Page 3 model: Woollen jumpers

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

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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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