Capstan Equation Calculator (Rope Friction Round a Post)
How much load a small hold force can restrain when a rope is wrapped round a post, bollard or drum, how much force a given load needs, and how many turns it takes.
Results
The dashed line marks the value you entered.
What this tool does
Friction builds up all along the rope in contact, so the tension grows exponentially with the angle wrapped: T_load = T_hold × e^(μφ). That is why a person can hold a boat on a bollard with two or three turns, and why every extra turn multiplies the grip again — the chart shows how quickly. The result depends strongly on μ (wet, worn, or rope materials change it a lot) and the equation assumes the rope is slipping or about to slip. It is for understanding; do not use it to rate rigging or lifting equipment.
Formula
T_load = T_hold × e^(μ φ) with φ = 2π × turns (capstan or Euler–Eytelwein equation) · turns = ln(T_load ÷ T_hold) ÷ (2π μ)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mu | Friction between rope and drum | — |
nt | Number of turns around the drum | — |
fh | Force you hold the free end with | N |
w | Load you want to hold | N |
LD | Largest load that force can hold | N |
MU | Multiplication of the hold force | — |
FH | Hold force needed for that load | N |
NT | Turns needed to hold that load with your force | — |
Worked example
- Friction between rope and drum0.3
- Number of turns around the drum2
- Force you hold the free end with100 N
- Load you want to hold5000 N
- Largest load that force can hold4338 N
- Multiplication of the hold force43.38
- Hold force needed for that load115.3 N
- Turns needed to hold that load with your force2.08
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.