Brian, when you say, "...the volume of the sunken cup, reduced by the
volume of water formerly displaced by the cup's weight when it was
floating empty," it seems to me you are making exactly the same error
you accused me of making. You are comparing a weight to a volume of
water. I was equating a mass to a volume of water. At least in my case
the numerical values are the same within several significant figures
because grams of water and cubic centimeters of water are almost the
same numerical value. But dynes (or newtons) of water is a very
different numerical value than cubic centimeters of water. I thought it
was convenient to work in mass because of (1) the numerical similarity
between grams of water and cubic centimeters of water, and (2) because
the volume of the cup's material will likely be found by using a balance
and "massing" the cup alone then "massing" the cup while it is hanging
submerged in water.
Also, I don't understand the desire to eliminate the word "difference."
The question is whether the addition of the second block will make the
water rise, fall, or stay the same. The sinking cup by itself will make
the water lever fall. The added block by itself will make the water
level rise. To find the net result we have to sum these two effects.
Before we can sum them, we have to determine the size of each effect.
To determine the size of the water lowering caused by sinking the cup,
we need to know the "difference" between the volume of water displaced
by the floating cup and the volume of water displaced by the sunken cup.
This difference is the volume of the cup's material (the final sunken
state) minus the volume of water that has the same weight as the weight
of the cup (the initial floating state).
The last sentence in the paragraph above is the same as my original
statement except I have replaced the original words "cup's mass" with
"volume of water that has the same weight as the weight of the cup."
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Chair of Sciences
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu