A coin spun on a table will eventually end up flat on the table, ignoring the cases where it remains balanced and comes to rest on its edge (or falls off etc).
Before the coin comes to rest on its side, there will be a stage where the coin has fallen and is now 'rolling' in a tight circle, with decreasing period (increasing frequency of the "ringing" sound associated with this).
Consider a point on the coin's edge $P$ - every 'cycle' of the coin rolling around on its edge, the coin as a whole rotates (with respect from above). $P$ will move around the axis of rotation in the same direction as the point of contact between the coin edge and the table, relative to the center of the coin in the table's rotational frame. I assume this is because the circle drawn out by the point of contact is smaller in circumference than the coin itself, approaching the coin's circumference as the coin comes to rest.
I am ignoring air resistance for convenience.
- If the coin and table were made of materials such that the coin bouncing off the table is a perfectly elastic collision, would it be possible for the coin to ever slow down, or would it be able to 'roll on its side' forever like that? Intuition says no, but I can't explain why that is.
- The rotational speed of the coin when viewed from above (average angular speed of P) - does this change or remain constant? If it remains constant, why the discontinuity when the coin comes to rest?
- The point $P$, every round-trip of the contact point between the coin and the table, will bounce off the table. This vertical speed appears to increase without bound as the period of the round-trip decreases, which in turn means the acceleration due to each bounce is much higher (from $-v$ to $v$ with increasing frequency and increasing $|v|$). This would imply the forces in the coin are growing as it's slowing down, but that can't be right... I can only assume the speed doesn't increase, merely the distance in the bounce decreases? Wouldn't that also mean that the bouncing speed actually decreases as the coin slows down?