More Tolkein Physics:

How Does Gollum See in the Dark?

How Many Fish Does Gollum Need?

I wasn't going to do another post on The Hobbit, but I can't stop. Let me look at the scale of hobbits. Really, it is one of my favorite ideas. The idea that things that are scaled down from something else aren't just smaller versions. You would think that size doesn't matter but it often turns out that it does.

First, a look at hobbits and dwarves. Let me try to get an estimate for the size of a hobbit from this image.

I will start with Gandalf. Of course, Gandalf is played by Ian McKellen who is 1.8 meters tall (according to IMDb). Ok, a couple of pre-emptive notes:

Yes. I know Ian McKellen is just an actor and not really Gandalf. He couldn't possibly be Gandalf because he is really Magneto.

Am I trying to say that Gandalf is 1.8 meters tall? Is that what Tolkien said? I don't know what Tolkien said about Gandalf's height (if he said anything at all). However, I have to start somewhere.

Even in The Hobbit, Gandalf isn't actually a human. You know that, right? Yup.

Why do I have to try and ruin The Hobbit? What did The Hobbit ever do to me? This is an important point. I look at the physics of hobbits and dwarves as a way of showing my respect. Yes, I know none of this is real. That won't stop me from thinking about the physics of it.

Back to the image. If Gandalf is 1.8 meters tall then Balin would be about 1.3 meters tall. Perhaps Balin is about the same size as Bilbo. That's just a guess. Now for some physics. There are a couple of assumptions I am going to make.

Hobbits have the same body proportions as a full-size human. They look just like humans except smaller.

Hobbits have the same kinds of muscles as humans.

Speaking of muscles. How do they work? A quick warning. I have no clue what I am talking about here. I might be making stuff up. However, if it is made up stuff it should at least make sense. Here is a diagram of a muscle.

I like to think of muscles as a bunch of fibers in parallel. These fibers can contract and pull on bones and stuff. The more fibers in parallel, the greater the muscle force. This would imply the following fundamental model of muscle strength:

In this model, the force is of course the force the muscle can pull on something. The other constants:

A m is the cross sectional area of the muscle (in m 2 ).

is the cross sectional area of the muscle (in m ). σ f is the fiber area density (in units of fiber/m 2 ).

is the fiber area density (in units of fiber/m ). k is a constant to represent the muscle fiber strength (units of N/fiber).

Now for hobbits muscles. Here is a diagram of an arm (and muscle) for a human and a hobbit.

I apologize for the confusing labels. I called the length of the human muscle L g (g for Gandalf) and the hobbit muscle is L h . It's not my fault that humans and hobbits both begin with an "h".

Now, suppose that humans and hobbits have the same muscle fibers as well as muscle fiber density. I can then write:

If a hobbit is 72% the height of a human, he would just have 52% the strength. Let's say a normal adult human male can bench press 150 pounds (just making this up people). That would mean the hobbit could bench press 78 pounds.

But what about DWARVES? Dwarves are the same size as hobbits, but aren't they stronger? Well, they are stronger than hobbits but they are probably bulkier. Maybe they have a greater muscle fiber density too.

Bonus: Flying Smaug ——————-

I didn't find a Smaug picture that I liked, so I made my own. It's an original.

Sometimes you can look at how things scale - but sometimes, that model is difficult to create. For instance, take a look at birds and wingspans. If you double the mass of a bird, do you also double the wingspan? I think not. If the mass (and thus the weight) double, wings need to be bigger - but not double the size. For complex cases like this, a plain data-based model might work best.

Fortunately, I have take the time to make such a data-based model. After looking up the approximate mass and wingspan of several different birds on Wikipedia, I made this graph of wingspan squared vs. mass of bird.

A linear function seems to fit well enough. From this, I get the following relationship between wingspan (w) and mass (m).

What about the non-zero intercept? Does this say that a bird with zero mass would still have a 60 cm wingspan? Yes. But stay calm. This is just a model, a model that clearly doesn't work for small flying creatures. However, don't worry. We will just use this model for dragons (which still might not be valid).

Question 1: I don't seem to recall an exact description of the size of Smaug in the book The Hobbit. Suppose that Smaug has a wingspan of 10 meters, what would be an estimate of his mass?

For this question, I simply need to solve for the mass in my above model.

Putting in a wingspan of 10 meters, I get a mass of 155 kg (341 pounds). That isn't very heavy. It's like the mass of a panda bear. Panda bears aren't that scary.

Question 2: Forget about the 10 meter wingspan. What if the dragon has the mass about the same as a large elephant? Based on the above model, how big would the wingspan be?

What's the mass of an elephant? Wikipedia puts the mass of an African Elephant around 6,000 kg. If I use the same equation above to solve for the wingspan (simply take the square root of both sides), I can get the wingspan of my dragon-elephant. With a mass of 6,000 kg, the wingspan would be 62 meters wide.

What would that look like?

There's your 60 meter wingspan flying and fire breathing elephant. You better run, ivory poachers.