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.
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