Ever tried to hold a rope that is pulled by someone or something, like a sail, a boat or a hanging weight…? If holding a pulling rope is something you’ve done a lot, you’ve surely soon figured out that if you wrap your end of the rope around something, like a ring, bollard or (as in the pic above – a winch), it’s much easier to hold the object trying to pull you.

You might also have noticed that the number of wraps around the object matters – the more times you wrap the rope around the winch, the more holding power you’ll have, i.e. the less force is required for you to hold the buckling rope still.

But how much less pulling force per wrap do you get…? That is, how much easier is it to hold the rope per wrap ?

Turns out there’s a mathematical formula for calculating that. It’s called “**Capstan Equation**” or “**Eytelwein’s Formula**” and looks like this:

**H = F * exp(-ua)**

where H is the holding force (i.e. the force you need to hold the object in place), F is the force pulling you, u the coefficient of friction, and a the amount of angle that the rope is wrapped around the winch, ring or bollard.

Notice that what we have is an **exponential function,** with both the friction coefficient and the wrapping angle as parameters. That is, **our holding force grows (or decays) exponentially** with both the friction coefficient and the angle. That’s very significant, since as you might have heard, anything growing (or decaying) exponentially, has some really surprising outcomes.

So, let’s look at what this means, by putting some numbers into the Capstan Equation: for the friction coefficient I’m going to use 0.4, which is fairly typical for rope against the type of surface on bollards, and for the pulling force F, I’ll simply use 1 Newton, so we can think of the holding force as a fraction of the pulling force.

The top graph shows the ratio of the pulling force you need to hold the pulling object static, over the wrapping angle. So, if you look at the x-value 360 degrees, that is, one full wrap of rope around the bollard, you see that you need about one tenth of the pulling force to hold the object. With two wraps (720 degress) you need one hundredth of the pulling force, and with three wraps, you need only about 1/1000th of the force pulling to hold the object.

Pretty impressive! And, as anyone who’s done any sailing would know, trying to hold a sheet without wrapping it around a cleat or winch is very painful, and totally impossible in even a light breeze, but a couple of wraps around a winch makes it a “breeze”.

*A good rule of thumb for the force multiplier, applicable for the gear you’ll typically encounter in marinas or boats is that 1 wrap reduces the required holding force to 1/10 th, two wraps to 1/100th, and 3 wraps to 1/1000 of the pulling force. Your exact milage will vary depending on the materials involved. *

**Moral of the story – never underestimate exponential growth! **

PS: an excellent derivation of the Capstan Equation can be found here:

Exponential growth is indeed powerful. However, the estimate of 0.4 is somewhat high for the coefficient of friction.

See https://www.samsonrope.com/docs/default-source/technical-bulletins/tb_coeffecient_of_friction.pdf?sfvrsn=408b2716_2#:~:text=What%20is%20Rope%20CoF%3F,side%20of%20a%20contact%20surface.

The value cited there is closer to 0.1 on average. This makes a very big difference in the answer. Instead of factors of 10, 100, and 1,000, we get something closer to 2, 4, and 8.