Capstan Winch Calculator
Work out how little force holds a big load when a rope is wrapped round a drum, post or bollard. Solve the capstan equation for the holding force, the load, the friction coefficient or the number of wraps.
How to Use
- Choose what to solve for: Holding force, Load, Friction (μ) or Wraps.
- Enter the load on the heavy side and the force on the holding side, each in N, kN, lbf or kgf.
- Enter the friction coefficient μ between rope and drum, and the wrap in turns, degrees or radians (one full turn is 360°).
- Read the answer, the load-to-hold ratio e^(μφ), the wrap angle and how much of the load friction carries. The drawing shows the rope round the drum.
- Show Work turns the wraps into radians and works through the exponential step by step.
Worked Example
Three turns on a wooden post. Load 1,000 lbf, μ = 0.3. The wrap angle is φ = 3 × 2π = 18.85 rad, so e^(μφ) = e^(0.3 × 18.85) = e^5.655 = 285.7. The holding force is 1,000 ÷ 285.7 = 3.50 lbf; friction carries 99.65% of the load.
A half turn over a branch. 80 kgf hangs on a rope over a branch (180°, φ = π) with μ = 0.3. e^(0.3π) = 2.566, so the other end needs 31.2 kgf to hold it.
The common mistake: putting turns straight into the exponent. φ must be in radians. Writing e^(0.3 × 3) = e^0.9 = 2.46 for three turns gives a holding force of 1,000 ÷ 2.46 = 406.6 lbf. Three turns are 3 × 2π = 18.85 rad, and the right answer is 3.50 lbf.
Show Work
Formulas
Sailors, Euler and Eytelwein
Sailors and dock workers used the effect long before anyone wrote it down: a few turns of rope round a capstan, bitt or bollard let one person check a ship that dozens could not hold with a straight pull. Leonhard Euler worked out the mathematics in the 18th century by balancing the friction on each small piece of rope in contact with the drum, which is where the exponential comes from: each piece multiplies the tension by the same small factor.
Johann Albert Eytelwein, a Prussian engineer, applied the result to belts and pulleys in the early 19th century, and engineers still call it the Euler–Eytelwein formula when sizing belt drives and band brakes. The same equation explains why a cleat needs only a couple of figure-of-eight turns and why a climbing belay device works.
About This Calculator
This calculator solves the capstan equation for any one of the four values: the holding force, the load, the friction coefficient or the wrap. Forces can be in newtons, kilonewtons, pound-force or kilogram-force, and the wrap in turns, degrees or radians. It also gives the load-to-hold ratio, the wrap angle and the share of the load taken by friction.
The equation assumes a rope that is flexible and has no weight of its own, sliding or about to slide on a drum with a single friction coefficient. Real μ values vary widely, so treat the typical values here as rough and leave a generous margin for anything that could hurt someone if it slipped.
All calculations run in your browser; nothing you enter is sent anywhere.
Related tools: Force Calculator, Torque Calculator, and Weight Calculator.
Frequently Asked Questions
What is the capstan equation?
T_load = T_hold × e^(μφ), where μ is the friction coefficient between rope and drum and φ is the total wrap angle in radians (2π per turn). With 3 turns and μ = 0.3, e^(0.3 × 6π) = 285.7, so 1,000 lbf of load is held with 3.5 lbf.
Why does one more turn make such a difference?
The holding power grows exponentially, not linearly: each full turn multiplies it by e^(2πμ). At μ = 0.3 that is 6.586 per turn, so one turn holds 6.6 times the pull, two turns 43 times and three turns 286 times.
Does the drum diameter matter?
Not in the capstan equation: only μ and the wrap angle appear, so a thin post and a large winch drum with the same surface and the same number of turns hold the same ratio. In practice a very tight bend wears and weakens the rope, which is why bollards and winch drums are not made too small.
What friction coefficient should I use?
Rough typical values are about 0.25 for rope on steel and 0.3 for rope on wood, but μ changes a lot with the rope material, wear, wet or icy surfaces and paint. Measure it instead: hang a known load, wrap one turn and read the smallest holding force. 100 N held by 10 N with one turn gives μ = ln(10) ÷ 2π = 0.366.
How many wraps do I need?
Turns = ln(T_load ÷ T_hold) ÷ (2πμ). To hold a 500 kgf load with 20 kgf at μ = 0.25 you need ln(25) ÷ (2π × 0.25) = 2.05 turns, so take 3 and keep a margin. Choose Wraps in the Solve For bar to do this.
How do I use the Capstan Winch Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Sailing winches
Three turns of a sheet on a winch drum (μ about 0.3) let a 1,000 lbf load be tailed with about 3.5 lbf.
Mooring bollards
Two turns round a steel bollard (μ about 0.25) hold 2,000 lbf with 86.4 lbf on the free end.
Rope over a branch
A half turn (180°) over a branch at μ = 0.3 cuts an 80 kgf load to 31.2 kgf on the holding side.
Belt drives
The same equation sets the tension ratio of a flat belt: 180° of contact at μ = 0.3 allows a tight-to-slack ratio of 2.57.
Measuring friction
Hang a weight, wrap the rope and read the holding force: the calculator turns the ratio into μ.
Last updated: