Wheels and Friction Part 1

The opening issue is about something that confused the hell out of me when I was a kid, and precollegiate teachers don't seem to do a proper job explaining. I'm talking about how friction acts on a rolling wheel. I got reacquainted with the problem during a physics tuition session, where we looked at a typical TYS question involving a toy car rolling down a ramp. For those of you who aren't Singaporean and probably don't know, "TYS" refers to "10-year-series", which is one of many books of exam questions taken from the past 10 years ultra-competitive Singaporean students use to prepare for their exams. Anyway, that question neglected friction between the car wheels and the ground, but got me thinking anyway.

So conventional wisdom due to our childhood teachers is that friction opposes motion and slows things down. But at the same time, a car needs friction to even start moving forward. So what's the deal?

I never got a satisfactory answer to that question until my 1st year at uni, during a freshman physics class. And the answer is actually counter-intuitive. Under friction, an ideal circular wheel actually never stops rolling forward. And you'd think friction killed all motion!

So let's see what really happens. First off, a primer about friction. To my knowledge, there are 2 types of friction: static and kinetic. Static friction arises between surfaces not moving relative to each other. Up to a threshold value, this friction is equal and opposite to any force directed along the tangent to the surfaces at any point of contact. On the other hand, kinetic friction arises between surfaces moving relative to each other. This friction is constant and opposite the direction of relative motion. Since the surfaces are either moving relative to each other or not, we can have only 1 type of friction between the surfaces at any 1 time. And that's all we need to know about friction to analyse this situation.

Let's first think about how a wheel can be in a steady state where its velocity is constant under friction, before going into how it got there in the first place. We have trusty Newton's 2nd law (or 1st law if you prefer) saying that for an object's velocity to be constant, there can be no net force on that object. Since there are no horizontal forces on the wheel other than friction, friction must be exactly 0! But how is that?

The only point of contact between the wheel and the ground is the bottom of the wheel at any instant. The only way for friction on the wheel to be 0 is if there is no relative motion between that point and the ground, so any friction would have to be of the static variety and 0 because there are no other horizontal forces acting on the wheel. But hold on a second! How can there be no relative motion between that point and the ground when the wheel is rolling forward? Well, it is precisely because the wheel is rolling forward that there can be no such relative motion. We just need the wheel's forward velocity to be equal to its rim's tangential velocity due to its rolling. Then the net velocity of the bottom-most point is the sum of the wheel's forward velocity and the equal and opposite tangential velocity and thus 0!

So what? We have a perpetual machine now? Nah, friction is 0, so no energy is lost to friction and its conservation is not violated. All we have is a wheel on a non-frictionless surface going on forever. Now everyone's happy, right? So why do cars need fuel to keep moving? I should be able to just burn enough gas to get up to whatever speed I want then cruise on forever, or at least until the next red light. Or if I'm cycling, I'd just need to cycle hard for a while, then shake my legs as my bike glides on effortlessly to wherever. If only. Don't forget about air resistance (and friction between other parts of your vehicle). Friction is not the enemy here - air resistance is!

I was supposed to tell you all how the wheel got into that steady state but let's leave that for another time, because I'm getting lazy and tired and I think you've read just about enough for now. Instead I want to go into some technicalities about the wheel's energetics when it's in steady state, purely for my interest and yours if you're curious enough. Specifically, the wheel's kinetic energy can be broken down into translational and rotational. Translational is just your usual 1/2 m v^2, representing the energy going into the wheel's forward motion, while rotational is 1/2 I w^2, representing the energy going into its rotation. I is the wheel's moment of inertia, which wikipedia gives as 1/2 m R^2, while w is the wheel's angular velocity. Since the wheel's forward velocity is equal to its rim's tangential velocity, v = R w, so the rotational energy is 1/4 m v^2, or half the translational energy. Interesting fact!

In the next issue I will talk about the processes involved in getting a wheel to its steady state. I'm already wondering how I can write equations in a more palatable form. I also need to figure out how to upload some Mathematica applets so you can play with these wheels. Comments!