# Run The Math, Or Try It Out?

I was reading Joshua Vasquez’s marvelous piece on the capstan equation this week. It’s a short, practical introduction to a single equation that, unless you’re doing something very strange, covers everything you need to know about friction when designing something with a rope or a cable that has to turn a corner or navigate a wiggle. Think of a bike cable or, in Joshua’s case, a moveable dragon-head Chomper. Turns out, there’s math for that!

Basically, the more you wrap a cable or rope around something stationary, the more friction you have to deal with. I put this to good advantage last Spring when my son and I were doing some random tree-climbing with ropes. Turns out that four or five loops of climbing rope against fairly frictiony bark is enough to hold the weight of a grown man, with nothing other than the weight of the rope itself on the other end, for instance. I was also using this effect in a recent wall-plotter-bot design that uses simple cable braid instead of the ubiquitous timing belt.

In none of these cases did I work out the capstan equation with my pocket calculator: four loops around is almost always enough™. But by digging into the math and physics, I got more insight. Basically, the friction is an exponential with the angle times the friction coefficient of your cable in the exponent. So what? So, that next turn holds exponentially more weight when you’re climbing. And the grippiness of the tree bark matters in just the same way. You might know this intuitively from experience, but it’s nice to have numbers.

And what’s even better about the insight from doing the math is what doesn’t matter. The radius of the tree falls out, so you can pick a fat branch or a skinny one, as long as it’ll hold your weight!) The fat branch bends less sharply, which gives you less friction per centimeter, but there’s also more centimeters per wrap, and they cancel out. So pick a fatter, rougher tree over a skinnier, smoother one. I would have never thought that!

Here at Hackaday, we’re big fans of rough-and-ready learning by doing. Will this work? Try it out! (If it shouldn’t, and yet it does, it’s a hack.) But there’s also a lot to be said for knowing the underlying math and physics too. Many people think of math as being about crunching numbers, but I find the deeper intuition about which variables in your problem matter, and which don’t at all, infinitely more useful. Learning by doing, guided by a good physical / mathematical intuition, is the best of both worlds. And it saves wear and tear on your mint HP-48G.

## 39 thoughts on “Run The Math, Or Try It Out?”

1. Foaly says:

Agreed, that was a very interesting piece. I didn’t know the Capstan equation, I would never have guessed the relation was exponential, I had never thought about it but intuitively I would have believed it was linear. Thanks for the article !

1. Jul13 says:

Even when you know that it is exponential, it’s hard to translate that to something we can use intuitively. Most things we grow up interacting with are almost always linear because in a way, exponential growth is …dangerous…? Fires and avalanches both grow exponentially. Disease. The speed of falling objects. We avoid all of these things and that makes it hard to grasp.

1. Dude says:

The difficulty of finding an intuitive comparison for something that is exponential is the fact that it tends to appear as an on/off thing. Either the rope will hold, or it won’t.

2. Ostracus says:

” Most things we grow up interacting with are almost always linear because in a way, exponential growth is …dangerous…? ”

Credit card debt. :-)

3. Sandro Magi says:

Exponentials are indeed dangerous, but falling objects are polynomial, not exponential. We’re quite good at predicting those since we can easily catch them.

4. Alex says:

“The speed of falling objects”?
Many people equate “exponential” with “quick” or with “nonlinear”. It’s not! Exponential growth is much more unintuitive in real life than any polynomial nonlinear growth.

1. Jul13 says:

Us non or semi technical people do use it to mean simply: non linear or “graph is curved, not straight”.

2. Jul13 says:

I heard an “educated” man tell “news” media that turning off our zoom cameras was an effective way to slow climate change. He went right into the math for them, about how the camera takes up like %83 of the power used by a video call. Seriously.

The fact that a five hour zoom calls takes about as much energy as running an electric car for 10 seconds… as far as he was concerned that was totally irrelevant.

Some people have a sense of perspective so undeveloped it is disgusting. It makes it really easy for them to delude themselves into whatever beliefs they prefer.

1. RW ver 0.0.1 says:

I would spitball that an all day conference call, on premises (Just to prevent staff from different departments mingling) with a dozen participants is using less electricity than if just one of them used the elevator to get to the conference room or boardroom.

1. Jul13 says:

Exactly. Most people dont understand the difference between the scale of energy for a signal, like light or sound, vs the scale of energy for doing work, like heating or moving something. It’s pretty much always absurd to worry about the former in the context of energy efficiency. To a philosopher… how could it ever be otherwise?

1. Dude says:

The signal might require very little energy, but running the laptop (or desktop) computer takes a thousand times more.

2. Dude says:

Elevators use surprisingly little energy.

75 kg by 20 meters is 14.7 kJ which is about 6 minutes on a laptop that is consuming 40 Watts (not to mention the routers and other infrastructure required for the call).

1. Dude says:

Which is btw. the reason they say the most energy consuming part about an elevator is the light bulb inside.

