Tension on a rope with significant mass

[RafaFutbol]

Homework Statement

Consider a rope that, unlike those usually studied in mechanics problems, actually has a significant mass "m". The tension at the right end of this rope is T2 and that at the left end is T1. (figure1.0) The rope has an acceleration a_rope to the right.

Complete the following equation for the acceleration of the section of the rope of mass m, taking the positive direction to be to the right.

F_rope = m*a_rope = ?

Give your answer in terms of T_1, T_2, and constants such as g.

The Attempt at a Solution

I'm thinking that the Force on the rope will be an addition between the tension difference that leads to the acceleration, and the force of gravity pulling it down.

So m*a_rope = (T_2 - T_1) + mg

However, since the answer can't include m, I'm kind of lost as to where to go from here. I might just be overlooking something simple, but some insight would be really helpful

Thanks!

 

[nasu]

You are writing the equation for horizontal acceleration. The weight is not along the horizontal.
At least not here...