Calculating lag in telephone wires (Math)

Factual Questions
1

Suppose you have a wire of length L strung between two poles separated by a distance of A. They are attached at the poles at the same height. What is the distance between the lowest point the wire lags (hangs down) and the point where the wire attaches to the pole?

My idea was that when the wire lags it would essentially be creating a semicircle. If I could create some triangles and using circumference ratios, trig, etc I could come up with an equation where H = f(A, L), but I just can’t do it.

For the record, this isn’t a homework problem. Just something that popped into my head.

Picture of my problem

Here’s an illustration of my idea

[edit] Oh, and please ignore some of the work around my illustration. I realize it’s incorrect.

2

To help you in your search, the curve so formed is called a catenary.

3

It doesn’t create a semicircle. It creates a catenary:

If it’s very shallow, the semicircle may be a decent approximation, but if there’s a large amount of lag, it won’t even be close.

4

Actually it is not a semicircle, but rather a catenary.

5

A catenary? Are you sure?

6

Oh, thanks guys. I knew there had to be name for it and it’s embarrassing because obviously if you had a 100 foot wire between two poles that are 1 foot apart it wouldn’t be a semicircle. :smiley:

Thanks!

7

I think the shape is called a catenary.

8

Rather than chiming in with any of the obvious jokes about deep spots in the ocean or songs named after Mexican rivers, I’ll point out that a catenary (functionally, a hyperbolic cosine) is also the optimal shape for a freestanding (that is, not supporting anything else) arch, such as the famous St. Louis Gateway Arch. Meanwhile, if the weight supported by an arch or cable is uniform horizontally (as, for instance, with a suspension bridge), the shape is a parabola, and if the weight supported at a point is proportional to the difference between the point and the maximum (or minimum) of the arch (or cable), as with a stone arch bridge, the optimum shape is a cycloid.

9

Which is a cross between a cat and a canary, as if you didn’t know.

10

What about freestanding guide bars on…treadmills?

11

Note that a catenary applies to infinitely flexibly cables. The inherent stiffness of wires changes things a bit.

So, technically, no right answer so far.

12

And… this is also why the 120V/240V power lines going to your house are aluminum and not copper. Ohm for ohm, aluminum is lighter than copper. If the cables were copper, you would need more poles to support the weight of the cables.

13

No, I think the major reason is that aluminum wire is cheaper than copper.

For the average house, the distance from the power lines is pretty short, so copper wires wouldn’t require more poles (they’d just hang a bit lower).

The disadvantages of aluminum wire which have pretty much removed it from use inside houses (loosening as power goes on & off, need for professional connections to prevent oxidation, and risk of connecting aluminum and copper wires) don’t apply much to the power feed to a house, so aluminum wire is still used there.

14

Well, that too. :slight_smile:

15

Not even that. The shape of a catenary is determined entirely by the location of the endpoints and the length of the cable. Gravity and the density of the cable don’t matter at all, as long as neither is zero.

16

Well, there’s also the issue of tension in the cable and the ability of the supports (power poles) to withstand that tension.

17

This post removed because ftg had already made the same point.

18

The wires go both directions from the pole so the tension isn’t a factor except for its downward component. At corners where the wires aren’t directly in line, guy wires are used to help support the pole.