How do varying forces on a massless rope affect tension?

[ForceBoy]

Σ

Homework Statement

A massless rope is pulled at from both ends. At end A, the force applied is a pull of 10N. On end B, the force applied is a pull of 5N. What is the tension of the rope?

Homework Equations

a=ΣF/m

The Attempt at a Solution

I understand that the rope would accelerate towards point A. What I don't understand is how you find the tension. Up until now I have only seen tension problems where the force applied at both ends is the same. It is easy to understand that the tension would be the same throuought the rope.
If I had to guess, I would say that tension at point A is 5 N towards point B. On B it would be 10 N towards point A. This means the rope would have two different tensions. This can't be.

 

[Orodruin]

You cannot have a net force on a massless object. It would violate Newton's second law. In the limit of the mass going to zero, the tension would vary throughout the rope, but the acceleration would tend to infinity.

If I had to guess, I would say that tension at point A is 5 N towards point B. On B it would be 10 N towards point A. This means the rope would have two different tensions. This can't be.

First of all, tension does not have a direction (it is actually the one-dimensional equivalent of the stress tensor, so technically it has two directions but this is besides the point and not helpful to think about at this stage). Second, the tension would vary the same way throughout the rope if it had any non-zero mass so in the limit of the mass going to zero, the tension would not be affected. You could therefore find this limit by considering a massive rope.

 

[Charles Link]

IMO, this one makes no sense unless you include a mass in the middle of the rope. If A is on the left, and B on the right, there would be an acceleration to the left. In addition, the tension in the rope to the left of the mass would be 10 N and the tension in the rope to the right of the mass would be 5 N. The mass would have a net force on it of 5 N and the system would accelerate with $a=5 N/m$.