What are the equations for tension in a massive rope under acceleration?

[bartersnarter]

Homework Statement

Here is my problem:

Homework Equations

Here is my work thus far and all the equations I've included.

The Attempt at a Solution

I'm not completely sure if my work is correct. I keep writing and erasing the tension equations because I can't decide which are right. Can anyone explain how I should go about writing the equations for the various tensions under these circumstances?

 

[ehild]

No, it is not right. The tension is never negative. Think: if you pull a mass M with a rope of mass m, and the whole thing accelerates, where is the tension greater? at the mass or at the end where you pull the whole thing?

 

[bartersnarter]

The tension is greater at the mass. I believe tension increases opposite to the direction of acceleration for a massive rope. If the tension is negative, I think I put the sign there to show that it was in the opposite direction. I meant to write T₂ - T₃ = 0. I believe the rest of the equations agree with the fact that tension increases opposite of the direction of acceleration. Still, I don't understand how to write equations with tension when the system is not in equilibrium in this way. Would it be similar to a pulley?

 

[ehild]

The tension is greater at the mass. ... believe the rest of the equations agree with the fact that tension increases opposite of the direction of acceleration.

NO. At the object, the tension in the rope has to accelerate only the mass of the object. The force at the other end (which is equal to the tension there) has to accelerate the whole mass of the system, the object and the rope.