Disclaimer: I did see a previous question touching on something like this but what I’m confused about was NOT addressed so hopefully this is allowed.
They say that the surface area volume ratio limits how big things can grow because surface area scales as a square while volume scales as a cube, so the ratio of volume to surface area goes up as you get bigger. Fair enough. BUT: how is this not just a matter of what units you’re using?
For example, a 1x1x1 ft cube has a surface area to volume ratio of 6sq. Ft to 1 cubic foot, so 6:1. A 1x1x1 meter cube has a ratio of 6:1 too but the units are meters. Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?
To bring it back to the actual question, wouldn’t your ratio be essentially the same no matter how big your object is? Imagine you expanded everything in the universe by the same amount but kept your unit of measurement the same, you wouldn’t suddenly hit some limit where it stops working right? Does it have something to do with the size of molecules and proteins etc? Please help I am so confused
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The thing you’re forgetting is how area and volume related to the ability of some structure to hold its own weight.
Volume correlates with weight. Switching units of volume isn’t going to change its weight. You can’t make an object lighter or heavier by deciding to measure its volume in different units.
The internal strength of an object correlates with area (specifically cross-sectional area). Switching units of area isn’t going to change its strength. You can’t make an object stronger or weaker by deciding to measure its area in different units.
As an object increases proportionately, its volume (and therefore its weight) increases by the cube of that factor but its area (and therefore its strength) increases only by the square of that factor. So its weight increases much faster than its strength, meaning at some point its weight will exceed its strength.
Specifically your 1m^(3) object is about 35 times as heavy as your 1 cu. ft. object, but only about 11 times stronger.
> Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?
Yes, but the laws of physics don’t depend on the units you choose to work with. Just because 1km is a smaller number than 10ft doesn’t mean it’s a shorter distance. The actual numbers are just arbitrary. This is why physicists often prefer to work with [dimensionless quantities](https://en.wikipedia.org/wiki/Dimensionless_quantities) which are independent of the units used to measure them.
Any particular material has something called it’s yield strength. That’s the maximum amount of stress (i.e force per unit area) that it can take before it is permanently deformed. If you double the size of an object, the volume becomes eight times larger (2^3), so the object now weights eight times as much. However, the cross sectional area of the object is only four times larger, so the material can only take four times as much load.
That means that as you scale an object up, the weight it has to support grows faster than the ability of the material to support that weight, and eventually you reach a point where the material cannot support the weight and will fail under the load.
> For example, a 1x1x1 ft cube has a surface area to volume ratio of 6sq. Ft to 1 cubic foot, so 6:1. A 1x1x1 meter cube has a ratio of 6:1 too but the units are meters. Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?
Sure. But we’re concerned with *growth*. You can use whatever units you want, but a 2x2x2 <unit> object is always going to have 4x the surface area of a 1x1x1 <unit> cube, and 8x the volume.
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