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The idea is to be able to weigh-up things, as with a "scales of justice" two-pan balance, except instead of weight being balanced, density is being balanced.
There's some interesting history behind this. Some pieces of metal were extracted from a piece of electronic equipment, and there was a serious question on whether they were made of zinc or silver. The metal was quite heavy, and shiny, so it was a valid question. The obvious solution would be to use the method described in How to Measure Density, but this would have required precise measurements, and the available equipment wasn't accurate enough.
So, the idea was: How about a pair of scales to measure density?
Here's how it's done: A pair of scales (two-pan balance) is made out of basic materials and balanced up. Then, the metal in question is put in one of the pans. Next, the other pan is loaded up with iron nuts and bolts to the same weight, so the scale is balanced.
(Any ballast material could have been used, but the nice thing about using iron nuts and bolts is that the density (specific gravity) of iron is 7.85 which is nicely between that of zinc (7.1) and silver (10.5)).
Now bearing in mind the two sides of the balance are of equal weight when weighed in air, it's a bit like in the question of which weighs more: a ton of lead or a ton of feathers? where the items are weighed again on the moon, or underwater, and found to be different. What happens next with the zinc/silver versus iron nuts&bolts is that the scale is lowered into water (or at least both of the two pans are). So now which weighs more?
Although the two items were the same weight in air, the scale no longer balances underwater. The side which has more density sinks and so the scale reads that it is heftier. With the zinc/silver versus iron nuts&bolts, the iron was heftier than the zinc/silver, and therefore it was concluded that it was zinc, not silver.
Repeating the same experiment with a lump of lead versus nuts&bolts (which had to be adjusted because the lead didn't happen to weigh the same as the zinc/silver), the scale was balanced up in air to make sure the iron nuts and bolts exactly matched the weight of the lead. Then, the apparatus was dunked in water, and the lead sank better than the iron. This showed that the lead was measurably heftier than the iron, and that is a similar result that would have been observed if the zinc/silver had turned out to be silver!
If all this seems silly, and you're thinking that the scale should balance perfectly well the same underwater as in air, consider this: What if you had a big sackful of corks on one side of the balance and a small ingot of brass of the same weight on the other side. In air, it would balance, but underwater, the brass would weigh a similar amount but the corks, well they'd float, and would by no measure be as hefty as the brass, when balanced underwater.
Here's the method again:
1. You set up a two-pan balance with the test sample on one side and some ballast made of a different material on the other. You adjust the balance so the weight is apparently equal when weighed in air.
2. You submerge the apparatus in water and see how the balance tips.
3. The balance tips so the denser material appears to weigh more.
So there you have it, a set of scales that can weigh density rather than weight. Notice it works by a "difference of a difference" method. However, what makes it easy is that the initial amount of ballast is adjusted to make the initial difference zero. Then when it's submerged, a new difference is encountered. It won't be the same difference as before, but a different difference! (Well, not unless the sample and the ballast are of the same density).
Another way to think about this is to consider the buoyancy, the Archimedian upthrust. This will cause the less dense sample to float more than the denser one, even if neither of them are actually light enough to float.
Incidentally, the difference of the differences is not big. The apparent weight difference is: The weight of water displaced by the difference in volume of the sample and ballast.
In practice, with a very basic set of scales made from a coat-hanger, string, and two bottoms of lemonade bottles, the different apparent weight was determined by which hits the bottom first when lowered carefully.
* You don't need a water tank big enough to sink the entire two-pan balance. Two buckets will do. It doesn't matter whether the buckets are the same size or whether they have the same level of water in them. They just need to have the bottoms at the same floor height.
* Try to avoid using this method to measure the density of sodium, cats, mobile phones, etc.
* When choosing a ballast, choose something that's about the same weight whether it's wet or dry. Nuts&bolts, lumps of lead, aluminium ingots, etc. Dry sand would be tricky, because if you use dry sand, it will become wet sand, and will not dry out quickly, so it will be difficult to repeat the experiment.
* In this experiment, the density of air is neglected. This is reasonable as air has a specific gravity of 0.01225
* To measure the actual density and get an answer as a figure, see how to measure density. In contrast, the "density scales" just measure which of two things is denser than the other. For example: Is that a piece of zinc or silver? Is this piece of uranium more dense than this lump of iridium? etc.