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What is the tension required for a rope with a finite mass per unit length, hanging between two fixed equal altitude points under gravity, so that the rope is perfectly horizontal (without any "slack"). Assume other properties of rope such as elastic constant, total mass, length, mass per unit length to be finite. And mass per unit length is uniform every point.

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Just for your information, let me start by saying that the form of a rope hanging between two, say, equally-hight (exactly) vertical sticks, is a catenary (just as a rope hanging between two points that are not at equal height).
The rope can never be in an exactly horizontal form, no matter how great the tension. Gravity will always be present to introduce a bend in the rope.
As you said in your question, the properties of the rope such as elastic constant, total mass, length, mass per unit length are finite. This suggests we have to do with a real rope. For the rope to be perfectly horizontal we have to apply an infinite force to the rope, in the horizontal direction. Obviously, the rope will have snapped before reaching (the impossible) infinite force.

Even if the rope was an idealized one (unbreakable, with constant length), it wouldn't be possible because an infinite force doesn't exist. The rope would be exactly vertical in form (the horizontal deformation caused by gravity is overwhelmed by the infinite vertical force) in this case, but because of the simple fact that it is impossible to pull with infinite force on both ends of the rope, also this scenario would fail.

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  • $\begingroup$ The 2 endpoints don't need to be at equal heights, you still get a section of a catenary. That follows because "the tension forces on each segment of the chain are directed along the chain. They must be equal and opposite, and the forces at each mass must sum to zero." So if you have a symmetrical catenary you can attach a new support at any point along the chain without affecting the curve. See physics.stackexchange.com/a/421965/123208 $\endgroup$ – PM 2Ring Jul 9 at 13:35
  • $\begingroup$ @PM2Ring The form of the catenary has no relevance for the answer. I just added it as extra information. I know the two endpoints don't have to be at equal heights but in this question that is the case. You could have asked the same question for a rope that is attached to two non-equal heights points. $\endgroup$ – Deschele Schilder Jul 9 at 13:48
  • $\begingroup$ The catenary tends to a straight line in the limit, where its length is equal to the length between the two points. I think a more realistic scenario is where the rope is attached on one end (say the left), it goes over to the other (say right end), and some length of it hangs over. Then one can adjust the length between the left and right ends by adjusting the tension from the hanging end (to the right of the right "pivot"). I know this isn't what OP asked for, but at least it has a well-defined answer that might interest OP. $\endgroup$ – Stratiev Jul 9 at 14:00
  • $\begingroup$ @Stratiev Yes. But the rope will not be perfectly straight, as the OP asked. The pull of gravity will always be there, so the rope's form is not straight. If you would hang the rope between two buildings and start walking on it, the rope (if the tension has not already the snapping value) can go up and down while you walk on it. Because the rope is not yet subject to the maximum possible tension (but it is subjected to a high tension) the rope will move up and down very fast when walking on it. But yo have to be very cautious not to let the rope move too much. Now, thát's a challenge! $\endgroup$ – Deschele Schilder Jul 9 at 14:21

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