Q: I would like to make certain I understand why the ideal rope is removed from the diagrams. Both sides of the rope are being pulled equally, so there is equal and opposite forces on the x axis that cancel out. In the Y axis, the rope would, if it had mass, have a weight vector, but due to mass being 0, there is nothing on the y axis as it is suspended. Because there is no net force, is that why we omit the rope?
A: You understanding is good! The main reason for which the ideal rope is omitted in the analysis/diagram is that it has ZERO mass! Therefore, according to Newton’s second law, we don’t need consider the gravity on the ideal rope. Also, the forces pulling the 2 ends of a same ideal rope must equal in magnitude/strength to match the zero net force on the rope!
In addition, according to Newton’s third law, the tension forces in a same ideal rope (pulling objects connected) must be constant in magnitude/strength, since the external puling forces on the rope equal in magnitude/strength; see figure.
