I formed a square of 11 rows and 11 lines of the bags, numbered 0...10. First weighing has 0 coins from each bag on line 0, 1 from each on line 1, etc, 10 from each on line 10. Second weighing does the same for row 0, row 1, etc. On each weighing, there are 11 x (0 + 1+ 2+ ... + 10) coins weighed, which is 565g of coins; but false coins cause the weight to be a bit over that. Subtract 565g from each value and multiply by 10 gives integer values m and n, and those are the coordinates of the bag with the false coins.
I wonder if we refer to a different kind of measuring device. Digital balance scales that I know of simply have stuff put on top and give an accurate measurement of the weigh on a display, usually a LED.