A line, a post, a limitThe Turn That Holds the Ship
Wrap an ideal rope around a post and test the capstan equation yourself. Change the turns, friction, and post radius, then watch a numerical integration expose both the exponential grip and the model’s limits.
How much load can one unit of hold resist?
The rope drawing is schematic. It separates successive turns along the post for legibility, while the calculation treats the contact as a planar arc with zero axial pitch.
The browser is evaluating the equation.
one full turn = 2π radians
a chosen model value, not a rope fact
changes geometry, not ideal ratio
Tload / Thold = exp(μθ)
The number above is a ceiling. If the load-to-hold ratio stays below it, static friction can supply whatever smaller difference equilibrium needs. At the displayed ratio, every part of the contact is at the model’s limiting friction and the rope is on the verge of sliding. The equation does not summon extra force by itself.
At the default setting, the ratio is in the hundreds. Make the rope slicker without changing the turns and watch that advantage collapse. No single coefficient belongs to “rope”: fibre, construction, post finish, contamination, water, wear, pressure, and whether the line is already moving all matter.
Friction does not merely add here
A smart objection says that each patch of post contributes another small friction force, so the total should grow linearly with wrap. The missing fact is that each patch receives a different tension.
This derivation models the contact as a planar, zero-pitch, effectively massless line. It neglects rope self-weight and every other distributed load, so the local balances ds = R dθ and dN = T dθ apply to that ideal contact.
The two nearly tangent tension vectors leave a small inward resultant. To first order, the post’s normal reaction on the element is local tension times its tiny turning angle.
Limiting Coulomb friction is μ times that normal reaction. It supplies the tiny rise in rope tension across the element.
Every increment acts on the tension produced by all the increments before it. Divide by T and integrate around the contact.
The logarithm of the tension ratio grows linearly. The tension ratio itself does not.
Tload/Thold = eμθ
Make the multiplication visible
The plot integrates dT/ds = μT/R along model arc length s. It does not call the closed-form exponential. Each marker is a later point on the same ideal, zero-pitch contact line.
Numerical integration pending.
2³ through 2¹² slices, fourth-order Runge-Kutta
The radius that disappears
Use the post-radius control above. The drawn post and the model contact length change, but the limiting ratio does not. The arc integration shows the cancellation in length units:
dN = (T / R) ds and ds = R dθ
therefore dT = μT dθ
A larger post provides more metres of contact, but bends the rope less per metre. In the thin, perfectly flexible rope model, those effects cancel exactly. Radius still matters to real ropes because stiffness, thickness, crushing, wear, minimum bend radius, and the post’s structural loading were excluded to obtain this equation.
A small atlas of wraps
Every cell is generated in this browser from exp(μ × 2π × turns). Read across to see turns compound, or down to see why an unmeasured μ cannot support a promised capacity.
| μ \ turns | 0.5 | 1 | 2 | 3 |
|---|
The check
- Measured constant: none. The interface coefficient μ is a free input here. The default 0.30 is a demonstration choice, not a measured value for every rope and post.
- Geometric inputs: total contact angle is turns times 2π. Radius sets the planar model arc length. Axial pitch is zero. The ideal closed form contains no radius.
- Model result: the readout is the maximum load-to-hold tension ratio at impending slip under Coulomb friction. It is not a rated working load and must not be used for rigging or life safety.
- Numerical check: a Runge-Kutta integration of dT/ds = μT/R is compared live with exp(μθ). The offline verifier repeats this independently across a grid and several radii.
μ is not universal
Surface pair, finish, water, mud, ice, wear, speed, and pressure can change effective friction. Static and sliding coefficients need not match.
The rope is idealised
The derivation assumes a thin, perfectly flexible, inextensible, effectively massless line with no bending strength, on a rigid circular cylinder, with uniform Coulomb friction. Contact is planar with zero axial pitch, and rope self-weight is neglected.
The post is not invincible
The equation predicts a tension ratio, not material strength. Rope, bollard, attachment, and whatever holds the bollard can fail first.
Honest apparatus
- Historical record. The relation is commonly called the Euler-Eytelwein or capstan equation. Euler’s Remarques sur l'effet du frottement dans l'équilibre, Eneström E382, appeared in the Berlin Academy volume labeled 1762. The Euler Archive record catalogs it as written in 1758 and published in 1769. Eytelwein’s Handbuch der Statik fester Körper followed in 1808. The page keeps those dates separate because “Euler 1762” alone conceals the publication history.
- Derivation and limiting condition. Stephen W. Attaway’s The Mechanics of Friction in Rope Rescue derives dN = T dθ and dT = μdN explicitly under impending slip and no bending strength, then names condition and contamination limits. MIT OpenCourseWare’s belt and chain lecture independently frames the capstan problem with static friction.
- Non-ideal rope. Harmeet Singh’s open-access 2022 analysis of an elastic rod around a circular capstan shows exactly why finite thickness and bending elasticity require a richer model and can permit unequal end loads even without friction.
- Reproduce the numbers. Run
node research/the-turn-that-holds-the-ship/verify-the-turn-that-holds-the-ship.mjs. It evaluates the formula grid, the default case, an independent arc-length ODE integration, and the radius invariance without browser code or dependencies.
This is a mechanics instrument, not operational advice. Actual mooring, climbing, rescue, lifting, and belt design require measured system behaviour, rated equipment, suitable safety factors, and domain-specific practice.