Holding a Load with Two Fingers: The Capstan Rule for Well Work, and Where Wet Rope Breaks It

When a rope carrying a load is wrapped around a fixed post, a windlass drum held fast, or a pin, friction between rope and surface allows a small holding force on the free end to restrain a much larger load. The capstan relation, holding force equals load divided by e raised to the product of the friction coefficient and the wrap angle in radians, quantifies the effect. This study applies that relation to the loads of well work — suspended masses of 50, 100, 150, 200, 300, and 500 kg — under four stated friction states: dry rope on wood at a coefficient of 0.30, wet rope on wood at 0.15, rope on steel at 0.20, and wet rope on steel at 0.10. It asks how many wraps are needed to hold each load with a comfortable one-hand force of 50 N, and it tests the workshop saying that two wraps will hold anything. The saying does not survive. Two wraps multiply a holding force by 43.38 in the dry-on-wood state, but by only 6.59 when the same rope is wet, by 12.35 on steel, and by 3.51 on wet steel. At the stated maximum one-person holding force of 150 N, two wraps therefore restrain at most 663.2 kg dry on wood, but only 100.7 kg wet on wood, 188.8 kg on steel, and 53.7 kg on wet steel. A 200 kg load on wet wood demands a two-wrap holding force of 297.9 N, a ratio of 1.99 to what one person is assumed able to apply. The paper replaces the saying with a wrap-count rule computed from the load and the wettest friction state that could plausibly occur: for a 50 N holding force, the loads examined need 2 to 3 wraps dry on wood, but 3 to 5 wraps wet on wood, 2 to 4 on steel, and 4 to 8 on wet steel.