Why does the bike fall when its speed is very low or close to zero and is balanced when going with a high speed?
The surprising answer is that the stability of the modern bicycle has little or nothing to do with centrifugal force or gyroscopes or any of that. Look up "bicycle stability" on Google. Experiments show that the sloped angle of the front fork is very important, e.g. If the fork pointed backwards it is very difficult to stay upright at any speed.
At higher speeds a very slight turn of the handles moves the bicycle under the center of gravity of the rider quicker, so that the dynamical stability is improved. As usual experiment corrects theory here.
See the answer of Tristan at Does leaning (banking) help cause turning on a bike? as well for an even better answer
See the comment of nibot below for a reference to an actual definitive article.
A report appeared in Science today which addresses this exact question: Kooijman et al., Science 332 (6027): 339-342, "A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects."
The abstract reads:
There is also a blurb in ScienceNOW that you can read without subscription.
Here is a free-to-read preprint.
When you walk on stilts or skate, you don't balance by being very careful. You don't even balance. You're continually out of balance, and you keep moving your point of support so that you arrest your fall in one direction and start falling in another.
If you're on a bicycle moving very slowly, you do the same thing. You keep moving your point of support left or right to arrest your fall in that direction. If you're moving slowly, it takes more steering motion to accomplish this, so you "wiggle about". At higher speed, it takes less steering motion to do that. That works even in the absence of gyroscopic precession, caster, or rake angle. Just watch a scooter with tiny wheels, or a ski-bike, or a unicycle.
Now, throw in rake angle. Turning the handlebars to the right moves the point of support to the left, even if you're moving very slowly, so that helps.
Now, switch to a high-speed motorcycle with a nice, heavy, gyroscopic front wheel. When it's traveling at a good speed, that thing precesses, no matter what people say, and its precession goes in exactly the right way to powerfully maintain stability.
So it's not an all-or-nothing single-explanation deal.
The basic concept (at least, as I've heard it) is angular momentum. As a bike wheel turns, it has an amount of angular momentum proportional to its rotational speed, associated with the plane of rotation of the wheel. This makes it act basically like a gyroscope: it "resists" any change in the amount or direction of that angular momentum, in the same sense that mass "resists" any change in the amount or direction of its velocity. This basically slows down the tipping of the bike to the point where you are able to prevent it by pushing down on the opposite side pedal.
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Here's an interesting talk on the subject, from the point of view of control theory:
oneat's answer is correct (I would have commented but I think I'm going to need more space)
Imagine a vector (line) starting at your center of gravity. The line represents all the forces acting on you. When you are standing still, the direction of the line is straight down (gravity is the single force present).
To not fall over whens standing still on a bike, you have to keep the point where the line intersects the ground (let's call it point A), between the two wheels of the bike. If you don't, you'll start tipping over.
When standing still, the only way to affect that point is to move your center of gravity which you do by shifting your weight.
Now let's say you're moving. If you're moving in a straight line, at a constant speed, everything is the same, the only force acting on you is gravity. But if you change direction, you get centrifugal force (as oneat correctly pointed out), the same as what you get when you make a sharp turn in a car moving at speed. The value of that force is proportional with your speed, your weight and the speed of the turn.
This centrifugal force is added to the gravity, and changes the direction of the resulting force acting on you.
Remember point A? If you're riding your bike and it starts to lean to the right, point A starts to move to the right and the bikes leans even more, and so on. But, you instinctively know to turn your bike to the right. This causes a centrifugal force, pointing left, to appear). If point A is still between your wheels then you're fine.
If you're moving slowly, the centrifugal force is small, so you have to take the turn more sharply to compensate. If you're moving fast, you might only need to nudge your bike a little to compensate.
It's explained in more detail here. (I actually thought of looking it up in wikipedia only after writing this answer, I don't have time to read the article now, hopefully I'm not too wrong)
We have a series of papers on exactly the topic of this discussion, but a bit more narrowly defined. That is, how and why can a bicycle balance itself?
In short, how does a moving bicycle balance itself? For a variety of complicated reasons it steers in the same direction as it falls. And, if you will excuse the sloppy informal physics language, because of the resulting curved path, thecentrifugal forces, push it back upright. What complicated reasons? Partially from the trail (or castor effects), partially from the angular momentum of the spinning wheels, and partially from other effects related to geometry and mass distribution. But there is no simple single necessary mechanism that we know of. For example, our paper in Science Magazine shows that a bicycle can be self-stable without no castor (no trail) and with no spin angular momentum of the front wheels.
We have written several papers and supporting documents. And we have in these a pretty exhaustive coverage of the literature. So if you want to know what we think, what others have thought, and what we think about what they thought, it's all there. I don't think you will know of some important reference that we have not reviewed and described. You can start with my www page http://ruina.tam.cornell.edu (or google ruina bicycle or google schwabb bicycle.
The www site includes photos and videos including simple explanations of some of these things.
The answer is "Centrifugal force"
The biger your speed is the biger this force is too.
You can notice that when you steer left you make your bike's slope on left side. And Contrifugal force don't let you fall (when your steer angle is constant at the end bike will make a circle). Then when you make your bike steer more left your bike returns to the balance because you increase that force (it comes from equation).
When your speed is smaller contrifugal force is smaller and bike is harder to steer so you can fall easier.