Spin is fantastic and ubiquitous! Wheel spins, fan spins, the earth spins, and atoms spin! Spin-related phenomena may be counterintuitive. For example, if you put one end of a fast spinning wheel’s axle on your finger, which may be dangerous, the axel can keep almost horizontal! Here’re videos about this phenomenon Wheel momentum, Anti-Gravity Wheel?.
There are many toys made from the idea of spinning, such as spinning tops, hula hoops or fidget spinners. Apart from these, there are also wrist balls, Euler’s disks and chatter rings. The latter three puzzles me a lot!
First, let’s talk about the wrist ball. It is a tool for exercising one’s wrist. There is a ball inside. You can speed up the rotation of the ball to 10000 rpm by moving your wrist with a much smaller frequency. That is amazing! An explanation goes as follows. If you push one side of a fast rotating ball’s axle, the touching point will move perpendicularly to the direction you push due to angular momentum conservation. As a result, the touching point slips along the surface of whatever that pushes it. The radius of the axle is so small that even a slow relative motion would lead to a high frequency rotating.
Secondly, there comes the Euler’s disk. It’s a heavier version of a coin, so let’s just talk about coins. If you spin a coin on a table, its center of mass will become lower and lower as time goes on. The frequency of the sound made by the coin would be higher and higher until it falls totally onto the table. Wikipedia gives an explanation on this, showing that the frequency of the touching point between the table and the coin is inverse proportional to $\sqrt{\sin{\alpha}}$. $\alpha$ is the angle between the coin and the table. In the Wikipedia explanation, the coin’s center of mass stays steady. At every instant, the coin rotates along the line from the coin center to the point of contact. That is a strange motion because the instantaneous axis of rotation moves in relative to the coin. Luckily, this motion can be fractured into two simpler rotations. And with the help of two equations, one from geometry (no relative movement) and the other one from physics (angular momentum conservation), one solves the motion of the coin.
It is more interesting to start the explanation from a motion that the center of mass isn’t steady. Just consider a coin rotating along its most symmetric axis. If $\alpha=90^\circ$, the coin just rolling on the table forever. Otherwise, the coin would orbit a second axis perpendicular to the table while it spins. If $alpha=0$, every orbit will subtract 360$^\circ$ from the coins’ spin. For example, if a coin of radius $r$ rolls over a distance of $4\pi r$, it spins for $720^\circ$. If it rolls inside a circle of radius 2r, the touching point also moves over $4\pi r$, then the coin would spin for only $360^\circ$. This ‘spin-orbit’ effect becomes more pronounced when the radius of the orbit becomes smaller. And finally, the two rotations offset each other, the rotation frequency of the touching point may be high, but every point on the coin keeps almost rest.
The third story is about the chatter ring. This is also a magic toy that you can make it spin fast. The real problem is more complicated as this time the rings moves more randomly. Anyway, the plane of the small ring has an angle with respect to the big ring, due to the torque of gravity force. The relative motion speeds up the smaller ring.
Hmmm, an additional story. The Dzhanibekov effect. This video gives an impressive introduction. It says a Russian astronomer discovered this bizarre phenomenon in outer space and Russians kept it a secret for 10 years, and Feynman was wrong.