This has been bothering me for years. I'm sure it can be done but every time I do a search it's always original gravity this and final gravity that. All I want to do is have a rough estimate of how much alcohol is in a concentration based on weight. Assume water/sugar mash. How do you do it?
You cannot do that based upon the weight of a sample of your final beer, because then you have a mix of water, alcohol and final sugars:
Final weight = Weight of water + weight of alcohol + weight of sugars
This is one equation with three variables, which you can combine to obtain the same final weight with different values for the water, alcohol and sugars.
You could obtain the alcohol value in your beer based upon samples of weight of your original wort and your final beer, but that is just a different way of measuring your original gravity and final gravity.
E.g. 1 litre of wort might measure 1.060 kg, and then 1 litre of beer might weigh 1.012 kg at the end of fermentation. This way you obtain the same figures as by using a hydrometer.
However, a hydrometer is much more precise and handy: how much spread would there be on weighing 1 litre of wort or 1 litre of beer, when the final decimals are so small? And handy: you can use the same small device for getting the final gravity of a batch of 10l or 10000l.
You might also start from the total volume of your wort and beer, but then you also need to have a very precise measurement your volume, and a precise measurement of the weight of this volume. Which kind of scale would you use for that?