Thursday, February 3, 2011

Why that darned contraption with two wheels won't fall over

When I was a senior in high school, I took my first physics class. It was A.P. Physics from a man named Mr. Soandso. It was a great class. We had lots of fun labs and it made me realize I might want to be a physics major. But here I am six years later, remembering something he told us that was incorrect.

One day somebody asked him, "How can you stay balanced so easily on your bike when it's moving?" It's hard to balance while you're standing still on a bike, but ridiculously easy if you're moving. He told us that it was because of the wind moving on either side of us that kept us balanced as we moved. That didn't make much sense to me, but I accepted it and moved on in life. Years later I realized that was wrong, or at most has only a small effect on the balance.

It really is quite amazing that we can ride on two very narrow wheels and not just tip over. The tires on my Bianchi are only 2.5cm wide, so they're definitely not helping me have a nice stable footing. If you remember my recent post about the conservation of angular momentum of a swivel chair, the problem of a bike's stability is also an angular momentum problem.

If you own a bike, take off the front wheel and do this experiment. If you teach a class, show this to your kids. Take the wheel so you're holding the axle in both hands. Spin the wheel on the ground or have a friend spin it for you. If you hold it in front of you so the wheel is still vertical, it will feel about like you'd expect. It feels no different from holding the wheel up in the air while it wasn't spinning. Now try to turn the axle and be careful. The wheel will resist your attempt to turn it.

That's exactly what's happening when you ride your bike. Your wheels start spinning around an axis and then they resist turning when you try to fall to one side. Now if you wear clip-less pedals, you can certainly feel the lack of resistance when you slow down to a stop, forget to clip out of your pedals, and promptly fall to your side in front of the pretty girl in the car next to you. Angular momentum helps you get the ladies.

Angular momentum also helps you turn on a bike. When you're going slow, you just turn your wheel and the friction causes your bike to turn-that's boring. But when you're going fast and try to turn, you'll notice that you're barely even turning the front wheel, if at all. You just lean to one side and you turn that way.

Here's another experiment you can do that helps you understand how you turn on a bike. Grab your front wheel again and get it spinning while holding both sides of the axle. Now take one hand off and just hold one side of the axle with one hand. First of all, you'll notice that it stays pretty much upright. That's pretty cool, but you'll also notice that its weight has caused it to tilt to the side just a bit. The last thing you'll notice is that the wheel actually starts to turn. You'll feel really silly because you'll have to keep repositioning your body so the wheel can keep turning. By allowing the wheel's weight to tip it to the side a bit, the wheel naturally turns.

This is the same thing that's happening when you turn on a bike. You tilt the wheel just a little bit and it applies a torque to the whole bike, making it turn.

Here's the technical version of why this works:
The civil engineers know that torque is r cross F. The mechanical engineers and physics peeps know that torque also equals the time derivative of the angular momentum vector. The civils are left tapping their heads because they've never seen a lower case t in an equation before. But all it means is that if you change your angular momentum, you get a torque. The direction of the wheel's angular momentum is along the axle. If you change the angle of the axle, you will get a torque in the direction of the change (note that the L is a vector, not a scalar). So if you tilt the axle to the right, the wheel will turn to the right. Get it?

Hopefully this tidbit of knowledge will help you remember to unclip from your pedals when you're at an intersection next to a pretty girl (unless you have mad trackstand skills). Hopefully this will also prevent you from inventing a bicycle that has two wheels side by side. It won't work.

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