A dockhand holds the tail of a mooring line wrapped round a bollard or capstan. Each turn multiplies the grip by eμ·2π, so a few wraps let a light hold stop a heavy ship — until the rope slips, overheats or parts.
Tension against extension of the line: the loop is the energy the rope loses as heat each cycle; the whole loop drifts right as creep lengthens the rope.
Runs the full simulation at each wrap count with your current ship, weather, rope and hold force, and compares the first count that never slips with the closed-form prediction n = ln(Fpeak/Thold)/(μ·2π).
Compares the code with a strip-by-strip derivation of the capstan law, the steady sliding speed solved by bisection, a rope-spring period and the heat bookkeeping.
The science
Capstan equation (Euler–Eytelwein). A rope wrapped through a total angle θ on a rough post: tension changes along the wrap as dT = μ·T·dθ, so
Tload = Thold · eμθ , θ = 2π·(number of wraps)
With μ = 0.35 the factor is 9 for one wrap, 81 for two, 740 for three: the hand needs only Thold ≥ Tload·e−μθ. The post radius cancels; the wrap angle and friction decide. Sliding starts when Tload exceeds Tholdeμsθ; once sliding, kinetic friction (taken as 0.8μs) gives Tload = Tholdeμkθ, so a rope can stick–slip.
Load on the ship. Wind and current are quadratic drag:
F = ½ρCdA v2 (air 1.225 kg/m³ on the area above water, water 1025 kg/m³ on the hull below it)
The ship's sway is (m+ma) x'' = Fwind + Fcurrent − T with added mass ma = 0.9m. The line is a Kelvin–Voigt spring with a creeping permanent set, T = k(x − s − xp) + c·(v − s'), k = EA/L with EA = breaking load / breaking strain, creep rate xp' = α(T/MBL)². Several identical parallel lines share the load equally.
Heat. The slipping wraps dissipate P = (Tload − Thold)·w (w = sliding speed). Half goes into the rope in contact (lumped, uniform temperature), the rest into the post and air; past 70 % of the melting/charring temperature the rope glazes: its sliding friction falls and its strength drops by up to 60 %.
Safe working load is taken as MBL/5. Breaking loads are 3-strand catalogue values (MBL ≈ c·d²) rounded; a real line is also weakened by knots, chafe, age and UV.
Windlass. With the pawl engaged the drum cannot turn backwards, so the wraps are the only thing that lets the rope slip; the crank drives the rope in at 0.12 m/s as long as the drum torque exceeds R·(Tload − Thold). Release the pawl and the free drum (J = 14 kg m²) simply pays rope out: the hold is then just Thold, not eμθThold.
Honest simplifications: sway only (no yaw/surge, breast lines at right angles), no wave motion in the loads, linear rope stiffness, one lumped temperature, constant friction coefficients (real ones vary with speed, wetness and wear), a one-hand grip limit with no fatigue, rope stock of 30 m on the quay. Real ships use many lines, winches and fenders, and nobody should hold a ship by hand: this is a physics demonstration. Friction values and breaking loads are typical rounded figures, not manufacturer data.
Credits. 3D models generated with Meshy AI and processed in Blender (bollard, capstan, rope coil, rowing boat); ship, dockhand, ratchet, fenders and the baked stone texture built in Blender 5.2; Three.js r160 runs the live part.