# Static and Sliding Friction -- Are you safe?

A popular book of physics puzzles (Mad About Physics, Jargodski & Potter, 2001) states that if a car rolling down a slippery ramp locks its rear wheels only (while not braking the front wheels) it will turn 180° to slide down rear first. The reason given is that because the front wheels are in “static friction” with the surface (because they are rolling, not sliding), they have a greater force on them than exists on the rear wheels, which are in “sliding friction” mode. What is wrong with this argument?

Consider the parallel case of a car rolling on an icy level road. As you may know from driving (or being driven), as long as you don’t accelerate, brake, turn, get hit by wind, etc. your car will continue to run straight – Newton’s First Law of Motion. If you then braked only your rear wheels, what would happen? You would slow down, i.e. accelerate backwards, proportional directly to your rear tires’ load and coefficient of sliding friction, and inversely to your total mass. You would NOT go into a spin. For the same reason, if only your bicycle rear brake works, it will slow you down, but NOT throw you over your handlebars. So what’s different about rear and front brakes?

It’s the geometry. Consider first the analogy of a shared stick. If you and your friend are each pulling at opposite ends, with equal forces, you can do this easily with your eyes closed. The geometrical arrangement of the forces (direction and location applied) is stable. But if you try this each pushing on the stick, and close your eyes, it’s much harder; the stick is likely to twist, and you fall. For the same reason, the stick is stable, hanging down from your hand (you can hold it with your eyes shut); but to balance it upright on your palm, you need to actively respond to its every movement, which you can’t do with your eyes shut. We might say the stick is “metastable” if balanced perfectly, but with even a small disturbance (“perturbation”) it becomes unstable, nothing counteracts the increasing lateral component of gravity as the stick leans more.

So, with the car: the momentum of the car acts as if from its center of mass (somewhere near the middle), in the forward direction (it’s not a force, though it acts like one for this discussion of geometry); and tire frictions operate in the rear direction, at the tire location, for both the rear and front tire sets. The direction of the rear tire friction and the car momentum is opposite, but they are stable (pulling apart), the direction of the front tire friction and the momentum is also opposite, but they are unstable (pushing together). This is why braking with your front wheels tends to let you spin under slippery conditions in your car, or throw your over your bike handlebars (if the front brake is much stronger than your momentum – you are unstable in the vertical plane).

So, what strategy is best for braking a car under slippery conditions? First, of course, don’t travel faster than your ability to stop safely under unexpected conditions. (If you haven’t watched internet clips of cars zipping along icy interstate highways and piling up into each other by the dozens, go take a look. The drivers assumed that because everyone else was driving 70 mph on an icy road, they would be safe. Similar belief operates (especially in coastal California) for driving through patches of dense fog. If nobody brakes at all, no collisions. But let even one person slow down in the fog, and the pileup begins. And then there’s deer. And puddles. And so on.)

Ideally, you want to not allow any wheel to begin sliding. But you need to allow safety margins – in speed, distance, driving habits, etc. as road conditions dictate. You can’t brake at all if you’re hydroplaning, for example; you need to slow down before you hit water on the roadway. Perhaps your car has “antilock breaking system” (ABS), and/or 4-wheel drive. They can’t keep you from collisions if you’re travelling way too fast; they simply operate your brakes (or power) so as to prevent individual wheels from sliding. Car commercials showing a car racing through a suburban street, then swerving violently around a child who runs a tricycle into the street (thanks to the car’s equipment) – that’s a poor driver who didn’t properly slow for a restricted visibility area, not a great car!

I personally have driven a front-wheel-drive van (very slowly) on an icy, hilly road and passed a dozen 4-wheel drive trucks in the ditch. Was the difference in their superior equipment? Think about it now: would you need to drive slower uphill, or downhill, on curvy, icy roads if you were worried about starting to slide off the road (hints: if a hill is too steep, can you go up? down? What are the effects in each case?) [I welcome answers as comments!]

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Stanton D.

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