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I think it's clear enough that if you turn your bicycle's steering wheel left, while moving, and you don't lean left, the bike will fall over (to the right) as you turn. I figure this is because the bike's momentum keeps it moving in the direction you were going, and since your wheels have friction against the ground, the top of the bike moves forward relative to the bottom of the wheels. The top of the bike going north while the bottom of the wheels go northwest will understandable cause you to topple.

So to counteract this and keep you from falling over, leaning into the turn is necessary.

But is there also a causal relationship -- that leaning will cause the bike to start to turn? If I start leaning left, I will turn left... but maybe that's because I know that if I don't turn the steering wheel left, the bike will fall over (to the left). I experimented with unruly turns of the steering wheel when I was a kid, and got my scrapes and bruises. Now that I'm a cautious and sedate adult I'm not anxious to experiment that way. :-)

(I also want to ask why airplanes bank into a turn... they don't have the same issues as a bike, i.e. the bottom part has no special friction against the ground. But that would probably make the question too broad.)

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Why don't you try it? Of course, you'll have to pay the hospital bills yourself... –  muntoo Nov 3 '10 at 5:17
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I'd quite like to see a quantitative answer here actually. A bit of Lagrangian mechanics should demonstrate the effect. –  Noldorin Nov 14 '10 at 19:14
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Seems to me the gyroscopic effect should have something to do with how the bike steers, however I just read part of an article suggesting this wasn't the case. I did write an answer for this but deleted it when I realised it may be wrong. –  Dom Nov 24 '10 at 10:23
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This link should be promoted from the comments on @Tristan's answer. It is a physics today article which goes into a lot of detail as to why a bike is stable at all. –  Chris Mueller Mar 27 at 23:50
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This Youtube video about bricycles might be of interest. –  Qmechanic Apr 6 at 13:52

7 Answers 7

up vote 15 down vote accepted

The simple answer is that the angle between the front fork and the vertical causes the force from the ground to create a moment about the axis of rotation that turns the wheel in that direction. This has nothing to do with actually riding the bike, and it will happen even if the bike is stationary.

Basically, if you project the axis of (steering) rotation all the way through the wheel, top to bottom, it will not be coincident with the point of contact with the ground. When the bike leans over, the upward (normal) force from the ground is not in the same plane as the axis of rotation, which causes a moment about that axis.

When the bike begins to turn, the frictional component of the contact force will cause the force to go back into the same plane as the axis of rotation, which causes the wheel to hold its position steady.

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OK, that sounds pretty plausible. I think between @steve's inductive and your deductive reasoning, we have a winner. The answer to the original question being "yes". And conservation of angular momentum is not required. –  LarsH Nov 17 '10 at 20:39
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A question, though... what causes the wheel to turn when you lean: the angle between the front fork and the vertical, or the fact that the axis of steering rotation does not pass through the wheel's point of contact with the ground? (Since these are independent properties of a bike's structure. We can imagine bikes with either of these properties without the other.) Or are both required? –  LarsH Nov 17 '10 at 20:44
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@LarsH: You can probably find the answer in, I believe, a Scientific American article many, many, years ago where they built bikes where they changed the design so that it didn't automatically turn the wheel in the direction you leaned. These were impossible to ride. They also built bikes where they changed the design the opposite way, so the wheel would turn too far. These bikes were so stable they even without a rider they would go for some distance without falling, but they had some other severe drawbacks (I forget exactly what). –  Peter Shor Nov 21 '10 at 4:00
    
@LarsH: And here is the original article in Physics Today which is the more technical counterpart to the Scientific American article I remember. –  Peter Shor Nov 22 '10 at 21:55
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This link is not behind a paywall google.de/… –  Georg Oct 12 '11 at 16:30

if you just lean, the steering wheel will turn in order to conserve angular momentum!

Have you ever tried to ride your bike not holding the steering wheel with your hands? If you try not moving just on a straight line, leaning your weight left or right is exactly how you will curve your trajectory.

Try it! I guarantee no trips to the hospital will be needed. I do it on regular basis when, for example, I drink or stretch my back during long bicycle rides!

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You have a good point. The fact that this happens would seem to be proof that leaning does indeed cause the bike to turn. I used to ride hands-free quite a bit and was able to make gentle turns, though obviously not as controlled as when hands are on the steering wheel. –  LarsH Nov 12 '10 at 20:33
    
Not sure what you mean about conserving momentum. Do you mean angular momentum? Conserving (linear) momentum I would expect would mean that the bike tends to keep going straight. –  LarsH Nov 12 '10 at 20:34
    
yeah sure we were talking about angular momentum! my bad! It's quite a translation problem.. in italian when we think about "momento" is the agular momentum.. the linear momentum is ""quantità di moto".. sorry again –  Steve Nov 13 '10 at 16:07

What makes anything travel in a circle? You need a continuous lateral force toward the center. Without that, it will just travel in a straight line.

You've seen vector diagrams, right? Look at you and the bike from directly behind. There's a lift vector starting where the wheels contact the road, extending directly up through the center of mass. If you and the bike are traveling in a straight line, the vector points perpendicular to the road, doing nothing but supporting your weight. If you and the bike are traveling in a curve, you are leaning into the curve, and the vector also points into the curve. If you project that vector onto the ground, it has a horizontal component. That horizontal component of the lift vector is the force that is accelerating you inward toward the center of the curve. Without it, you would not be turning.

