Scale Factors -II

Page Two
The reason that scale factors are important is that some physical properties of bodies depend on area, and some depend on volume. This has important consequences for the sizes of those bodies, and how they function.

Mass

The mass of an object depends on its volume. For example, if you increase an object's volume by eight times, its mass will also be eight times greater.

Here's a spider that has grown in size by a factor of three. The big one is three times as long as the small one.

From the previous page, you know that this will give the bigger spider a volume that is 27 times the volume of the small spider. This means the bigger spider's mass is 27 times the mass of the smaller one. The big one is 27 times heavier.

This principle holds true whenever you enlarge someting. If a dog were to grow to 5 times its normal size, it would weigh 125 times as much. The weight of an object increases as the cube of the scale factor. Another example? Suppose an ant were to grow 200 times bigger. It would weigh 200³, or 8,000,000 times as much.


Strength

The strength of an organism depends on its cross-sectional area. For example, you might measure your strength by how much weight you could lift with your biceps muscle. If you wanted to get stronger, you would exercise that muscle to make it bigger. But what will increase is the cross-sectional area of the muscle. It will get wider in the middle, but won't get longer.

For example, a muscle that is twice as wide might be four times stronger.
The strength of an organism increases as the square of the scale factor.
Another example? A leg that is twice as big will support four times as much weight. An ant that is 200 times bigger will be 200², or 40,000 times stronger.


Do you see the problem now? As an organism gets bigger, its weight increases as the cube of the scale factor, but its strength increases only as the square. Its strength isn't increasing as fast as its weight.

Let's use an actual example to make it clear what's happening. We'll use the example of an ant that has grown 200 times in length.


Let's make the normal ant's strength 1 unit, and its weight 1 unit. We can then make a ratio of strength to weight by dividing these numbers. We get 1/1 = 1. We'll call this ratio the relative strength ... for the normal ant it's 1.

Now let's work it out for the big ant.
The big ant is 200 times as long, so it will be 200², or 40,000 times stronger.
The big ant's weight is 200³, or 8,000,000 times heavier.
The big ant's strength to weight ratio is 40,000/8,000,000 = 0.005, so it's relative strength is 0.005, or 1/200^th.

Although the ant's size has increased, its strength hasn't kept pace with its weight. It has only 1/200^th of the strength needed to support itself. The result would be an giant ant that couldn't stand up! In fact, since its body shell strength won't have increased enough either, it would end up as a squashed puddle on the floor, totally unable to support itself or its internal organs.


Oxygen Requirements

Cells in living organisms require oxygen to function. If all the cells in an organism were to grow by a factor of X, their surface area would now be X² times as large, but their volume and mass would be X³ times bigger.

The amount of oxygen required depends on the mass of the cells, so the bigger cells would need X³ as much oxygen. However, oxygen gets into the cells through their surface membranes, which have only increased by a factor of X².

The bigger cells' oxygen-aquiring ability won't have increased as fast as their need for oxygen ... so all the cells will die. An organism that increases in size will suffocate!

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You may be wondering how any large animals can exist at all, given what we've shown you. For example, when a young animal grows to adulthood, how do its bigger cells breathe?
The answer is simple ... the cells don't get bigger ... they divide. As the organism grows, the number of cells grows, but the cell size remains about the same.

Consider animals that are really large, like the elephant. What do you notice about its body? An elephant isn't shaped like a dog or mouse that's been expanded many times. It's legs are much thicker, in proportion to its body size, than smaller animals.

In order for an organism to be this big, its supporting bones and muscles in the legs must be bigger in proportion to the rest of its body than organisms that are small. Otherwise it wouldn't be strong enough to stand up.

If a spider were as big as an elephant, it would have to look something like this. Notice the very thick legs ... far thicker in proportion to its body than a regular-sized spider.

But of course this spider could never exist. Not only would it suffocate because its cells couldn't get enough oxygen, but it would also be unable to breathe because, although its shell would have to be proportionately thicker to support its inner organs, its area would have only increased as the square of the scale factor, while its mass will have increased as the cube of that value. And a spider breathes through its skin.


So despite what you see in really bad movies, if anyone ever manages to come up with a device to instantly make insects a hundred times bigger, or to enlarge a baby to skyscraper height, none of these enlarged organisms could ever survive. They'd be unable to stand up, and they'd suffocate!