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:

 

rock

 

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.)

 

ladder

 

A simplified view of the important components is here.

 

diagram_capstan

 

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.