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The Friction Circle

The Friction Circle -- a diagram showing total friction and its directional components
The Friction Circle -- a diagram showing total friction and its directional components
Drawing By The Author

Anyone who's ever driven a car knows that a tire can only hang on to the road so long before it loses its grip. After that, it stops gripping -- possibly even rolling -- and starts sliding, whether you've locked the brakes or gone around a corner too fast. Or both at the same time.

Have you ever wondered why you can't push on the brakes as hard when you're going around a turn as you could in a straight line? That's because a tire's tread can only generate so much friction. Further, the tread doesn't know the difference between going straight, braking, turning or accelerating -- it only experiences a total force in a given direction.

Engineers graph this "combination limit" on what they call a "friction circle," which is shown in the picture accompanying this article.

As with any point on a circle, its location on the graph can be broken down into a triangle -- an "X" direction (horizontal), a "Y" direction (vertical) and a vector sum, also known as the "hypotenuse" of the triangle. In the case of a "Friction  Circle," the X-Axis -- the left-and-right direction -- shows lateral force. That's what you feel when you go around a corner and slide toward the car's door. The "Y-Axis" is the fore-and-aft acceleration -- up for braking and down for acceleration. Again, think of "up" as "forward" and the direction as which way you'd slide if you suddenly hit the gas or the brakes.

For the most part, the graph is circular, with a flat bottom in the "acceleration" direction (downward on the Y-Axis) showing the limits of friction due to wheel spin. The only other limit to the size of the Friction Circle graph, in terms of the G-force you feel is how much grip the tire has on the road. Dry asphalt has a lot of friction, polished concrete less, and tires-on-ice very little.

Now back to the original question: Why can't you go as fast around a corner under braking as you could if you were just turning the circle? And why can't you use as much braking when you turn? Because, as stated above, tires have a limit of friction, represented by the hypotenuse of the triangle shown on the graph.

You can go around a corner at maximum grip if that's all you're doing. Similarly, you can brake for all the tire is worth if you are only going in a straight line. When you start trying to do both at the same time, you still have the same total grip, but now you're trying to use it in two directions at once. When that happens, you start giving up some grip in each of those directions.

This all goes back to basic plane geometry. That, if you remember, is written,  x^2 + y^2 = r^2 , or "X-Squared + Y-Squared = R-Squared. That looks a lot like our old friend, the "Pythagorean Theorem," doesn't it? Only in this case, "R" is a constant, which is the radius of the circle. Or the limit of adhesion on a Friction Circle graph.

The thing to keep in mind, as you contemplate the formula, is that "X" and "Y" can be almost any two numbers, provided they add up to the constant, "R." If X = 0, then all your grip is fore-and-aft. If Y = 0, then all your grip is sideways, as you go around a turn. So, when you go around a turn, trying to brake or accelerate and turn at the same time, something has to give. Grip -- the "R" direction of the radius -- remains constant, but X and Y are diminished proportionately to keep R a constant number. Put another way, you can walk four steps forward and three to the right, or four steps to the right and three forward -- or any other combination like that -- but you are still going to be five steps diagonally from where you started.

Mathematics and Nature's limits will not be exceeded.

Now to dispel another persistent myth -- "Four Wheel Drive Gives You Extra Traction."

Put simply, no, it does not. Each individual tire and road surface combination has its limits of adhesion, as shown in the Fiction Circle and you cannot "make more friction" simply by adding power to more wheels. What four wheel drive will allow you to do is to make better use of what traction you do have. That way, if one end of the vehicle -- or even one tire -- has more traction than the others, power can be apportioned to that wheel to help pull you out of trouble.

But once you're stuck, you are stuck, period. Depending on what kind of differentials you have in your vehicle, it is quite possible for one with four-wheel-drive to get stuck with two wheels on solid ground and the others on ice. An "open" differential will allow one wheel to spin while the other has traction. Only when you limit the differential action by use of friction plates or a "locking" mechanism can you use power on both sides, or transfer it from the spinning wheel to the one with grip.

Otherwise, to quote an automobile industry inside joke, "all four wheel drive does is get you a few more yards deeper into trouble before you get stuck."

Think about that before you spend a lot of extra money for the four wheel drive option on your next vehicle. And don't go asking your poor little tire contact patches to do more than they're capable of. Unless, of course, you really want to go sliding all over the place and making the acquaintance of two truck operators.


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