# Facile Physics

## My Personal Blog

#### Tags: physics nudity Archimedes buoyancy

If you made it through elementary school without hearing the story Archimedes and the king’s crown, pull up a chair (if you have heard this story, just skip to the next section).

A long time ago, in the city of Syracuse, the king had commissioned a crown of pure gold. However, once the crown had been made, the king became suspicious. Was the crown truly pure gold? He’d paid for pure gold, but the smith could have mixed in some cheaper metals and pocketed the difference. Kind of like when you’re not sure if the mechanic actually replaced a head gasket or if they just charged you $500 to replace a$7 spark plug.

Now, there was no Internet back then, so the king didn’t have a blog where he could complain about unscrupulous artisans. However, he did have the extraordinary advantage of having access to Archimedes, the smartest man in antiquity. The king asked Archimedes to tell him whether or not the crown was pure gold.

Of course, the answer to that problem was a no brainer. Break of part of the crown, melt it down, and see if it’s gold. You don’t need the smartest man in the world to tell you that. However, the king had spent a lot of, well, gold on this crown and wasn’t particularly keen to see it broken. Especially since there was a chance that the crown was made entirely from gold and this would only ruin a perfectly good crown.

It’s at this point that most of us would have just given up. Just tell the king that the crown looks like gold and he should stop worrying about it. Archimedes, however, decided to figure it out.

As the story is told, he finds the solution during his evening bath. As he lowers himself into the tub, he watches the water level rise around him. He bobs in and out of the tub a few times, watching the water level go up and down. Everything clicks together inside his head in that way that normally only happens in poorly made detective films. He then runs stark naked through the streets of Syracuse, shouting to anyone who could hear that he’d solved the problem. (I’d like to take a moment to thank Archimedes for giving me an excuse to tag one of my posts with the keyword “nudity”. I’m sure this will wind up being my most popular and most disappointing post).

Later, and presumable clothed, Archimedes fills a large container to the brim with water. He then places the crown into the container, causing some of the water to spill out. He takes the crown back out.

He then asks the king for a lump of pure gold that weighs just as much as that super expensive crown that he’s been so worried about. The king pulls said lump out of his back pocket, for that’s walking money when you’re a king. Archimedes places the lump into the water and the crowd gasps as the water level doesn’t reach the top of the container. Archimedes says “The goldsmith’s story doesn’t hold water” and puts on his sunglasses.

## Reality Time

Alright, so I started sliding into artistic license near the end of the story. I’ll defend myself in the classic seven-year-old fashion: everyone else was doing it. Parts of this story never made any sense.

First, the king has to have a large enough gold reserve to have essentially a spare crown’s worth of gold sitting around. Of course, it’s good to be the king, so that’s a possibility. However, the next question is how well does this actually work. After all, the first thing that the goldsmith would do is point at the water level and say, “Yeah, that’s totally at the brim”. He’d then argue that the crown was left when they took it out, so the water level wouldn’t go all the way back up to begin with. He’d then hire a lawyer who’d do a far better job of arguing all the points I’m missing.

The point is, this isn’t really a good way of measuring how much gold is in a crown. Furthermore, if I’m smart enough to see this, Archimedes certainly would have seen it. His toenail clippings were smarter than me. So he probably had a much better idea than just dropping the crown in a bucket.

We also know that he loved levers. When the man saw a battle ship, his first thought was “Hey, I could totally lift that with a lever”. Then he did, because that’s the sort of thing you do when you’re the smartest man in the world. The reason I mention this is because it seems out of character to Archimedes to not throw in a lever here. Well, a lever or a spiral, but I couldn’t find a good way to work a spiral into the solution here, so we’ll just ignore it.

## Alternate Solution

Here’s an alternate solution to Archimedes problem that’s more in character with what we know of him as a scientist. When I first started writing this blog post, I was quite proud of myself for figuring all of this out. Over the course of my research, I found that Galileo proposed this same adjustment to the story, so it’s not exactly a novel idea. On the other hand, it’s still not included in the elementary school version of the tale, so it’s still worth spreading - I just won’t get any credit. Darn.

Moving back to the physics, let’s say that crown had a weight $W_{crown}$ and a volume $V_{crown}$. Let’s also say that the king donates a tiny piece of gold with weight $W_{gold}$ and volume $V_{gold}$. Archimedes can then hang both pieces from opposite ends of a long straight rod.

       |
|
___|____________________________
| d              D             |
|                              |
|                              |
|                              |
|                              |
|                              |
|                              |
crown                           gold



As I’ve marked above, $d$ is the distance from the crown to the hanging point of the balance. Similarly, $D$ is the distance to the sample of pure gold.

Now, as Archimedes knew about levers, for a balanced lever, the distance times the weight has to be the same on both sides. Therefore, $d W_{crown} = D W_{gold}$.

So far, all we’ve done is hang a crown and piece of bold off of a stick. The beauty comes when we submerge this the whole thing in water.

       |
|
___|____________________________
| d              D             |
|                              |
I   |                              |     I
I~~~|~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~|~~~~~I
I   |                              |     I
I   |                              |     I
I   |                              |     I
I crown                           gold   I
I                                        I
I========================================I


Here were going to take advantage of a scientific law known as Archimedes Principle (see how that might be relevant?). The principle states that, when something is submerged in water, its weight is decreased by the weight of the water that would have been in the space that it’s taking up. This is why boats float - they take up so much space that they don’t weigh enough to push all of that water out of the way.

So our equation from before changes. For the scale to remain in balance, we now need $d (W_{crown}-\sigma_{water} V_{crown}) = D (W_{gold} - \sigma_{water} V_{gold})$, where $\sigma_{water}$ is the density of water. I’m mostly going to try and avoid putting too many Greek letters into the math in this blog, but I felt I should include one here, in honor of Archimedes.

Now, I haven’t talked much about density up till now. Density is the relationship between mass and volume. We’re going to be a bit sloppy here and say it’s the relationship between weight and volume, because Archimedes wasn’t really in a situation to care about the difference between mass and weight. I’m pretty sure that he did all of his measurements on Earth. Pretty sure.

So the value $\sigma_{water} V_{crown}$ is just the weight of an amount of water that takes up the same space as the crown. Similarly, $\sigma_{gold} V_{gold}$ is the weight of water that takes up the same space as our test piece of pure gold.

If we play with the algebra for a bit, we can find that $\frac{W_{crown}}{V_{crown}} = \frac{W_{gold}}{V_{gold}}$. Since the density is the relationship between the weight and the volume, that just becomes $\sigma_{crown}=\sigma{gold}$. In other words, if the two sides of the balance remain even after everything has been lowered into the water, then the density of the crown and the gold must be the same. We can then go a step further: if the goldsmith added in less valuable metals during the making of the crown, the two sides will lose balance when everything has been submerged.

This makes for a far more compelling courtroom drama. Archimedes carefully balances the crown and the gold. He then makes a little mark where he balanced the gold. The crowd holds their breath as the scares are submerged in the water, only to gasp as the weight of the tiny piece of gold raises the debased crown.

“And that’s why Justice carries scales.” say Archimedes, while putting on his sunglasses.