How do sailors stop a huge ship with just a few wraps of rope round a post?
― Each wrap multiplies your hand's strength
At the docks, a ship weighing many tonnes is held back by a sailor with just one rope. The trick is wrapping the rope two or three times round a post on the quay. Friction in the wrapped section doesn't add up — it multiplies. Wrap it three times, and your hand's strength is boosted roughly 300-fold.
At a fishing harbour, a small boat approaches the quay. Someone on shore catches the rope thrown from the boat.
They don't pull against it with brute force. They simply wind the rope round and round a short iron post on the dock.
Even though the boat is still moving a little, holding just the end of the rope in one hand brings it to a smooth stop. You don't need to be strong to do this. So how does it work?
Just two reasons
Pulling on a bent rope presses it against the post. The harder it presses, the harder it is for the rope to slip.
If you divide the wrapped section into tiny segments, each segment cuts the force by the same fraction. Because it's a fraction each time, the effect stacks up multiplicatively.
Combine these two, and each extra wrap multiplies the force you can hold back. Let's look at this step by step.
The harder you pull, the harder the rope presses
Push down on a book on a table with your finger while moving it, and the harder you press, the harder it is to move. This "force that resists sliding, proportional to how hard you press" is friction.
With a rope wrapped round a post, the rope itself creates that pressing force. When a taut rope bends, it presses inward on the curve — toward the post. The harder the ship pulls, the harder the rope presses against the post. And that makes the friction stronger too. The harder it's pulled, the harder it's held — a very convenient relationship.
Friction multiplies — it doesn't add up
Imagine dividing the wrapped rope into very short segments. At the ship's end, the rope is pulled hard. Moving along the post from there, each segment takes over a small share of that force through friction.
The key point is that every segment takes over "a certain fraction" of the force at that point. Where the force is large, the rope presses harder on the post, so more is taken over. As the force shrinks, so does the amount taken over each time. It always drops by the same fraction.
This is the same shape as compound interest at a bank. If one turn cuts the force to about 1/6.6, then two turns cut it to 1/6.6 of 1/6.6. Look at the bar chart on the right in Figure 1. The bars grow by the same height with each turn because the same multiplier applies each time. Wrap it three times, and the sums work out so that 20 kg in your hand can hold back about 5.7 tonnes.
Another surprise: the thickness of the post doesn't matter. A thick post means more rope contact, but it also weakens the pressing force. The two effects cancel out exactly, so all that matters is how many turns you wrap.
Large ships have a thick rotating drum for winding rope, called a capstan. The sailor just holds the end of the rope lightly. Friction against the turning drum hauls the rope in with hundreds of times the force. Ease off, and the rope instantly slips on the drum. The force can be fine-tuned with just the palm of a hand.
In rope-handling work, a basic rule is never to wrap rope round your wrist or fingers. A wrapped hand gets clamped just as tightly as the post, multiplied many times over. If the boat lurches suddenly, your hand gets pulled in with it. Always hold the rope loosely in your palm, never wrapped.
Summary
Wrapping a rope round a post means the harder it's pulled, the harder it presses on the post, and the stronger the friction becomes. Because friction cuts the force by the same fraction at every segment, the effect multiplies with the number of wraps. Dockworkers know this from experience, and use it to stop huge ships with very little effort.
Friction multiplies with every wrap.
That's how 20 kg in your hand can stop a ship weighing tonnes.
For a story where friction at your feet decides the outcome, see "Does the stronger team always win at tug-of-war?"
- Fill a 2-litre plastic bottle with water and tie a string to it. Rest a broom handle across the backs of two chairs.
- Loop the string half a turn round the handle and pinch the other end between your fingers. Note how heavy the bottle feels on your fingers.
- Now increase to one turn, then two. By two turns, a single finger should hold it easily.
Keep the bottle just above the floor while doing this. A wooden handle, or one wrapped in rough cloth, shows the effect more clearly than a smooth one.
Want to know more? ― Terms, formulas, and textbook connectionsWe mark clearly which level each part belongs to, from middle-school science to university specialist courses
- MSCovered in middle-school science
- HSCovered in high-school "Basic Physics" / "Physics"
- HS+High-school advanced content, or textbook sidebar material
- UnivNot covered in high school — university specialist courses (mechanics, machine dynamics)
- ResearchNot yet settled even at university level — an active research topic
MSTerms: this phenomenon has a name
- Friction force: a force that resists the sliding of two surfaces in contact. It grows with the force pressing the surfaces together.
- Tension: the force pulling on both ends of a taut rope or string.
- Capstan equation: an equation stating that the ratio of forces at the two ends of a rope wrapped round a post depends only on the friction coefficient and the angle wrapped. Also called Euler's belt formula.
MSHSCheck with the formula: how many tonnes can 3 turns hold back?
