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not only does it affect the weight of the bow but also its elastic ability. the more elastic the material the less force the archer needs to use in order to pull the bow, or give you more pulling effect for the same energy.
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And if we're being picky then we can talk about efficiency losses depending on material too - coefficient of restitution etc. are probably quite relevant here. I seem to remember that the maximum return on the energy you put into a bow is on the order of 1/3, and that the rate of return is in some way dependent on the length of the bow.