I've been getting help from sparknotes.com and I've come across an example that has stumped me.

The page is at http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section4.rhtml

I'm at example 2 (toward the bottom of the page).

The problem comes when they use Newton's second law to solve for the tensions. It says that the sum of the forces = 3mg - T1 = 3ma. I don't understand how mg suddenly turns into ma and (-T1) disappears.

If any could enlighten me, I'd appreciate it!

Ok, here is your misunderstanding. On mass 1, there are two "forces" acting. The tension in the rope and the weight of the mass. We don't know the tension of the rope, so we called it T1. But we DO know that the weight is 3 mg. Now, we know that T1 represents SOME force, which means we can state it as 3m(something). So when we solve it, we get: 3mg-3m(something)=3ma where a is equal to g-(something). Oh, and (something) is an acceleration of some sort. The a in the equation is a variable, not a defined constant.

Hope that clears things up.