According to Insultingly Stupid Movie Physics, it’s only a matter of time before America’s favorite rapidly expanding rhino-hipped faux populist is crushed under the weight of his own snack-inspired density swell:

It’s an old movie gimmick; a misguided scientist, radioactive fallout, pollution, or some other folly of mankind abnormally shrinks or expands someone or some creature. While we must admit to being entertained by such gimmicks, the physics are another matter.

Let’s start with the density problem. Ordinary matter is mostly empty space, and so it is conceivable that an object could be shrunk or expanded by somehow adjusting the amount of empty space inside it. Unfortunately, this would leave the weight exactly the same.

Expanded objects or persons would have such low densities that they would be blown away in the wind like big balloons. Tiny people would suddenly exert huge pressures under their little feet since the area of their feet would be miniscule but their weight the same.

For instance, a normal-sized person exerts a pressure of about 2 pounds per square inch with their feet when they are standing on both feet. If their weight stayed the same and they were shrunk by a factor of 100, a six foot tall person would now be about 0.72 inches tall. Their foot pressure, however, would rise by a factor of 10,000 or in other words become 20,000 psi.

Such a person would instantly sink if they stepped on mud. The pressure under their feet would exceed the compressive strength of concrete (typically 3000 to 4000 psi) and would likely mar the surface of sidewalks. How could such pressures be possible? To explain it we must first look at the mathematical model for pressure:

P = F/A

Where P is pressure, F is the magnitude of the force (in this case weight), and A is area (in this case the area of the bottoms of a person’s feet). Note that when weight remains the same and area decreases pressure increases. Since pressure and area are inversely proportional, decreasing area by a factor of 10,000 increases pressure by a factor of 10,000.

Reducing a person’s size by a scaling factor of 100 decreases the area of their feet by a factor of 10,000, since area scales up and down with the square of the scaling factor. If this seems strange, then consider the fact that the area of a rectangle is the width times the length. If both dimensions are decreased by a factor of 100 then the new area is decreased by a factor of 100 times 100 or 10,000.

The density problem could perhaps be solved by removing molecules when reducing size and adding them in when increasing size. This would be an inordinately complex process, because it would be extremely difficult to make sure that molecules were removed in exactly the correct proportions.

However, if we assume that this problem could somehow be overcome, serious problems would still remain. A creature’s legs (human or otherwise) are similar to the columns which hold up Greek temples. Their strength is directly proportional to their cross-sectional area. This, in turn, is proportional to the square of the radius of the column, according to the equation:

A = πr²

Hence, the strength of legs scales up (or down) with the square of the scaling factor. For instance, suppose we scale up an ant by a factor of 1000. This increases the ant’s length from 1/8 inch to about 10.5 feet It increases the strength of the ant’s legs by a factor of 1000 squared, or 1 million. This sounds very reassuring until we look at the ant’s increase in mass and weight.

Each segment of the ant’s body is roughly similar to a sphere whose weight is proportional to its volume given by the equation:

V = (4/3)πr³

With constant density, the weight therefore increases with the cube of the scaling factor. Hence, weight increases by a factor of 1000 cubed, or 1 billion. This means weight increases 1000 times faster than leg strength. In other words, the ant would probably collapse under its own weight.

The ant’s mass, and hence inertia, also increase about 1000 times faster than muscle strength. So, if the ant could still stand, it would barely be able to move.

Scaling downward, or shrinking, avoids some of the weight problems. However, it has problems of its own, especially for warm-blooded creatures.

Heat loss is related to the ratio of bodily surface area to mass. In other words, a creature with a high ratio of surface area to mass will cool off much faster than one with a low ratio. Such a creature would need a higher metabolism rate and to eat more food to maintain body temperature.

Small creatures have high surface-area-to-mass ratios, which explains why shrews must eat several times their bodyweight in food everyday. They must do so to maintain body temperature.

Surface-area-to-mass ratios scale up or down inversely proportional to the scaling factor. In other words, shrinking a human by a factor of 100 would increase the surface-area-to-mass ratio by a factor of 100. Such a person would have to eat continually or risk death from hypothermia even in 70-degree Fahrenheit weather.

None of the above discussion even mentions the fact that the design of lungs, hearts, brains, blood cells, etc. is very specific to relative size and does not scale up and down well if at all. Physiology changes dramatically with size due to basic laws of physics. Although it might be entertaining, the prospects of big bugs and tiny humans will have to wait until the laws of physics are substantially altered.

Of course, if the laws of physics don’t do in the Lard of the [Onion] Rings, it appears those veggiehumpers at PETA just might. Which would be a Pyrrhic victory for them in any case — after all, they’d still be PETA, and those would still be carrots stuffed up their rectums. But it’d be, like, so totally bitchin’ for the rest of us…

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