For decades, movies have paraded before us incredible monsters like Godzilla, King Kong, giant squids, eight-legged freaks (i.e., giant spiders), and others. What’s interesting is that they just end up looking like bigger versions of ordinary Earth-dwelling animals—their arms and legs aren’t any thicker relative to the torso, they’re just directly scaled up.
In this quiz, we put movie monster designs to the test using basic, logical reasoning about forces. The iconic creatures of the silver screen are products of the human imagination, but in case you weren't sure whether a building-sized lizard could wreak havoc on a city, we'll examine the balance between the forces that keep her standing and the mass that weighs her down.
If we look around the physical world at land-dwelling animals, generally the smaller ones have features that are quite different from those of the largest ones. For example, Komodo dragons don't exactly look like larger copies of a grass lizard.
In this quiz, we are going to imagine a super-sized lizard that grows larger indefinitely to discover a relationship between weight and bone support in animals. This kind of scaling argument is an idealization (because nothing can grow larger indefinitely) but is used in often physics to bring important relationships to the forefront without doing any actual modeling.
To simplify the argument, we will assume the scaled-up lizard is just like most movie monsters: the larger version is exactly similar to the smaller versions. In other words, the scaling is proportional. This makes it simple to relate the monster at different scales.
For example, if the tail of the grass lizard in the image above is \(20\%\) of its length, what percent of its total length is its tail after it is scaled up proportionally?
The weight of an animal, large or small, is supported by rigid bones in its legs. If we had a powerful microscope, we could zoom in on a piece of bone and look at the microscopic structures that provide this support.
With a powerful enough microscope, we would see individual calcium atoms stacked in a crystal lattice of chemical bonds roughly running up and down the length of the bone. You can picture these calcium-calcium bonds as plastic rods with a definite length, and there are billions of them in a microscopic piece of bone. No single rod is strong by itself; it warps or breaks under stress. But taken together, they distribute the weight of the animal and can support quite a bit without snapping.
However, every material, if it's loaded with enough weight, has a breaking point....
Although we generally try to avoid it, bones break sometimes. Applying too much force in the wrong direction can fracture the crystal lattice we saw on the previous pane.
Imagine you're comparing the two bones shown in the image below. They are the same length, and assume they have microscopically identical crystal lattices—the spacing between pairs of calcium atoms and the lengths of the bonds are the same.
The only difference is their cross-sections: the larger bone has double the width \((b)\) and thickness \((t)\) of the smaller bone. Compared to the smaller bone, about how much more weight would you expect the larger bone to support before it fractures?
We've discovered that the amount of support a bone provides scales with (i.e., grows in proportion to) the area of a cross-section of bone. Now we can consider what it takes for Godzilla to stand under his own weight.
Godzilla has starred in more than 25 movies, and since his first screen appearance in 1954, the king of the monsters has been successively larger in each film. Yet, his basic shape has changed very little.
Let's compare two realizations of Godzilla: one is \(10\) times taller than the other, but similar in every other way. How many times larger is the cross-sectional area of the larger lizard's leg bone compared to the smaller serpent's?
If he is setting off to stomp around the metropolis, Godzilla's leg bones must be able to support his weight. Of course, the larger reptile is heavier and requires more support. As long as Godzilla's weight scales by the same factor as the support his bones provide, he won't have any problems as he terrorizes the city residents.
If the larger version of Godzilla is \(10\) times bigger in each dimension (height, width, and thickness), how much bigger is his weight compared to is older, smaller representation?
In the last two questions, we've worked out how Godzilla's skeletal support and his weight scale with his overall size. Now, by comparing these scaling behaviors, we can decide whether Godzilla has even a fighting chance in future movies when he's even larger.
Unless his weight and bone support scale in exactly the same way, he will eventually succumb to one of two outcomes. If his weight outpaces bone support as he grows, his leg bones will painfully break under his weight. If the opposite is true, his bones will grow so heavy that the muscles won't be able to move them.
You might be surprised that it's even possible to deduce Godzilla's fate from a single assumption about how bone support is related to a bone's cross-section. Due to its brute strength, the scaling approach we used in this quiz will return predictably throughout this course.
To conclude, if Godzilla were to continue growing, what would happen to him?