Capstan
Equation and Reality: Ideal Geometry on“Synthetic Rock,” and a More Realistic Geometry on Real Rock
Ever wonder how much the force you feel, as a top belayer, is reduced (or increased) by friction as you run the rope over a rock? Here’s a simple experiment you can do with tools you have around the house!
First, the ideal case. The Capstan Equation is often used to rationalize effects observed in climbing and working with ropes.
The synthetic “rock” is made from a mix of 4 lbs dental
plaster (Kerr Suprstone), 1 lb fine garden sand, and 1.25 lbs water, poured into
half a round gallon cider jug to get a half-cylinder shape:
The rough edges of the bubbles were smoothed with a file, and a very slight rectangular notch was indented on the equator to help guide the rope. When it was cured, the rock would easily support my weight.
The beautiful test apparatus appeared as below. I used 8mm 100% PET Maxim
8mm rope (actually more
like 8.3mm), an aluminum step ladder, a 30 lb dumbbell (which with its
biner
and holding rope came to 31 lbs), and a 660 lb calibrated hanging
scale. I
climbed the ladder, and pushed down with my foot in the footloop
slowly, till the force
stabilized. (Click (desktop) or tap (pad or phone) small image to see
full-sized image.)
The Capstan Equation is given below, along with the parameters are for our system. T_pull is the tension needed to pull the weight over the rock, and T_hold is the tension at the weight, which is consistently 31 lbs. The "mu" is the coeficient of friction (CoF) between
the PET rope and rock; this value will be determined in the experiment,
as it is virtually unreported in the literature. There are few reports of the CoF between rock
and rock, and they range between 0.3 and 0.8. The coefficient of PET rope on steel is generally around 0.3 - 0.4.
Note that the equation does not include the diameter of the rope, nor the diameter of the cylinder over which it is pulled. That's because in the assumption that the diameter of the rope is small compared to the cylinder, any dependence disappears. So when you see people apply the Capstan Equation to climbing, always ask if that assumption is correct. One of the main, real uses of the Capstan Equation was to measure CoF, when the conditions of the experiment met the assumptions of the equation. The normal derivation of the Capstan Equation is a bit sketchy, but more rigorous derivations give the same result.
From measurement, pulling the weight up (by pushing down on the foot loop) required about a 3x increase in force; inversely, letting the weight back down requires only about 1/3 the force (~10 lbs). The Capstan Equation is reasonably appropriate here (because the diameter of the cylinder is much larger than the diameter of the rope) and we calculate a coefficient of friction of about 0.393 for the rope-on-rock.
Now a Real Rock (decent agreement)
So now I replace the synthetic rock half-cylinder with this:
(zoomed on rock at right, click on small image to see more detail)
This is a porphyritic basalt cobble with three roughly flat sides, slightly bowed on each edge so the rope will be held in place. The right-facing facet is ~40 degrees from vertical, and the back (hidden) facet is ~32 degrees from vertical. I used the same 31 lb weight; pulling down on the right side gave 72 pounds, and letting it back gave 11 pounds, not all that different from the synthetic rock example.
There is no honest, neat way to model this with the Capstan Equation; but we can model three contacts providing the friction:
