This isn't as trivial a problem as it might seem. Contrary to appearances, a multitude of forces affect a moving bicycle's behavior. The most common factors that draw attention are the gyroscopic effect and fork geometry. As it turns out, these aren't the key elements, although they do have a real impact. It turns out that the topic has been addressed seriously, and thanks to this, I can write this article, which will provide a simplified answer to the question of why a bicycle doesn't tip over.
David Jones's experiment
Dr. David Jones was a chemist, but he was eager to step outside his field, combining a scientific approach with incredible creativity. He also had a knack for writing brilliantly and entertainingly. Besides inventing "extraordinary" devices like a black hole garbage can and an atomically powered pogo stick, he foresaw chemical lasers and the possibility of creating incredibly durable structures from carbon atoms, surprisingly resembling the fullerenes synthesized almost twenty years later. And it was he who set out to explain why a bicycle doesn't tip over.
You can read the full article here , which I highly recommend. It's well-written, and Dr. Jones approached the topic concretely, building and testing models (actually, modifying his own bike) and creating a computer simulation (in 1970!).
The gyroscope effect is not responsible for why the bike doesn't fall over!
When we release a bicycle wheel to ride on a flat surface, it will remain vertical for quite a while. This effect is precisely what happens when we release a speeding bicycle – it will travel another dozen or so meters. Jones studied this phenomenon in a wonderfully simple way – he attached an additional wheel to the bicycle, one that didn't touch the ground. He spun it in the opposite direction to the "basic" wheels during riding, thus eliminating the gyroscopic effect – the bicycle would fall over almost immediately. However, when the wheel rotated in the same direction, it remained stable for much longer, even at low speeds. Interestingly, when the bicycle was loaded with a rider, the difference in ride was not noticeable. This leads to the conclusion that the greater the load, the greater the role of other factors.
A matter of geometry
Almost every modern bike (except those designed for niche sports, such as acrobatics) has a fork mounted at an angle, and the wheel axle is slightly forward of the steering axis. Even a small change in this angle and distance (called lead) can dramatically affect the bike's handling. However, it always remains within limits so that the lean exerts pressure on the side of the wheel, which... straightens it, shifting the center of gravity. This also helps the bike lean and generate a stabilizing centrifugal force in a turn. This phenomenon (though without centrifugal force) is easy to observe when riding by the saddle. This is also why it's so difficult to ride without holding on to a bike with the headset turned too far.
On some bikes, however, you can rotate the fork 180 degrees (every mechanic has probably seen one), and riding it is perfectly possible. Interestingly, a bike with negative offset, when left unattended, handles much better than one with a properly installed fork, but in everyday use, it's quite unpleasant to ride. It's worth noting, however, that this solution was occasionally used on speed record-breaking bikes. As I mentioned earlier, you can also ride these bikes with zero offset on a vertical fork, although it's more difficult. In fact, you can ride them without holding on, which is a common occurrence in acrobatics. So, it's a supportive element, but not the most important one.
Why doesn't a bicycle fall over? Because it's a trailer.
I know it sounds absurd. Let me explain. Everyone agrees that the front wheel directs a bicycle. The fact that it's the rear wheel that transmits power and carries most of the load is irrelevant (or at least notable). Think of a shopping cart. If you try to push it with one hand, it will constantly need to be corrected. However, if you pull it, it will simply follow you. Just remember that in this case, it's you, not the cart's wheels, who are directing the direction! Or a car with a trailer. Driving forward isn't difficult, but keeping it straight in reverse takes a lot of practice.

While this may seem strange, from a handling and geometry perspective, the entire frame, including the rear wheel and rider, follows the front wheel and fork, which provide direction. If you accelerate the bike backwards and let go, it will almost immediately tip over. Therefore, riding backwards on a fixed gear bike requires significantly more skill. However, it is possible thanks to the other factors mentioned above, such as centrifugal force, fork geometry, and, above all, active balance.
You do it
Maintaining balance comes naturally to us, which is why anyone can hold a bike upright while riding very slowly, even for a few seconds. Of course, anyone who can ride a bike can, but for most people, this isn't a difficult skill. And the faster you go, the more stable the bike becomes. Balance is easily demonstrated by doing a stand-up, where you hold the bike upright while standing still. This takes some practice, but once you get the hang of it, it becomes natural.
As you can see, the issue is complex, but manageable. Eliminating individual components, as in the case of inverting the fork or riding backwards, doesn't prevent riding, but it significantly complicates it. Eliminating all components, however, i.e., placing the bike stationary without a rider, results in it tipping over. So, the conclusion? Cycling is meant to be ridden.
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