Capstan
Equation over “Rock”
If you are belaying, and the rope runs over a rock, how does that affect the force you feel? Here’s a simple experiment you can do with
tools you have around the house!
The “rock” here is a synthetic, 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 vinegar 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
apparatus looked as below; using 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.)
A simplified view of the important components is here.
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.
Note that while the Capstan Equation is often used to model
systems where the rope is close to or greater than the diameter of the solid it
is draped over – such as over a carabiner or belay device – those models cannot
be quantitative, even though some of the same forces are in effect.