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Wednesday, November 29, 2023

How Strong Is King Kong? And Could He Even Stand Up?

It's time for Godzilla vs. Kong—a classic battle between two impossibly giant creatures. I've only seen the trailer, and it looks like a fun movie. But movies aren't just for fun, they are also for physics. In particular, this is a great chance to consider the physics of scale—what happens when we make small things into big things? For instance, what happens if you take a normal gorilla and make him into a giant gorilla and then you name him King Kong?

How Tall Is Kong?

If we want to see what happens when you have a giant gorilla, the first thing is to find out how tall he is. Oh sure, I could just look this value up somewhere—but that's not fun. Instead, I'm going to see if I can estimate his size based on just what I can see from the trailer. I love the challenge of just using a trailer. It's sort of like real science. Sometimes you have to struggle to get some nice data, and other times, boom, it's just there. In this case, I'm lucky. There's a shot of Kong and Godzilla both standing on an aircraft carrier. Assuming this is a Nimitz-class carrier, I can use the size of it (around 330 meters) to measure Kong.

This gives a rough height of 102 meters—since it's just an estimate, I'm going to go with 100 meters. Oh, it looks like Godzilla's tail is around 110 meters long. Wow.

How Much Would He Weigh?

OK, I need another assumption. Let's say that Kong is made of the same stuff as a regular-size gorilla. I will also assume that Kong is the same basic shape as a normal gorilla—you know, both animals have legs that are the same ratio to their total height, and the width of their arms compared to the total height is the same. I mean, it looks that way, right? He looks just like a big gorilla.

If Kong is a big gorilla, then he would have the same density as a gorilla—where we define density as the total mass divided by the volume. But what's the volume of a gorilla? Actually, we don't need to know that. Instead, let's just use an easy shape like a cylinder. Suppose I have two cylinders of different size, but with the same proportions (radius to length ratio).

Let's find an expression for the density of the smaller cylinder. Remember the volume of a cylinder is the area of the base (a circle) multiplied by the length. Oh, I'm using the Greek letter ρ (rho) for the density—that's what all the cool physicists use.

I can use this density to find an expression for the mass of cylinder B, but before I do that, let's talk about volume. Suppose cylinder B is twice as tall as cylinder A. That would mean that B's radius would have to also be twice as large as the radius for A in order for them to be the exact same shape. So, let's compare the volume of cylinder B to the volume of A for this double height example.

Check it out. If you double the length of the cylinder, you increase the volume by a factor of 8. This is because the volume depends on the length and the square of the radius. If you increase all of these by a factor of 2, you get three factors of two or 2 cubed (which is 8). What if I increased the height by a factor of 3? Then you would increase the volume by a factor of 33. So, if you increase the height by a generic scaling factor s, the volume would increase by a factor of s3.

Now we can put this all together. What is the mass of a cylinder that's increased in height by a factor s? If the density is the same, then it's mass would increase by a factor of s3.

Notice that I don't actually need to know the density of the cylinders—just that they are the same. And here's the cool part—it doesn't even matter if the objects are cylinders, spheres, or gorillas. As long as the proportions are the same (same shape), the mass increases by a factor of s3.

So, what is the mass of Kong? I only need to know two things—the mass of a regular gorilla and the height of a gorilla (I need the height to calculate the scale factor of s). According to Wikipedia, a Western gorilla has a height of 1.55 meters with a mass of 157 kg (346 pounds). That means that Kong has a scale factor of 100/1.55 = 64.5. Here is the answer (as a Python calculation so you can change the values).

Yes. Kong is MASSIVE—42 million kilograms, or 93 million pounds. Ummm … news flash. That aircraft carrier that Kong is standing on has a mass of 100 million kilograms. He is about half of that mass. Oh, what about the mass of Godzilla? That one is tougher to calculate since there isn't a normal-size Godzilla to use for calculations, but I would guess that he would be around the same mass as Kong. But either way, I'm not sure that aircraft carrier would stay afloat with those two monsters fighting on it. Good thing this is just a movie.

How Strong Is King Kong?

If we can scale up the mass for a large animal, what about strength? We can at least try to estimate this, right? Let's start with a model of muscle strength. One simplistic version says that the strength of a muscle is proportional to the muscle's cross-sectional area. So, if you have a muscle in your arm that's twice as thick as another one (twice the diameter), then the cross-sectional area and therefore the muscle strength would be 4 times greater. Yes, this is just an approximate strength model, but it's at least plausible. The idea is that a wider muscle has more muscle fibers that can contract and exert a force. The more fibers working in parallel, the greater the force. Let's use the following equation for strength (as a force).

In this expression, A is the muscle cross-sectional area, and c is just a proportionality constant. I don't actually know the values of c or A for a gorilla, but that's OK. The one thing that I can roughly estimate is the strength of a gorilla. According to this site, a fully grown gorilla can lift (bench press) 4,000 pounds (1,810 kg). Let's use the same scaling factor (s) from the weight estimation. If Kong is s times taller than a gorilla, then his muscle cross-sectional area would be s2 times larger—assuming Kong is the same shape (and proportions) as a normal gorilla. With this, I can calculate his strength (F1 is the strength of a normal gorilla).

If Kong has a scale factor of 64.5, his strength would increase by a factor of 4,160. That means that Kong would be able to bench press 16.6 million pounds (74 million Newtons). So, don't mess with King Kong. Don't. Do. It.

Could Kong Even Stand Up?

But wait. Even though King Kong would be super strong, he would also be super heavy. For instance, let's take the ratio of bench press strength divided by weight for both a normal gorilla and Kong (it doesn't matter what units you use since they cancel). Note, I am using Rg for the gorilla and Rk for Kong.

Even though King Kong is much stronger, he's much much more massive. His strength to weight ratio is way worse than it is for a normal gorilla. Could he even stand up? Maybe—I think it would be close. If his legs are stronger than his arms, he could do it—but he would probably get tired fairly quickly. This ratio calculation is for his bench press strength, and maybe his legs are even stronger (or maybe they aren't). But still, it's quite clear he wouldn't be running around like his smaller cousin.

The problem is the dimensions. His weight is proportional to his volume—so that depends on s3. His strength is proportional to his cross-sectional area—that goes like s2. So, as the scale increases, the weight increases faster than the strength does.

This is all part of the physics rule that says "big things are not like small things." For instance, if you bake a muffin, smaller muffins cool off faster than bigger muffins. This is because the total amount of thermal energy depends on the mass of the muffin (that goes as s3), but the muffin cools off by radiating from its surface area (that goes as s2). So this smaller muffin will have a larger surface-area-to-volume ratio and will cool off faster.

Something similar happens to meteors as they enter Earth's atmosphere. The momentum of the object depends on the mass, which depends on the volume (s3), but the drag force depends on the area (s2). So, if you have two rocks entering the atmosphere with the same speed the smaller one will slow down more (and land at a different place).

So, what would a realistic King Kong look like? Well, he wouldn't be just like a normal gorilla except bigger. Since he's so massive, his arms and legs would have to be way thicker compared to his body than you would expect. He would probably look super weird with such huge arms. And this is exactly why he doesn't look like that. It would ruin the fun of the whole movie.

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