June 2013  Cheap Gear and the Capstan Equation

What is the Capstan Equation and how does it apply to rock climbing? In this edition of UJCC, Uncle Jacko gets technical, offering insight into why knots and belay devices work. He also discusses cheap gear and reveals how he decides between buying in-store versus online.

Tribology

The time has come to provide some insight into why knots and belay devices work. My long-time friend Duncan Hall offered some useful advice about friction, knots and climbing ropes many years ago, which I have always wanted to publish and will now paraphrase. 

It turns out that the ratio of output to input forces on a climbing rope going around a karabiner is equal to the exponential of the product of the coefficient of friction and the angle of the bend:

… where Tin is the tension felt by the belayer, Tout is the tension put on the rope by the climber (his body weight if he is dangling on the rope), μ is the coefficient of friction, ϴ is the angle of the bend of the rope going around the karabiner in radians and e is Euler’s Number, 2.71828…

If the coefficient of friction is, say, 0.7 (for dry nylon on stainless steel) and the angle is π (half-way round the karabiner) then the tension multiplication is e2.2, which is close to 9. By bringing the rope around a full circle, 2π (i.e. 360°), the tension multiplication is just over 81. Two turns gives you a multiplication factor in excess of 7,000. This explains why a relatively small person can easily belay a much larger person and even quickly arrest a fall. Sailors have long known about this effect, and routinely use it to raise anchors and pull in sheets on sailing boats with two loops around a motorised or hand-winch capstan (for a multiplication factor in excess of 7,000), which is why this formula is called the Capstan Equation. Note that the formula shows that the diameter of the bend does not affect the friction multiplication factor, which is not what I would have immediately guessed.

If the rope is particularly slippery – a wet monofilament nylon fishing-line, for example – you might have to make quite a few passes around itself so that the higher angle component compensates for lower coefficient of friction component. And if you’re a kid with particularly wriggly feet, you might have to add more turns to your shoelace knots by “double knotting” them.

In short, we learn from the Capstan Equation that friction doesn’t add up, but multiplies. Lead climbers leaving the rope zig-zagging up the pitch behind them soon find themselves dragging hard to get the rope through the runners (three 120° bends act the same as one 360° bend, even if in different directions). For this reason, climbers on difficult routes (and so using lots of protection) often use two ropes in order to minimise the angles the ropes subtend.

There’s also an interesting relationship between the Capstan Equation and so-called imaginary numbers: e to the power of pi equals the reciprocal of the square root of minus one (i) to the power of two i – which is also almost equal to twenty plus pi:

Duncan thinks that this is the only occurrence where all the really cool numbers (1, i, e and π) come together in a simple equation.

Cellphone Coverage

Having made some bold assertions that cellphones often work on hills and mountains overlooking the plains, I now find that this is not always the case. When tramping in the Eyre Mountains recently, I could see a number of hilltops where I knew there were Vodafone cellsites, but my iPhone wasn’t having a bar of it (excuse the pun). The most likely explanation is that the cellsite electronics were set up so that mobiles too far away couldn’t answer in time to establish a link (light is pretty quick, but it’s not as fast as you may think). Alternatively, perhaps no antennas pointed in our direction, which would be sensible considering how few people there are in the Eyre Mountains at any given time. Be warned!

Cheap Gear

I accidently discovered the world’s largest on-line shop a few months ago, that sells everything from LED torches to sleeping bags, to Chinese Army boots and much more (www.aliexpress.com). To be frank, I would be cautious about some of their products; but on the other hand, I’m not keen to pay a premium price to New Zealand retailers who import exactly the same stuff from China themselves. 

The escrow service means that your money is safe until you get the goods and the free postage and lack of GST (for items under NZ$400 including freight) is a bonus. Downsides include specifications that are sometimes impossible to fathom and ambiguous translations. I guess there’s also a message to manufacturers here: get your gear made in China and maybe you can buy it cheaper from Aliexpress than you can from your outsourced factory? If you really want to buy in bulk, go to the sister site www.alibaba.com, but be warned about traps for young players when you start paying with Telegraphic Transfers (TT) instead of VISA.

I have a rule of thumb: if the local retailer provides me with service, customisation, fitting and advice then I buy it from them, even if they are more expensive. There is, after all, real value in buying well-fitting boots. However, if the shop only marks up the price and doesn’t add any extra value, then I’ll shop on the Internet. Getting good advice in person from a shop and then buying on the Internet is a form of stealing, or at least dishonesty.

Hakili matagi,

Robin McNeill

r.mcneill@ieee.org

This column was originally published in the June 2013 FMC Bulletin. We will be regularly re-publishing a number of stories from Uncle Jacko’s Cookery Column here on Wilderlife. 

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