1. Dude says:

Which reminds us to the point that when all the participants to the conference call are in a separate rooms, they will all have their own light bulbs…

2. Jul13 says:

But lighting a room is clearly work. Lights used for signals are generally smaller and possibly efficient. Even then it seems like that saying is based on old incandescent bulbs.

2. Dude says:

And before you complain: the elevator cab is counterbalanced.

1. Jul13 says:

It is counter balanced, but I’m not sure they’re balanced that well.

2. David says:

I heard somewhere on the internet – so it must be true – that an elevator’s counter-balance is actually balanced for when it is over half full. So it actually does more work with one person in it than when it is half full. (don’t mind me, I’m just spreading baseless rumour).

3. Dude says:

Well it’s not going to be hundreds of kilograms off balance. That would be a safety risk.

4. Foldi-One says:

Most elevators will have to be 100’s of KG off balance, at least some of the time – they are generally rated to 10 persons or more from what I’ve seen, and 10 people is (even if they are all tiny) well into the 100’s of KG…

Sure they are counter weighted, which helps control the forces involved, so a weaker motor can do the work – you don’t want to have to lift the tonnes of box around the people on the motor.

Even if it was perfectly weighted (which it won’t be) you still have frictional losses, which over long elevator rides are going to be meaningful. Which still doesn’t make an elevator inefficient, its just not as simple as lift x mass person up x floors of height, boom there’s the energy used, there’s plenty of losses in there, and the counter balance would never be perfect.

3. Dude says:

> it seems like that saying is based on old incandescent bulbs

It was true still when the incandescent bulbs were swapped for 12 Watt fluorescent tubes. The light is on for 24 hours which is about 288 Watt-hours, which is a Megajoule of energy. This much energy would lift the average person almost a mile up in the sky.

1. RW ver 0.0.1 says:

You’re missing some of your favorite facts about electric motors, they’re efficient at one end of their range or the other. I guess the 15-20 kilowatt motors on the top end of a lift shaft might get close to your single person figure times the maximum person capacity of the lift… when it’s at maximum capacity.

2. Dude says:

> they’re efficient at one end of their range

In terms of speed. The efficiency is low during the initial acceleration, but since the elevator quickly reaches its designed traveling speed, it can operate near peak efficiency for most of the time.

2. Foldi-One says:

In many ways it is totally irrelevant that driving using more energy, its a different problem altogether – to communicate effectively to a group or as a group you don’t need video, just audio will do nicely (usually). So cutting down on processing power, network bandwidth, the sensor itself is a saving you can easily make. So yes reducing personal vehicle use would be a bigger gain, though how you’d manage to go buy your locally sourced fresh veg if you can’t travel to the farm shop for example – its not like that can be easily avoided – transport and communications are not a directly comparable problem! (though they can be linked)

And with everyone and their dog on video chats these days if they all went audio only that would be a massive saving in energy – millions of users multiplied by a few watt hours adds up to turning off the odd power stations on its own. The important detail to all these things is even small gains multiplied out across millions (even billions) of users is still highly significant, and simple easy changes like not using a camera can make a huge difference to the total power use, while not changing your lifestyle much, or requiring large infrastructure changes (which being able to avoid personal cars would need) to gain that future efficiency…

Same thing with music videos… Though modern pop music seems to be universally terrible, so I assume the only attraction is the video… But its supposed to be music why are we streaming video?!?! For that matter why are we streaming at all, yes the 500 odd GB my music library has reached does require material investment to store locally, but zero extra energy to listen to compared to engaging multiple networking devices along the way to the server It would be streamed off, to play on the same computer that can just hold the data.

Streaming and internet services are useful, and can be greener – far more efficient to stream a movie you will never want to see again, or for a very long time at least, than packing up a polycarbonate disc and shipping them around the world to you. Almost certainly still vastly better than the old (now defunct entirely as far as I can tell) disc rental systems, where discs might be used 100’s even 1000’s of times, which is great for its green credentials, but the travel back and forth to the store isn’t so great, probably enough to undo that gain many times over..

Though streaming in power consumption terms is terrible compared to broadcast TV and radio, again its the many thousands of users tuning in to that one big transmitter so effectively is milliwatts per user vs those same users firing up many networking devices on the way to the server to start a stream so multiple watts per user (and that’s assuming they use the same end device, so same powerdraw – which we all know they don’t, even smart TV’s are vastly lower powerdraw than a normal computer, and a radio can run months on a single battery while still being handheld if you want it to now)..

1. Dude says:

It’s a moot point to complain about the energy use of your laptop computer when 2/3rds of domestic energy consumption is hot water and hot or cold air. Put triple glazed windows in and you save a thousand times more energy than turning off video when on skype.

