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I've heard it said that the force a muscle can exert is proportional to its cross-sectional area. It seems to me it should be proportional to length also. If you have someone pulling on a large block, you could increase the force by getting another person to pull in parallel, OR by attaching a rope to the block and having both of them pull in series on that one rope. So shouldn't muscle strength be proportional to both area and length, making it proportional to volume?

Does the answer have to do with the fact that material strength is proportional to cross-sectional area, so this puts a limit on the amount of force that muscles can exert without breaking?

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  • $\begingroup$ sounds more like theory than reality. Sure, there is correlation between sectional area of muscles and force exerted, but proportionality? I think anyone hitting a gym knows better. $\endgroup$
    – Umaxo
    Commented Dec 13, 2019 at 5:36
  • $\begingroup$ @Umaxo Well, there are other factors of course that would have to be considered in a real situation. When talking about a proportionality between two variables you assume that everything else is held constant. $\endgroup$
    – CyborgOctopus
    Commented Dec 13, 2019 at 5:41
  • $\begingroup$ "you assume that everything else is held constant" and of course changes are small enaugh to use first term of taylor series:) I just think that in reality such simple relationship cannot hold. This is in contrast to the proportionality of p~T in ideal gass. Yes, you need to keep volume and particle number constant and it is not the most exact formula, but it can be used in real situation pretty well. It just seems to me, in the case of muscles, such simple relationship is useless anywhere else than textbooks. But i might be wrong $\endgroup$
    – Umaxo
    Commented Dec 13, 2019 at 6:29
  • $\begingroup$ @Umaxo I think it would be useful when roughly estimating strength difference between animals of hugely different size, like insects and humans, because the change in area is so great that it makes a really big difference. It's probably not super useful for figuring out strength differences between humans or anything like that. $\endgroup$
    – CyborgOctopus
    Commented Dec 13, 2019 at 6:37

2 Answers 2

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you could increase the force [...] by attaching a rope to the block and having both of them pull in series on that one rope

No, this doesn't work. In the series arrangement, their forces don't add. However, the work they do does add.

This is all exactly in analogy to the way muscles work. The force a muscle can do is proportional to its cross-sectional area, but the work it can do in one contraction is proportional to its volume.

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  • $\begingroup$ Okay, I think I get it now. I was really just confused about which arrangements were series and which were parallel. $\endgroup$
    – CyborgOctopus
    Commented Dec 13, 2019 at 5:05
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After doing a bit more research, I believe I can answer my question. In the case of muscle cells, the rope analogy I gave is incorrect. A better analogy would be a line of people lying horizontally who have linked arms and legs to form a human rope. This rope will then have a constant tension which all of the people in it maintain through the force they exert, but adding more people in series doesn't increase the tension (pulling force) of the rope. Someone please let me know if this isn't a good way to think about it.

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