Our experiment - We actually used four big machine nuts from our shop, each of them about 1-1/16" square and 5/8" thick, with a threaded hole about 1/2" in diameter. All together, they weighed 9.0 ounces (255 grams). Those four nuts were our "anchor". Here's a picture of our boat and lake. (You can see half of our anchor, two of the four nuts we used.)

When we dropped the "anchor" over the side of the Rubbermaid boat, lo and behold, the water level went down by about 10 mm.


After we had done the experiment, we thought about it a while and decided that it was "obvious" that the water level should go down. (There are many cases in the history of physics where the result is obvious - at least to the theoretical physicists - but only after they know what actually happens.) Why is it obvious? Here, we found it useful to do what physicists so often do: examine limiting cases. Suppose the anchor were made of very, very dense material, even denser than steel. Think of it as "infinitely dense", so that all the weight, x grams, of our "anchor" was the size of a mathematical point. If we dropped that "anchor" overboard, it would occupy no space at all, or only a tiny bit of space if it were made of steel (or uranium, say, which is much more dense than steel). So having the anchor at the bottom of the lake would make the water level go up not at all (or only a tiny bit), but the boat would now be a good deal lighter; the boat would float a bit higher in the water, so the actual water level in the lake would have to go down a bit. Obvious!

Here's another potential "limiting case". What if the anchor were made of some light material, say wood? If we threw that anchor overboard, it would float, almost totally submerged. Not much of an "anchor"!

But that gives us an idea. Suppose the anchor were made of something just about the same density as water. (Dense wood, oak, say, not balsa wood.) Now it will float with nothing sticking out above the water (in fact, it could be anywhere below the surface), so it's displacing an amount of water exactly equal to its own volume. But when it's in the boat, Archimedes' principle tells us that it will displace an amount of water that weighs as much as the anchor itself does -- and because we supposed that the anchor and the water have exactly the same density, that amount of water has a volume exactly the same as that of the anchor. So we see that in this case, the water level will be the same whether the anchor is in the boat or has been thrown overboard.

If our anchor were made of steel (pretty dense, but not infinitely dense), we would still expect the water level to go down when we threw the anchor overboard, but not as much as in the limiting case. And if our anchor were made of, say, aluminum - which has a density intermediate between those of steel and wood, we would expect the water level to go down, but not as much as when we used a steel anchor.

We can get quantitative about this and start writing equations, assigning symbols to things like the area of the "lake" and the density of steel, and then invoking Archimedes' principle. We did this, but it does get a bit tedious, so we're omitting our algebra and leaving this as an "exercise for the reader". Suffice it to say, though, that after measuring all the relevant dimensions, our calculation of how much the water level would go down came out to be 10.5 mm. Amazingly good agreement with our observation of 10 mm, since some of our measurements were pretty rough. (If we did the usual freshman lab "error analysis", we would have to say that our prediction was for a water level fall of 10.5 mm +/- about 1 mm.)


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