Page 3 model: Woollen jumpers

How did your favourite sweater become so big? And how big can it get?


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.

Wool unstretched (A), and wool stretched (B)

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
$$\frac{\mathrm{d}\ell}{\mathrm{d}t} = \alpha-\beta l,$$
where $t$ is time, $\ell$ is how much the jumper has stretched, and $\alpha$, $\beta$ are constants depending on the mechanical properties in states $A$ and $B$.

A jumper being stretched downwards

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:
$$\ell(t) = \frac{\alpha}{\beta}\left( 1- \frac{1}{\exp \left( \beta t \right)} \right).$$
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, $\ell \approx \alpha/\beta$.

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.

Jakob is a PhD student and mathematician from London and works mainly in differential geometry. In his spare time, he likes to draw, and think about mathematics in art.
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