All context is given here. I am trying to understand a few things about what Morin is doing.
1) How is proving that f(x) is a linear function, which I understand the proof of, relevant to the claim that the torques at the left end cancel? Also, the introduction of the function makes little sense to be as well. Clearly in this example ##f(a+b) = a+b## and ##f(a) = a## and ##f(b) = b##, so why bother?
2) Why does Morin seemingly assume the answer to the problem at the very beginning of the proof (2.8). Claim 2.1 states ##F_3a=F_2(a+b)## to prove, and then he calls it a "reasonable assumption" in the first line, except it's inside a function, which is the same thing as distance.
Claim 2.1, (2.8), (2.9), (2.10), and the concluding statement: f(x) = Ax.
The Attempt at a Solution
Technically there is already a solution in the book, I am just trying to understand it.
I am brainstorming and thinking that we eventually prove that what f(x) is IS the distance, because f(x) = x (so A=1, but he just says A is irrelevent; why not come out and say it's one?), but in the first line he literally says that we are related the "forces and distances," so that thought doesn't make much sense. Also, yes I do see the footnote (3), but I can argue with that, because they aren't being applied at the same point, so I'm not sure why he's even saying that - so that's about where I am at this point.
Is this much confusion bad news for me? How can I understand this better and not have to ask you everything? I'd prefer to not brood over this for >1 hour when most people get this instantly.