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...

 

[baba89]

I would like to clarify that the equation provided in the problem, F_rope = m*a_rope, is already in terms of the mass of the rope (m). This means that the answer should not include m, and instead should be in terms of other variables such as T_1, T_2, and constants like g.

To find the acceleration of the rope, we can use Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration. In this case, the net force is the sum of the tension forces acting on the rope and the force of gravity pulling it down.

So, the equation would be:

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

Note that the tension at the left end of the rope is in the opposite direction, so it is represented as a negative force.

Now, we can rearrange this equation to solve for a_rope:

a_rope = (T_2 - T_1)/m + g

This is the final equation in terms of the given variables and constants. I hope this helps clarify the solution.