (or have clarified) some of these comments:

It sounds to me that you believe the axis about which an object will spin is most strongly determined by the stability of its axes. I was going to approach this problem by finding the eigenvectors of the its moment of inertia tensor - is that reasonable?

You seem to believe that the coin enters precession immediately after being spun. Here are some observations I have made about the inconsistent precession of the coin, and I would love to know how others interpret these observations:
• If you observe a coin spinning, you will see that the angle of the rotational axis and the z-axis can vary quite a bit, with the coin "wobbling" to a comparatively large angle between the two axes and then "recovering" to a small (zero?) angle. This makes me believe there is something in the motion of the coin that is standing it up as it spins rapidly.
• As an experiment, I colored the bottom half of a coin black and watched it as it spun. What happened was the black portion of the coin quickly stabilized and the coin rotated on a single point on its edge instead of rolling along its edge. Occasionally the edge would roll slightly then stabilize again (corresponding with the wobbling effect I described earlier). Eventually the coin would slow to a point that it would roll constantly - at this point the side that it would land on was determined; however, I believe that prior to this point the coin had the opportunity to land on either side.

Thanks for the response - I really appreciate it.

let me see if I can clarify (or have clarified) some of these comments:

It sounds to me that you believe the axis about which an object will spin is most strongly determined by the stability of its axes.
I did not make that statement. It is half true. An object can only have a stable rotation about its main inertial axes with the larges and the smallest moment of inertia. Otherwise small deviations quickly build up.

I was going to approach this problem by finding the eigenvectors of the its moment of inertia tensor - is that reasonable?

The coin is a special case, where this is probably not helpful. If we ignore the embossing, then the largest moment of inertia is around axis through the centre of the coin's faces pointing perpendicular to the surface. Apart from the rim, this is the reason why coins roll so well and don't tend to tip especially when they roll fast.

The smallest axis of inertia and the second smallest are degenerate (this is why the coins is special). Any axis from the rim through the centre of the coin has the same moment of inertia, this also stabilizes the coin against tipping sideways and makes it indifferent against rolling / shifting of the axis along the rim.

You seem to believe that the coin enters precession immediately after being spun.
Yes due to friction and slight tilting.
Here are some observations I have made about the inconsistent precession of the coin, and I would love to know how others interpret these observations:
• If you observe a coin spinning, you will see that the angle of the rotational axis and the z-axis can vary quite a bit, with the coin "wobbling" to a comparatively large angle between the two axes and then "recovering" to a small (zero?) angle. This makes me believe there is something in the motion of the coin that is standing it up as it spins rapidly.
• As an experiment, I colored the bottom half of a coin black and watched it as it spun. What happened was the black portion of the coin quickly stabilized and the coin rotated on a single point on its edge instead of rolling along its edge. Occasionally the edge would roll slightly then stabilize again (corresponding with the wobbling effect I described earlier). Eventually the coin would slow to a point that it would roll constantly - at this point the side that it would land on was determined; however, I believe that prior to this point the coin had the opportunity to land on either side.
This is pretty complicated. It may or may not be the case, that the coin has a tendency to stand up. There is a tendency for the coin to resist tipping to the side, but no tendency to prevent movement along the rim. But a general tendency to lift the centre of mass does not exist. In the end the spinning coin is a complicated object. When it tips only the outside edge touches the surface producing off axis forces and torque, some of the spinning motion will go into rolling motion or some pure "wobbling". I am sure that there is a long treatise somewhere, probably from the late eighteen-hundreds early nineteen-hundreds. Maybe you can find something. Rotator theory is complicated, especially if there is external torque from friction.

If you want to see something cool look up rattleback.