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When we were taught physics at school and the topic was the Inertia and Flywheel, they used as an example that the moment of inertia of the bicycle wheel is the reason why we are able to keep the bicycle in the upright position.

Is this true?

If I take into account the lightweight of the rim and tire, its moment of inertia seems nothing compared to the mass of the bicycle and the biker. And I am even able to keep in the upright position a micro scooter where the wheels are really tiny.

How can I compare the significance of the rim as a flywheel with the other factors?

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I do not know what you mean with this moment exactly. Maybe you think of stabilization by gyro effects? Bicycle dynamics was explored exhaustively by Prof David Jones (1970) in the following:

http://socrates.berkeley.edu/~fajans/Teaching/MoreBikeFiles/JonesBikeBW.pdf

He built several URB's (unrideable bicycles), showing that gyro effects are not relevant (among other facts) for bicycle riding.

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