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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1$\begingroup$ Try it yourself and see $\endgroup$– RC_23Commented Jan 15 at 5:51
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$\begingroup$ My intuition tells me no, but surprisingly I don't have anything that's elastic enough to show the difference... $\endgroup$– Number BasherCommented Jan 15 at 5:52
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$\begingroup$ But it's possible that it is true, perhaps due to equal tension of string; but I don't see how it directly connects to length. $\endgroup$– Number BasherCommented Jan 15 at 5:54
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3$\begingroup$ Voting to reopen. The question is perfectly clear and already has a good answer below. $\endgroup$– gandalf61Commented Jan 15 at 9:55
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1$\begingroup$ Is this rubber band a theoretical perfect circle of equal width and thickness with homogeneous rubber throughout? Or is this a thin sector sliced off an innertube with scissors in the real world ? $\endgroup$– CriggieCommented Jan 15 at 20:50
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1 Answer
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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.
answered Jan 15 at 7:19
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2$\begingroup$ What if the rubber band is not even? $\endgroup$ Commented Jan 15 at 7:21
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2$\begingroup$ If the rubber band is not even then we imagine it as a series of small even bands with lengths $\ell_i$ and force constants $k_i$. Then when we apply a force $F$ each band expands to a new length $\ell_i(1 + F/k_i)$. Now sum up all the new lengths before and after your mark to find where the new position of the mark is. $\endgroup$ Commented Jan 15 at 7:25
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1$\begingroup$ @NumberBasher As an extreme case, an uneven rubber band of length $l$ might behave like a metal rod of length $l/3$ connected to a rubber band of length $2l/3$. Obviously in that case the first part doesn't stretch as much as the second part. $\endgroup$– JiKCommented Jan 15 at 17:01
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3$\begingroup$ I think that this breaks down right next to where it's being held. I.e. if you put the mark 1% of the distance, it's going to end up less than 1% of the stretched length. $\endgroup$ Commented Jan 15 at 18:26
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1$\begingroup$ @Acccumulation I'd be interested in hearing more about your thinking here. It seems to me that all the molecules in the band that are not constrained by "fingers" (or whatever) would be free to move just like all the others along the length. What you describe sounds like a "boundary layer" (like airflow over a wing), but I'm not convinced. In that case, there is friction between the aircraft skin and the air molecules right next to it, and then between that layer of molecules and the next, and so on. How does something similar happen here? $\endgroup$– SteveCommented Jan 16 at 0:12