Spinning mass has rotational inertia. In this case the bicycle tires are rotating quickly so they have lots of inertia. The more inertia something has the harder it is to change that inertia. With the rotating mass it’s happy to move along the plane that it’s rotating in (ie forward of backward) because that’s a small change to inertia. Twisting the plane of rotation requires lots of force because it’s a big change to inertia. Falling over requires you to twist the plane of rotation from vertical to horizontal.
/u/Veritasium explains it well in [https://youtu.be/9cNmUNHSBac](https://youtu.be/9cNmUNHSBac).
Essentially bicycles are design to “self correct” and stay upright when in motion.
When the bike lean in one direction, the front wheel will turn to counter the motion and keep them upright – as long as they are moving forward.
[edit: fix user mention]
When things are moving they want to keep moving. Say if you push a block on ice, you can imagine it will just keep going. It’ll actually take force to stop it.
The physic term is called momentum which things can ‘have’. So the faster the block is moving, the more momentum it has, the more it wants to just keep moving the same speed. Objects in motion stay in motion.
Now bicycles have this momentum but each wheel also has momentum. As in each wheel wants to keep on spinning, the momentum the wheels ‘have are in spinning orientation, called angular momentum.
The thing that is so special about angular momentum (the wheels spinning) is that they are very resistant to tilting, or falling over. It would be like trying to stop that block from moving on ice because just tilting the wheel, would bigtime change the angular momentum, which goes against what the object wants which is to stay in its same motion.
So when you are on a bicycle that is not moving, the bicycle wheels have no angular momentum. But when you are moving, the spinning wheels want to stay spinning and upright, so they do some of the balancing for you.
This idea of angular momentum is used a lot in mechanical design. Gyroscopes are a great example and worth searching videos of them.
IMHO based on trying to teach my kids to ride it is partially the rotational interia of the wheels but mainly the brain. The brain senses the movement of the bike and the small resistance of the rotational inertia and is able to correct it via the handlebars. This takes time to learn. When the bike is not moving there is no rotational inertia and nothing to sense or push against.
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