How Is Tension Calculated in a Pulley System with Unequal Masses?

[Notyou]

Homework Statement

Two objects are hanging from a pulley. One has a mass of 12kg and another has a mass of 10kg. Initially, the two masses are at rest and friction is ignored. Once in motion, find the tension in the string and the acceleration of each of the two masses.

Homework Equations

F=ma

The Attempt at a Solution

Ok... another tension problem. I'm really having an extremely hard time understanding the concept of tension. For this problem I drew the free body diagrams of both masses. The mass of 12kg is m1 and the mass of 10kg is m2.
For m1, the force in the negative y direction (pulling the object to the ground) is the weight, or the mass times acceleration. The mass pulling on m1 in the positive direction is the tension.
For m2, the force pulling in the negative y direction is the weight. The force pulling in the positive y direction is the tension and the weight of m1.
The net force acting on m1 would be... m1g-T=(m1 + m2)a
The net force acting on m2 would be... T+m1g=m2a

Something in that train of thought is wrong. I believe it is the net force acting on m2 or maybe even the force acting on m2 in the positive y direction. Any help would be greatly appreciated.

 

[Notyou]

Figured it out <_<. The force was acting on the objects independently. What I don't really understand though, is how come in this scenario the same force is acting independently on the objects but if I laid the two blocks on a flat surface and pulled the heavier block with a string the force would be acting on their combined weight?