## La Bilancetta

The theory of Galileo's method for determining specific weight is derived below.

According to Archimedes' Law of the Lever, the two masses M1 and M2 balance when they are on arms L1 and L2 related as M1/M2=L2/L1.

If M1 is placed in water, it is effectively lighter by the weight of the water displaced, Mw. To balance this lighter weight, M2 must be moved in by a distance y, where (M1-Mw)/M2=(L2-y)/L1.

Now subtract the second condition from the first to find Mw/M2=y/L1, or multiplying through by M2/M1=L1/L2, find Mw/M1=y/L2. The quantity on the left is the reciprocal of the specific weight of M1. If we take L2 to be the unit of length, this just says that the specific weight of M1 is 1/y. For example, if the specific weight of M1 is 3, y=1/3: you have to move M2 in by 1/3 of the length of its arm to balance M1 in water.

Galileo's own description now tells you what to do. You can take samples of "gold" and "silver" (actually brass and aluminum) and determine their specific weights, represented as LOCATIONS on the right hand arm of a balance. Once you have done that, wind a cord neatly and tightly around the arm, starting at one location and ending at the other, to make a sort of "ruler" for determining proportions in an alloy. Then determine the specific weight of a composite body (made by joining some "gold" and "silver" together). It should be some intermediate value. Galileo says that the new position of the moveable counterweight divides the "ruler" in the same ratio as the total mass is divided between "gold" and "silver". For example, if half the mass is "gold", the counterweight will be right in the middle. In your case, what fraction (by weight) is "gold"? Where on the "ruler" does the bilancetta come to equilibrium?