Originally Posted by Technatic
The system certainly calculates different Swingweights for these different cases.
It certainly does not since the W head and W throat in your formula are the same for all three cases! (You can try it)
The "innovation" of calculating swingweight from balance and weight appears on Talk Tennis at least once a year, so you are not alone in this misconception. The problem is that it works as long as the weight is evenly distributed over racquet, but you can never tell when it doesn't. It also makes the concept of swingweight meaningless since it just a rearrangement of weight and balance. If you don't want to measure swingweight (which is not that difficult) it is better just to talk about weight and balance as such.
Finally, if you look at the definitions of swingweight (J) and balance (R) for and object of length L and weight M:
You can see that both depend on the distribution of mass m(x). But since the integrals are different, you can never use R to calculate J without knowing m(x).
(I have used the one dimensional case to simplify)