I was reading Joshua Vasquez’s marvelous piece on the capstan equation this week. It’s a short, practical introduction to a single equation that, unless you’re doing something very strange, covers everything you need to know about friction when designing something with a rope or a cable that has to turn a corner or navigate a wiggle. Think of a bike cable or, in Joshua’s case, a moveable dragon-head Chomper. Turns out, there’s math for that!
Basically, the more you wrap a cable or rope around something stationary, the more friction you have to deal with. I put this to good advantage last Spring when my son and I were doing some random tree-climbing with ropes. Turns out that four or five loops of climbing rope against fairly frictiony bark is enough to hold the weight of a grown man, with nothing other than the weight of the rope itself on the other end, for instance. I was also using this effect in a recent wall-plotter-bot design that uses simple cable braid instead of the ubiquitous timing belt.
In none of these cases did I work out the capstan equation with my pocket calculator: four loops around is almost always enough™. But by digging into the math and physics, I got more insight. Basically, the friction is an exponential with the angle times the friction coefficient of your cable in the exponent. So what? So, that next turn holds exponentially more weight when you’re climbing. And the grippiness of the tree bark matters in just the same way. You might know this intuitively from experience, but it’s nice to have numbers. ..
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