Physics of the Catenary Cable

Rhett Allain
7 min readJan 17, 2021
Photo: Rhett Allain

In this modern society, cables are pretty much everywhere. It’s not just the stuff that connects your computer and monitor, but data and power cables need to be run all over the place. Often these cables are suspended between poles to keep them out of the way of cars and pedestrians.

Some of these overhead lines might look like they are straight from one pole to the next, but they aren’t. There is always some bit of sag. You can really notice this when the cable mounting points are quite a distance away.

But now we get to the physics of this. What shape does a cable make as it is supported between two points? The answer is a catenary curve. No, it’s not a parabola — even though it sort of looks like one.

Photo: Rhett Allain

The exact shape of the hanging cable depends on the linear mass density, the tension, and the location of the two mounting points.

OK, let’s derive this curve. It’s going to be fun.

I’ll be honest. This is one of those problems that at first seems too hard, then it seems straight forward, then you get stuck in some maths. We are going to work through it together.

Suppose we have a hanging cable. If we look at one particular segment of wire, it is at rest so that the net force (vector force) must be zero. Here is a diagram.

So, what’s going on here?

  • This is for a segment of length s and a uniform linear density of λ (kg/m).
  • There are three forces on this segment — the gravitational force and the tension on the left-end and the tension on the right-end.
  • Since this is NOT a massless string, the tension in the string is NOT constant.
  • I picked the left point of the segment such that T1 is horizontal.
  • The tension on the right (T2) is up at some angle, θ.

OK, three forces — I can add up the x-components and that has to be equal to zero. Same…

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Rhett Allain

Physics faculty, science blogger of all things geek. Technical Consultant for CBS MacGyver and MythBusters. WIRED blogger.