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Euro coins as standard weights

Created by Olli Niemitalo on 2007-05-21, last modified 2010-03-10

What if you suddenly need a scale, accurate to one gram? Here is the solution, in case you live in the euro zone. (Something like this has actually been done before!)

The physical mass of each euro coin is actually standardized. So they can be used to measure mass of other objects using a balance. Here's how I measured 3 grams of salt. See the figure 1 below. A thin black string goes around a round pen that may freely roll on top of two pens well leveled on a table using a pencil and a book. On each end of the string hang two bags, by hooks made of staples (was out of paper clips). One bag is for the coins and one bag is for the sample (an empty bag is included in the other end for balance). The first coin bag contains two 20 cent coins. That's 11.48 grams of coins. The other coin bag contains a 2 euro coin with a mass of 8.50 g. The difference is 2.98 g, near to what we wanted, 3 grams. Salt is added to the sample bag (next to the 2 euro bag) until the pen no longer rolls. I do admit that it is a bit cumbersome to use, but if you have nothing else, it still works. A balance made from a rod would be more reliable, easier to handle, and could be calibrated so that different weights are measured by moving the position of the standard weight.

Figure 1. Measuring 3 grams of salt with a cheap and lousy balance scale.

If you want to measure other weights (0.5-50 g), you can use table I as a reference. No 1 cent and 2 cent coins are used in the combinations, because those coins are not available in Finland. If you want the extra convenience and accuracy provided by them, have a look at table II. The tables also give error limits at two standard deviations. The error analysis is based on the presumption of 0.5 % systematic and 0.5 % standard deviation random error in the masses of the coins, agreeing with an analysis on Belgian 1 euro coins. The coin combinations in the tables were found by first systematically searching for the combination that gives the most accurate mass, and by then finding the most convenient coin combination that gives at most 25 % more error.

Note that this method of measuring the mass is better suited for weighing a wanted amount of a substance rather than for determination of an unknown mass.

Table I. Euro coins as standard weights for 0.5-50 g, excluding 1 cent and 2 cent coins

Standard weights Negative standard weights
and sample
Mass of sample that
balances the scale
Error
at 2 std.dev.
10c5c 1esample 0.5 g ±0.2 g
2e 1esample 1 g ±0.2 g
20c 10csample 1.5 g ±0.3 g
50c 20csample 2 g ±0.2 g
20c10c 1esample 2.5 g ±0.3 g
1e5c 2esample 3 g ±0.3 g
1e 5csample 3.5 g ±0.2 g
5c sample 4 g ±0.1 g
2e 5csample 4.5 g ±0.2 g
50c20c 2esample 5 g ±0.3 g
20c5c 10csample 5.5 g ±0.2 g
50c5c 20csample 6 g ±0.2 g
2e20c 50csample 6.5 g ±0.3 g
1e10c5c 2esample 7 g ±0.3 g
1e sample 7.5 g ±0.1 g
10c5c sample 8 g ±0.1 g
2e sample 8.5 g ±0.1 g
1e20c 10csample 9 g ±0.3 g
20c5c sample 9.5 g ±0.3 g
20c10c sample 10 g ±0.3 g
1e50c 10csample 11 g ±0.4 g
50c10c sample 12 g ±0.2 g
2e2e 5csample 13 g ±0.3 g
2e20c5c 10csample 14 g ±0.3 g
2e2e20c 50csample 15 g ±0.4 g
2e1e sample 16 g ±0.2 g
2e2e sample 17 g ±0.2 g
2e2e2e 1esample 18 g ±0.3 g
1e50c5c sample 19 g ±0.4 g
2e1e5c sample 20 g ±0.3 g
2e2e5c sample 21 g ±0.3 g
2e50c20c sample 22 g ±0.3 g
2e2e50c5c 20csample 23 g ±0.4 g
2e1e10c5c sample 24 g ±0.3 g
2e2e10c5c sample 25 g ±0.3 g
2e50c20c5c sample 26 g ±0.3 g
2e2e20c10c sample 27 g ±0.4 g
2e1e50c10c sample 28 g ±0.4 g
2e2e50c10c sample 29 g ±0.4 g
2e50c20c10c5c sample 30 g ±0.4 g
2e2e2e20c5c 10csample 31 g ±0.4 g
2e2e2e2e20c 50csample 32 g ±0.5 g
2e2e2e1e sample 33 g ±0.3 g
2e2e2e2e sample 34 g ±0.3 g
2e2e2e2e2e 1esample 35 g ±0.5 g
2e2e2e2e50c 20csample 36 g ±0.5 g
2e2e2e1e5c sample 37 g ±0.4 g
2e2e2e2e5c sample 38 g ±0.4 g
2e2e2e50c20c sample 39 g ±0.4 g
2e2e2e2e50c5c 20csample 40 g ±0.5 g
2e2e2e1e10c5c sample 41 g ±0.4 g
2e2e2e2e10c5c sample 42 g ±0.4 g
2e2e2e2e2e2e 10c5csample 43 g ±0.5 g
2e2e2e2e20c10c sample 44 g ±0.6 g
2e2e2e1e50c10c sample 45 g ±0.5 g
2e2e2e2e50c10c sample 46 g ±0.5 g
2e2e2e2e2e2e 5csample 47 g ±0.6 g
2e2e2e2e2e20c sample 48 g ±0.7 g
2e2e2e2e1e50c sample 49 g ±0.7 g
2e2e2e2e2e1e sample 50 g ±0.5 g

