Testing US coins for silver content, part 2

I'm not a gold collector, but I wouldn't mind having some.
This was an interesting project, but also a somewhat trivial pursuit.
I'll test more coins as I get them.

The coins minted by the US have always had very accurate composition. The problem is that to get an accuracy of 0.005 (i.e. 0.5%) in the silver content, you will have to measure the specific gravity to an accuracy of around 0.005 * 0.17 = 0.00085 or 0.085%, as the difference in specific gravities of copper and silver is only about 17%.

And then, because of the way you're measuring the coin weights in and out of water, to measure the specific gravity to an accuracy of 0.085% you will have to measure the weights to an accuracy of around 0.00085 x 0.10 = 0.000085 (since the weight in and out of the water differs by about 10%).

So to measure the silver composition as accurately as your smelter buddy, you're going to have to weigh them in and out of water with an accuracy of around 0.01%, i.e., you need to be measuring with 5 significant digits. In short, to get results that don't bounce all around you're going to have to invest in quite the scale.

Here's a table for silver / copper (gold/copper would be a lot easier):

5 digits: 0.5%
4 digits: 5%
3 digits: 50%

The above are worst case and are my back of the envelope guesses. You should be able to do better, depending on how lucky you are. (For example, if you have 3 digits accuracy you might measure 11.3 or 95.7. Then 0.1 is around 0.1% of 95.7 but closer to 1% of 11.3.)

As it turns out, this problem is one I've faced in my primary hobby, physics. This is why you need to give "error bars" on every measurement you make. And then, when you do the computations, you have to include the error bars with them.

I can type up an example showing how to do error bar calculations if you're interested. Doing calculations with error bars is a pain, so when I had to do this for hundreds of data items in one of my papers (on 100+ hadron masses), I wrote a java applet to take the data and do the calculations automatically for me.

I've been thinking more about the challenge of measuring specific gravity accurately, with inexpensive scales. I think there might be a way.

Part of the problem is that cheap scales tend to be for weighing things a lot heavier than a coin. A way to get around that problem is to use leverage.

Another problem I've observed with scale is that their measurements tend to depend on just how the item is being weighed, i.e. which part of the scale, etc. So a way to improve accuracy is to make the measurements of weight in and out of water using the exact same setup. This can be done by first measuring the weight while the coin is in water, and then letting the water drain out.

The mass of the halter that is underwater should be negligible. I'd ignore that part of the calculation completely. If you want to get more exact, you can always measure how much copper (I suppose) wire you use, and then calculate its mass from tables showing weight of copper wire as a function of gauge. But I'd use 30 gauge kynar wire wrap wire (without insulation) and I doubt you could possibly weigh it. For example, if you used copper and you had 2.23 grams of wire, then when you computed the difference in weight between in and out of water, you'd subtract 2.23 g / 8.94 since the specific gravity of copper is 8.94 (and therefore, the copper displaced 2.23/8.94 cubic centimeters of water).

To show how this works, an illustration might assist. So here's one:



The fulcrum is just some single point where the fishing pole rests on the scale. The objective is to not have it move around (which would change the weight). The pivot is just a place where the fishing pole can pivot around; use your imagination but don't prevent the rod from free movement.

A more convenient way to drain the water might be to set up a siphon. Fill the siphon before making the "in water" measurement so you won't have to disturb it.

Wow.

And I think Jaobler should flesh out his response and enter it in Wheezydog's contest.