If you ask someone why a bicycle doesn't fall over when you ride around a corner, they'll probably give you the standard answer, and that answer is wrong. The real reason is so subtle and unintuitive that even experienced cyclists may have trouble believing it when they first hear it.
The standard explanation says that when you turn the handlebars to steer around a curve, you must simultaneously lean into the curve. But when you think about it, that cannot be the whole explanation. When you are on a bicycle, you don't have any fixed surface to push off from. Sure you can learn your shoulders one way, but only by pushing the handlebars and seat the other way.
Try it! Cycle along a straight line, and try to lean your whole body and the bicycle in the same direction without diverging from the line. You can't do it! At slow speeds, you could shift your weight to one side to initiate a small lean, but that's not going to work for higher-speed turns.
So how does turning a bicycle really work?
If you're going to turn right, you must first turn the steering a little bit to the left. This makes you start “falling” towards the right. When you've fallen far enough to the right, you turn your steering to the right to “catch your fall”, and follow the curve.
At the end of the turn, you also need a special manoeuver. You nudge the handlebars a little bit more, to make a tighter curve. This brings the bicycle back under you and pushes you upright, and you can straighten up and come out of the curve.
This “counter-steering” technique is quite unintuitive at first, and it's why it is so hard to learn to ride a bike. Once you've learned to ride, you counter-steer automatically without realising that you're doing it. Countersteering is genuine, and all the doubters who try it out on a painted line, or who work through the vector math of the forces and accelerations, come to realise that there's no way to get a bicycle around a corner at a reasonable speed except by using counter-steering. And with that understanding, their cycling skill moves up a tiny notch.
I'll stop writing here, because I can sense that you, the reader, are in disbelief and itching to head for your bicycle right now. If you don't have a bicycle, you can see counter-steering filmed clearly in this video at 2:30.
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