1. Foldi-One says:

Indeed, more efficient houses will save alot, though how much isn’t always as clear cut – you go putting in triple glazed ‘airtight’ window you lower the heating cost, but in most older properties you will also then require lots of ducting, fans and heat exchangers added in as well to keep the air fit to breath, or those windows spend all their lives open being less insulating than the single pane they replaced – ironically a somewhat drafty house can actually be nicer and healthier to live in…

Its also a significant fiscal and energy investment for a more efficient future, which is great if you can afford it, but can you? Just turning stuff you don’t actually need off has no cost, and across huge numbers of people even the tiniest change helps significantly.

p.s. yes all that extra work to make an old house superbly insulated and good to live in should still come out more efficient in the end, but its significantly more challenging to do particularly in parts of the world where houses are generally not made entirely in wood, and cuts down on the efficiency gained compared to building a house around efficient heat management and air circulation in the first place – which for me should be the bigger target, make all new structures passively cooled and heated (yes its possible) or at least give a maximum heating/cooling energy budget, so proper use of convection etc means they are designed around efficiency. The older housing stock, despite its vast embodied energy is probably better of being scrapped and rebuilt slowly over time than bodged again and again, if its already lasted 100+ years that embodied energy could be considered well spent, as long as what you expend in replacing it is better, and will also last.

3. snarkysparky says:

Interesting philosophical quandary. I was once asked how many ‘other’ things could be plugged into the same outlet as our companies product and not blow the breaker. As the resident EE I pleaded my case that this was something better calculated than trial and error experimented. Manager was not an EE. So i started the company product and plugged in things until the breaker tripped. Breaker was labeled 20A but it took round 28 to actually trip this one. So what does one make of this result. Should we put in our product manuals that one can run other appliances to 28 Arms – our products RMS and be fine.
I’v seen it sooooo many times. Manager to me test it. No I can calculate it. No test it… Ok …… Now what the F to do with this one off data point ignoring the two hindered other interacting variables.

I trust theoretical analysis first that is then backed up by careful experiment, Not really necessary for the load experiment. Ohms law is soundly accepted science. Or is it a conspiracy!!

1. Dude says:

Was it a slow fuse or a fast fuse?

2. Grey Pilgrim says:

Unless you left the load on the breaker until it reached thermal equilibrium (could be 8-10 hours to really get there) even that data point is worthless.

Breakers trip longer at room temp and (in the USA at least) are normally calibrated at 40C. Plus you have a little bit of margin between the nameplate rating and the trip current.

So I’m not surprised it took >20 A to trip a 20 A breaker. And I bet it would be less if you tested it long term.

1. It’s bombproof even if done wrong — I never tie that bowline/loop at the end, just hang the remaining rope up over the tensioned part to keep it off the ground.

Thinking hard about it, though, there’s no reason not to tie a knot there and make it absolutely, belt-and-suspenders solid. Takes ten seconds. Thanks!

4. craig says:

I discovered the article’s premise as a small child after watching Indiana Jones. We couldn’t figure out how to get a “whip” to tie itself to a branch like Indy did in order to swing from it. Undeterred, after locating a suitable piece of rope and slapping it over some branches, it was pretty clear that as long as you could get it to wrap around several times, you could indeed swing from it. If you swung back and forth a few times, though, the rope would definitely work itself looser on each swing and you would pretty quickly wind up on your butt, and into some imaginary lava pit or crocodile’s mouth.

The rope has to be sufficiently skinny enough, too. A thicker rope didn’t exactly wrap tight enough, and a super skinny piece was either too weak to support our weight or too hard to hang on to. Thinking back on it, Indy’s whip is good and fat at the handle end and really skinny at the whip end. Smart guy, that Dr. Jones.

Fun to see, an undisclosed number of decades later, the math treatment explains our childhood discovery.

1. Alan says:

If it supports his weight, how come it detaches afterwards with a simple flick of the wrist?

1. Poetic license is exponential in the number of seconds that getting it right detracts from the overall story?

1. Grey Pilgrim says:

Elliot’s second law?

Test case anyone?

2. Dan says:

Possibly by de-tensioning it, removing the friction. If you send a small wave down the whip it’ll disengage?

3. Foldi-One says:

There is only significant friction when there is some tension on the rope/whip, there will always be some just from its own weight and stiffness, but the real grip comes from it being under load, so a flick to loosen the coils while its not under any load should let it loose relatively easily, as long as you don’t then pull the loose coils hard enough to bind back up.

The one flick release every time as effortlessly as the movie, probably not no matter how much you practice, but its certainly not impossible on paper. Would need to know more about the material properties of what its wrapped around and the whip itself to know if it would be one of those spend 5 mins shaking it loose, as just the one loop would hold a man its so grippy, or the so slippery and flexible that 20 turns just barely holds the weight.

Its pretty easy to get a feel for it with a pens/pencils in a vice and a shoelace/string – when you haven’t put a big load on the string it will slide with a gentle pull even with enough turns to latch hard under load. Would be fun to test such things with a whip just for fun, would also show up just how hard it is to accurately model such things, so many minute variables..

2. I bet the unravelling is just from it being a bit loosely wrapped, and the free end flopping up and over as the looseness works itself out. Then you lose a coil and fall on your butt.

If you could flop even more over, so that it could tighten up without losing a coil, I bet you’d be safe from crocodiles. I’ll have to go test it out at the local pit.

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