Dealing with pulleys and strings with mass

Question

So far, Ive been solving problems where the pulleys and strings were assumed to be massless. But how to write free body diagrams for a string/pulley with mass? This was the question I was trying to solve:

Two blocks of masses 6 kg and 4 kg connected by a rope of mass 2 kg are resting on a frictionless floor as shown in the following figure. If a constant force of 60 N is applied to 6 kg block, the acceleration of the system is:

I can draw the fbd for the two blocks. For the string what is feel is since the tension at any point on the string is same net force in horizontal direction is 0. But the string must be moving with som acceleration. Please correct me if im going wrong anywhere.

EDIT 1: Ok i realised that i was doing a few silly mistakes and i rectified them. Is this correct?

Answer 1

Since the string or rope follows translational motion with both the blocks the string along with the blocks can be considered as a single system. Therefore the total mass of the system is 12 Kg and the force acting on the system is 60 N. According to the Newton's second law of motion we get the acceleration of the system as 5 m/s². Thus the string will also move with an acceleration of 5m/s².

Answer 2

You are right - the string will be accelerating with the same acceleration as the rest of the system, so the total system has a mass of 4+2+6 = 12 kg, and the acceleration is 5 m/s$^2$.

This means there must be a net force of 10 N on the rope; and this means that the tension on the right of the rope must be 10 N greater than the tension on the left (and in fact it means the tension is varying linearly all along the rope - consider the rope made up of little masses connected by massless strings of zero length, and you can see that).

Hope that clears it up for you.