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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Yes, you could, but the physics aren’t unit dependent. The failures here come from things like “the rate at which one inhales has to grow too high for the tissues of the windpipe to survive” and “bones cannot physically support the larger weight”, and in both cases you’re talking about materials strength. Most of the structural properties of materials have units of force per area (=pressure), and area embeds your length unit into the failure point.
As an example, suppose you have a 1 meter square beam that can support 1 GN/m^(2) (i.e., 1 GPa, but I’m intentionally going to use the explicit force and area units here) of weight, with a 1 meter cube on top of it pressing down on it with that maximum of 1 GN of force. Now, double the size of everything. The cube is now 2x2x2 = 8 m^3 and has 8 times the weight, spread across 4 times the area since we scaled up the beam. That means the pressure on the beam is now 2 GN/m^(2), which is beyond its strength and the beam breaks or buckles.
Now, redefine new units. Introduce the splark (abbreviation sp), where 1 splark = 2 meters. The strength of our beam is now 4 GN/sp^(2). In the initial state, the cube is 1/2 spark x 1/2 splark x 1/2 splark and the area of the top of the beam is 1/4 splark^(2), and the cube still applies 1 GN of force to that 1/4 splark^(2), so it’s applying 4 GN/splark^2 – as before, it’s exactly at the limit of our streel beam. Double the length and we now have an 8 GN force from the cube now pressing down on a 1 splark by 1 splark beam, resulting in an 8 GN/splark^2 pressure and failure of the beam.
In either case, the beam fails above a volume-to-area ratio greater than (1 m^3 / 1 m^(2)) = 1 meter (the initial state), which is equivalent to ((1/8) sp^3 / (1/4) sp^(2)) = 1/2 splark. The value of the ratio embeds your unit, and the system fails at the same value (expressed in your unit of choice) regardless of measurement choice.
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