## 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?