Table II. Euro coins as standard weights for 0.5-50 g, including 1 cent and 2 cent coins

Standard weights Negative standard weights
and sample
Mass of sample that
balances the scale
Error
at 2 std. dev.
5c1c 20csample 0.5 g ±0.2 g
10c 2csample 1 g ±0.1 g
5c 1csample 1.5 g ±0.2 g
50c 20csample 2 g ±0.2 g
10c1c 5csample 2.5 g ±0.1 g
2c sample 3 g ±0.1 g
20c 1csample 3.5 g ±0.2 g
5c sample 4 g ±0.1 g
1e 2csample 4.5 g ±0.2 g
20c1c 2csample 5 g ±0.1 g
50c 1csample 5.5 g ±0.1 g
50c1c 10csample 6 g ±0.2 g
20c2c 1csample 6.5 g ±0.1 g
5c2c sample 7 g ±0.1 g
1e sample 7.5 g ±0.1 g
10c5c sample 8 g ±0.1 g
2e sample 8.5 g ±0.1 g
20c10c2c 5csample 9 g ±0.2 g
10c2c1c sample 9.5 g ±0.1 g
50c1c sample 10 g ±0.2 g
10c5c2c sample 11 g ±0.2 g
20c5c1c sample 12 g ±0.2 g
1e50c 1csample 13 g ±0.2 g
50c5c1c sample 14 g ±0.2 g
50c10c2c sample 15 g ±0.2 g
2e1e sample 16 g ±0.2 g
2e2e sample 17 g ±0.2 g
2e10c2c1c sample 18 g ±0.2 g
2e1e2c sample 19 g ±0.3 g
2e2e2c sample 20 g ±0.3 g
2e2e5c sample 21 g ±0.3 g
2e50c20c sample 22 g ±0.3 g
2e1e5c2c sample 23 g ±0.3 g
2e2e5c2c sample 24 g ±0.3 g
2e2e10c5c sample 25 g ±0.3 g
2e50c20c5c sample 26 g ±0.3 g
2e2e50c1c sample 27 g ±0.4 g
2e2e10c5c2c sample 28 g ±0.4 g
2e2e20c5c1c sample 29 g ±0.3 g
2e2e1e50c 1csample 30 g ±0.3 g
2e2e2e50c 1csample 31 g ±0.3 g
2e2e50c10c2c sample 32 g ±0.4 g
2e2e2e1e sample 33 g ±0.3 g
2e2e2e2e sample 34 g ±0.3 g
2e2e2e2e2e 1esample 35 g ±0.5 g
2e2e2e1e2c sample 36 g ±0.4 g
2e2e2e2e2c sample 37 g ±0.4 g
2e2e2e2e5c sample 38 g ±0.4 g
2e2e2e50c20c sample 39 g ±0.4 g
2e2e2e1e5c2c sample 40 g ±0.4 g
2e2e2e2e5c2c sample 41 g ±0.4 g
2e2e2e2e10c5c sample 42 g ±0.4 g
2e2e2e2e2e2e 10c5csample 43 g ±0.5 g
2e2e2e2e50c1c sample 44 g ±0.5 g
2e2e2e1e50c10c sample 45 g ±0.5 g
2e2e2e2e50c10c sample 46 g ±0.5 g
2e2e2e2e2e1e 2csample 47 g ±0.5 g
2e2e2e2e2e2e 2csample 48 g ±0.5 g
2e2e2e2e2e10c1c sample 49 g ±0.5 g
2e2e2e2e2e1e sample 50 g ±0.5 g

1 Comment »

  1. Salt?

    Comment by Feddy — 2015-01-11 @ 21:21

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