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If we place a typical coin on a table, it will almost immediately fall due to gravity. However, with a little push it will roll and not fall anymore until friction eventually slows it down enough to come to a halt. Since it is a round object (disk-like) it will roll and therefore gain a spin angular momentum. Since it has an angular momentum (thanks to the motion) it will precess and experience a gyroscopic effect which prevents it from falling, as we can see the wobbling effect as it slows down.

Though I know this is the underlying explanation for why the moving coin doesn't fall immediately, I fail to see the underlying force/torque diagram, namely, as soon as it is tilted, a torque rotates it back to the vertical position, but I fail to deduce the direction of the torque diagrammatically. Without the tilt, the angular momentum is perpendicular to the face of the coin (so in the horizontal direction), but when titled the angular momentum also tilts, but the gravity force remains along the vertical. So how do we get to the torque that goes against the tilt?

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Your question is related to the question of how it comes that a bicycle that is set in motion without someone riding it will keep going for a surprising long time. As long as it has sufficient velocity it doesn't fall over.

Henry Reich, the creator of the Youtube channel Minutephysics has a video about that case:
How dow bikes stay up?

In the case of a rolling coin:
Your suggestion is: "as soon as it is tilted, a torque rotates it back to the vertical position"

I think that is not what happens, I think the effect should be understood as described in the minutephysics video.

Specifically for the rolling coin:
Let me define three axes:
- rolling axis
- pitching axis
- swiveling axis

The name 'swiveling axis' comes of course from 'swivel chair'. The swiveling axis is the axis that is parallel to the direction of gravity.

As the coin rolls it will inevitably gain a little bit of pitching motion. In response to the pitching motion the rolling coin will turn a bit (swiveling motion). The swiveling motion turns the coin in the direction that it is pitching, and that turn brings the contact patch back underneath the center of mass.

So it is not that the coin is pushed back upright. The turn has the effect that it moves the contact patch back underneath the center of mass.

Henry (Minutephysics) demonstrates the gyroscopic effect as follows:
He has lifted the entire bicycle (both wheels of the ground) and he has put the entire bicycle at an angle such that when the front wheel is not spinning tilting the bicycle left or right doesn't make the front wheel turn. Then he has spun up the front wheel, and then when there is tilting motion of the wheel the response is swiveling motion.

For the explanation of the response I refer to my 2012 answer here on physics.stackexchange to a question titled: What determines the direction of precession of a gyroscope

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