The straw that broke the camel’s back

Oct 18 JDN 2459141

You’ve probably heard the saying before: “It was the straw that broke the camel’s back.” Something has been building up for a long time, with no apparent effect; then suddenly it crosses some kind of threshold and the effect becomes enormous.

Some real-world systems do behave like this: Avalanches, for instance. There is a very sharp critical threshold at which snow suddenly becomes unstable and triggers an avalanche.

This is how weight works in many video games, and it seems ridiculous: In Skyrim, for instance, one 1-pound cheese wheel can mean the difference between being able to function normally and being unable to move. Fear not, however: You can simply eat that cheese wheel and then be on your way.

But most real-world systems aren’t like this. In particular, camels are not. Yes, zero pieces of straw will not break a camel’s back, and some quantity of straw will. No, there is not a well-defined threshold at which adding just one piece of straw will kill the camel. This is one of those times where formal mathematical modeling can help us to see things that we otherwise couldn’t.

If this seems too frivolous, consider that this model need not be about camels: It could be about the weight a bridge can hold, or the amount of pollution a region can sustain, or the amount of psychological stress a person can bear. I think applying it to psychological stress is particularly appropriate at the moment: COVID-19 has suddenly thrust us all above our usual level of stress, and it’s important to understand where our limits lie.

A really strict formal model useful for engineering purposes would be a stress-strain curve, showing the relationship between stress (the amount of force applied) and strain (the amount of deformation of the object). But for this purpose there are basically two regimes to consider:

Below some weight y (the yield strength)the camel’s back will compress under the weight, but once the weight is removed it will return to normal. A healthy camel can carry up to y in straw essentially indefinitely.

Above that point, additional weight will begin to strain the camel’s back. But this damage will not all occur at once; a larger amount of weight for a shorter time will have the same effect as a smaller amount of weight for a longer time.

The total strain on the camel will thus look something like this, for exposure time t: (w-y)t

There is a total amount of strain that the camel can take without breaking its back. This has units of momentum, so I’m going to use p.

What is the amount of straw that breaks the camel’s back? Well, that depends on how long it is there!

w = p/t + y

This implies that even an arbitrarily large weight is survivable, if experienced for a sufficiently small amount of time. This may seem counter-intuitive, but it’s actually quite realistic: I’m not aware of any tests on camels, but human beings have been able to survive impacts of 40 g for a few milliseconds.

If you are hoping to carry a certain load of straw by camel over a certain distance, and need to know how many camels to use (or how many trips to take), you would figure out how long it takes to cover that distance, then use that as your time parameter to figure out the maximum weight a camel could carry for that long.

So what would happen if you actually added one piece of straw at a time to a camel’s back? That depends on how fast you add them and how long you leave them there!

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