While riding slow or being stopped, the bike is itself unstable, but when moving at a reasonable speed, even a riderless bike can keep itself balanced.
As to why a riderless bike can keep itself balanced when moving, that’s sort of still an open problem. Part of it is because the wheels of the bike act as gyroscopes. Part of it is that the point of contact of the front wheel is behind where its axis of rotation intersects the ground. Part of it is a complicated thing with how the shape of the wheel acts. But even those explanations together don’t give the entire answer
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