As the weather turns from grey to cold and grey, it’s time to get out your favourite woolly jumper. Only, it’s a bit longer than you remember. Has someone has been stretching it out behind your back? Instead of jumping to conclusions, let’s try to understand how this wool-d happen.
Your jumper is made up of yarn, made from woollen fibres spun together. Let’s assume the fibres are in one of two states: A, unstretched, or B, completely stretched out.
The jumper as a whole is much harder to stretch with the fibres in state B than in state A, because each individual fibre is already elongated.
As we start stretching the jumper, more and more fibres go from state A to state B, and the rate at which it stretches slows down. Let’s model this stretching over time as
dℓdt=α−βl,
where t is time, ℓ is how much the jumper has stretched, and α, β are constants depending on the mechanical properties in states A and B.
Assume that the jumper started off un-stretched (unless you got fleeced), so that l(0)=0. Then this equation has a unique solution, giving an explicit formula for how much your jumper has stretched:
ℓ(t)=αβ(1−1exp(βt)).
On the bright side, even if someone has been wearing it, this shows your jumper won’t go on stretching forever: as t gets very large, ℓ≈α/β.
This approximate solution was reproduced by experimental evidence from the appropriately named Wool Textile Research Laboratory, just in case you thought this whole ‘two-state’ yarn was a bit of a stretch.