Suppose you have a rubber band, and a point is marked on the rubber band at the 1/3 point. If you now apply force to the two ends of the rubber band to stretch it, will the point maintain its 1/3 position on the new rubber band? That is, is the marked point still the 1/3 point of the rubber band? If not, what is its position, and how do I calculate the position for the 1/4 point or 2/5 point, say?
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Assuming the rubber band is even throughout its length then yes the ¹⁄₃ mark will remain ¹⁄₃ of the distance as the band stretches.
To see this imagine taking three identical rubber bands of length $\ell_0$ and gluing them together to make a single band of length $3\ell_0$. The join between the first and second bands is at a position:
$$ x = \frac{\ell_0}{3\ell_0} $$
that is at ¹⁄₃ of the way along the joined bands.
Now apply some force to stretch the bands. Since the tension is the same everywhere in the joined bands each band stretches to some new length $\ell_1$ and the total length increases to $3\ell_1$. The join between the first and second bands is now at a position:
$$ x = \frac{\ell_1}{3\ell_1} $$
i.e. it is still ¹⁄₃ of the way along the band.
This will work for any fraction. Just imagine the band as made up from many smaller identical bands joined together, all of which expand in the same way under tension.
John Rennie
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