We find the ship-side force from the hand force and the angle wrapped. The angle is counted as 2 × 3.14 per full turn. The meaning of each symbol and its unit are shown in the table below.
| T_ship | Force pulling the rope on the ship's side (kgf) |
| T_hand | Force holding the rope's end by hand (kgf) |
| μ | Friction coefficient between rope and post (no unit) |
| θ | Angle the rope is wrapped (one turn ≈ 6.28. Unit: radians) |
| e | Euler's number. A fixed value of about 2.718 |
| In symbols | T_ship ＝ T_hand × eμθ |
| In words | Ship-side force ＝ hand force × (Euler's number raised to "friction coefficient × angle wrapped") |
| Where it comes from | It's derived by balancing, over a very short segment of rope, "change in tension = friction coefficient × force pressing on the post," then summing that balance over the whole wrapped section. It's credited to Euler in the 18th century. |
| Hand force | 20 kgf (a rough figure one hand can hold comfortably) |
| Friction coefficient | about 0.3 (a rough figure for rope on an iron post; varies with material and wetness) |
| Number of turns | 3 |
| Euler's number to the power 1.884 | about 6.6 (the multiplier for one turn) |
| Angle per turn | 2 × 3.14 ＝ 6.28 |
| "Friction coefficient × angle" per turn | 0.3 × 6.28 ≒ 1.884 |
| Multiplier for 2 turns | 6.6 × 6.6 ≒ 43.56 |
| Multiplier for 3 turns | 43.56 × 6.6 ≒ 287 |
| Ship-side force held by 3 turns | 20 × 287 ＝ 5740 kgf |
The sums work out so that 20 kg in your hand can hold back roughly 5.7 tonnes — about as heavy as four cars. Since each extra turn multiplies by 6.6, four turns give about 1900x. In practice, dock workers also let the rope slip slightly on the post to bleed off force gradually.
HSHS+Why does it drop by a "fraction"?
HSIn high-school physics, the friction force right before something starts to slip is taught as "coefficient of static friction × normal force." The normal force is the force with which a surface pushes back on an object. In a bent rope, the greater the tension, the greater the normal force.
HS+The relationship where, over a short segment, the drop in tension is proportional to the tension itself is expressed as a differential equation. Solving it produces an exponential function — mathematically the same form as radioactive decay, where a substance halves with each half-life. The post's radius drops out of the equation because the length of rope in contact and the strength of the pressing force are inversely proportional to the radius, in opposite directions.
UnivThe capstan equation and belt friction
This relationship is called the capstan equation, or the Euler–Eytelwein formula. In machine dynamics, it's taught as the basic equation of "belt friction," which sets the upper limit on the force transmitted between a belt and a pulley. Real ropes have bending stiffness and thickness, so they don't perfectly match the equation's assumptions. For that reason, corrections are said to be needed with thick ropes or narrow posts.
📖 For the derivation and further reading: Capstan equation (Japanese Wikipedia) ／ Friction (Japanese Wikipedia)
ResearchWhat isn't fully understood yet
- Why don't knots come undone? A knot also holds force through the friction of rope strands wrapping round each other. Which knot is strong, and why, has long relied on rules of thumb. In 2020, a study used fibres that change colour under load to examine the inside of knots, and predicted strength from the number of twists and crossings.
- The friction coefficient isn't constant. The real-world friction coefficient is said to vary with sliding speed, moisture, and how the rope is braided. Sometimes a wet rope grips better than a dry one, sometimes the reverse — it can't be pinned down simply.
- Handling stretchy rope. Synthetic-fibre ropes stretch a great deal. Because the amount of stretch varies from place to place within the wrapped section, predicting where and how fast slipping begins remains an engineering challenge.
In other words, this article too describes things only "as currently understood." The figure of 5740 kgf is just an estimate based on assuming a friction coefficient of 0.3.
Textbook connections (by level)
| Level | Subject/Unit | Where in this article |
|---|---|---|
| MS | Science 1 "How forces act" | Friction that grows with pressing force |
| HS | Basic Physics "Friction force" | Static friction coefficient and normal force |
| HS+ | Math III "Differential equations, exponential functions" | Why a fractional drop compounds multiplicatively |
| Univ | Machine dynamics "Belt friction" | The capstan equation |
| Research | Mechanics of knots | Predicting knot strength |
| ― | Everyday connections | Tying down cargo, looping a dog leash round a post, mooring a ship |
- Wikipedia, "Capstan equation" (キャプスタン方程式, Japanese)
- Wikipedia, "Friction" (摩擦, Japanese)
- Patil, V. P., Sandt, J. D., Kolle, M., Dunkel, J. "Topological mechanics of knots and tangles." Science 367 (2020)
- High-school Basic Physics textbooks, unit on "Friction force" (various publishers)
※This article is a general-audience science explainer. The figures given are approximations meant to illustrate the mechanism. For actual mooring work, follow rope specifications and on-site instructions.