I have found several discussions on how to calculate the sag of rope that is tie
ID: 1377154 • Letter: I
Question
I have found several discussions on how to calculate the sag of rope that is tied off at two points (like a tightrope), and I understand it to a certain extent. What I can't wrap my head around is how come it is impossible to pull the rope tight enough that the sag goes away completely? I'm talking about big thick ropes, like a tightrope that has to be strong enough to hold at least one person. I can pull a piece of string so that it's perfectly straight...does this just have to do with scale?
I like physics, but I'm not great at it, so if someone could answer this in as close to layman's terms as possible, that would be greatly appreciated.
Explanation / Answer
There are three parts to this:
In mechanical equilibrium, things go to their lowest energy state.
A straight line is the shortest distance between two points.
Whenever you've minimized something, it means that small deviations don't change its value (to first order).
Let's start with the third point, which is mathematical, and then look at the physics of the situation. Take a look at this hill:
enter image description here
At the bottom, it's flat, which the middle red line shows. When you look on the sides away from the bottom, it's sloped. So if you want to be at the bottom, you need to be somewhere where small steps in any direction don't change your height. When you minimize a quantity, small deviations don't change it.
On to the physics.
enter image description here
The black things are pulleys on supports. The brown thing is a rope. The grey things are weights. How heavy do the weights on the sides have to be to pull the rope straight?
The physics of a system like this is that the weights will try to fall as much as they can. Another way to say this is that at equilibrium (i.e. nothing is moving), systems go to their lowest possible energy state. (Or a local minimum, at least.)
There's a tradeoff here in terms of energy. You could pull the middle weight down further, lowering its energy, but that would yank the end weights up some, raising their energy. The system will have to find the right angle of the ropes so that the energy is minimized.
A straight string will never minimize the energy, though. It's the shortest possible path between the two supports. Since it's the shortest path, small deviations to that path don't change its length to first order. (That's the math point from the beginning of the answer.) That means that you could always lower the energy a little bit moving the middle weight down. The side weights don't go up because the distance between the middle weight and the posts isn't changing. Meanwhile, the middle weight is going lower down, so the total energy of the system goes down. That means the straight line is never the lowest energy state, and so can't be the equilibrium configuration