BTW, aircraft work exactly the same way. What banking does is tilt the lift vector, so its horizontal component provides a horizontal acceleration into the center of the curve.

Got that?

Now, how do you start a turn, say, a turn to the left? You do it by turning slightly to the right, which causes you to fall to the left. When you have fallen far enough to the left, you stop turning to the right, so you stop falling to the left, and now you are in a left turn.

Essentially, the way to stay up on a bicycle is if you start falling to the right, turn to the right, and it will stop your fall. Similarly for the left. By the same token, if you want to fall to the left, don't turn left, turn right. So by moving your point of contact left or right, you can always control your angle of bank.

It all happens very quickly so you might not notice, and you don't think about it because your cerebellum is doing all the work.

Another effect is the gyroscopic action of the front wheel. If you pull the right handlebar toward you ever so slightly, the wheel precesses to put the whole bike into a lean to the left. That also helps to start the turn.

Then there's the rake angle of the front fork, which has still another effect.

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What do others think about this? Is there really a "horizontal component of the lift vector"? By leaning into the curve, do I somehow cause the lift vector to lean inward?? –  LarsH Oct 12 '11 at 15:27
    
@LarsH: Check this out. –  Mike Dunlavey Oct 12 '11 at 15:50
    
I think Mike is right. I will try to realize the movements next time I ride a bike. Next time? Right now! Till later. –  Georg Oct 12 '11 at 15:52
    
@LarsH: It's easy to see the lift vector. Just hang a weight from a string and suspend it somewhere on the front of the bike where you can see it. –  Mike Dunlavey Oct 12 '11 at 15:59
    
@LarsH: I used to teach neighborhood kids to ride. We'd start on a gentle grassy slope so they wouldn't have to think about pedaling or skinned knees. Then I tell them "If you start falling to the right, turn right, and it will stop your fall. Same for the left." They wobble around for a while, but stay up. Pretty soon the cerebellum gets the idea, and they're off! –  Mike Dunlavey Oct 12 '11 at 16:03

Yes. A lean induces a side force that makes the bike follow a circular path.

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How do you know that? Can you give references or relate it to more general physical laws? –  LarsH Nov 15 '10 at 3:15
    
Seems like a mere "side force" would cause the bike to tip over. It would have to be a side force in the front or back in order for it to make the bike follow a circular path. Why would a lean induce a side force in the front or back? –  LarsH Nov 16 '10 at 22:22

I disagree with the answer, which currently has the most votes, stating that the normal force changes as the bike leans.

Consider this experiment: A hollow block of metal shaped like a text book with mass similar to that of the average physics student + bike. The spine (binding) of the book is rounded, similar to the rounded surface of the bike tire. Place the block with rounded side down on a hard smooth surface in a vacuum with low friction (similar to the friction a rolling bike tire experiences). Tilt the book the slightest bit to the right and give it a shove, perfectly applying force in only one direction (forward), accelerating the block to 50 kph, and allowing the block to slide on its own after the initial shove. What's to stop the block from falling to the right? Not much...just the friction of rolling the spine of the block on the surface. It's going to fall and it is not going to turn.

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Good point. I'm not sure where to go from there. What if your hollow block of metal were jointed, like a bike... would that change the result of the experiment? –  LarsH Nov 16 '10 at 22:23

What happens if you lean on bike while trying to ride straight?
It happens that you turn on same side you lean.
Why, because you have to generate centrifugal force and lean back up if you do not want to fall down. So if you lean left you have to make left turn in order to generate force which will turn you straight position.
If you turn too much left you will eventually fall on right because of too much centrifugal force kicking you other side.

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so your conclusion is that leaning does not directly cause the bike to turn, but it forces you to steer in that direction in order to avoid falling over? –  LarsH Nov 3 '10 at 1:34

If you just lean without holding the steering wheel, that will automatically turn in order to conserve the total angular momentum. If you hold it tight however, this is not possible and you will crash.

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Interesting ... why should it conserve the total angular momentum rather than letting you fall over and crash? Also, it is possible to fall over and crash when leaning while not holding the handlebars... Does your explanation still hold in that case? –  LarsH Nov 2 '10 at 22:15
    
@LarsH: That's at least my first intuition, see e.g. also Nutation. I'll try to elaborate this further tomorrow –  Tobias Kienzler Nov 2 '10 at 22:19
    
Since reading your answer I looked up conservation of angular momentum. Wikipedia says "Angular momentum is conserved in a system where there is no net external torque"; but when your bicycle is leaning I would think there is external torque due to gravity... so does this mean angular momentum has no "obligation" to be conserved? –  LarsH Nov 2 '10 at 22:23
    
@LarsH: The angular momentum vector of your bike is orthogonal to the wheels. When you tilt, the vector would also tilt up (or down, depending on the direction of your tilting). This causes Nutation, that is, a torque induced by gravity that changes the angular momentum vector forward (or backward, I'm too tired for a vector product atm), thus turning the wheel left or right if the steering wheel is left loose. –  Tobias Kienzler Nov 2 '10 at 22:31
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It is known since the 1970s that the angular momentum of bicycle wheels is negligible. See e.g. en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics –  Frédéric Grosshans Nov 16 '10 at 10:29

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