The hanging cable

A interesting problem was put forward by my physics teacher on Friday, relating to material properties - what is the longest possible constant-diameter cable, from a material with a given yield stress and density, that can be hung vertically in the Earth's gravitational field (with its end just touching the ground) without snapping?

In essence the problem is very simple. The point of maximum stress is right at the top of the cable, where it is hanging from some kind of steadfast loop capable of supporting the material in its entirety; at this point the stress is equal to the weight of the cable divided by the cross-sectional area. However when the materials used are strong enough to last kilometres into the air, the changing value of g starts to become a factor in the weight force experienced by the maximum stress point. I did some scribbling today, and came up with a little proof of a nice formula that takes into account the changes in g, with the use of a definite integral as the infinite sum of a series in its limit:

The most important insight to be gleaned from the final formula for the maximum height of cable is that the cross-sectional area of the cable is irrelevant to the height achieved, as long as it remains constant. It depends only on the radius of the Earth (constant), the universal gravitational constant (constant), the mass of the Earth (constant), the density of the material (constant for a given material) and the breaking stress of the material (constant for a given material) - note that it is assumed yield stress = breaking stress, since it is unnecessarily complicated to consider the plastic properties of the cable between yield and fracture.

An example of the use of the formula is to consider a material - steel is a popular choice for high-stress cabling. Wikipedia tells me the breaking stress of a certain type, 2800 maraging steel, is 2617MPa and the density is 8100g per cubic metre. The formula returns a H_max value of 33072.8304159 ≈ 33.0km. If the uniformity of the Earth's gravitational field was assumed, the H_max would be significantly different: 

Cross-sectional area of cable = 0.25πd²
Volume of cable = 0.25Hπd²
Mass of cable = 0.25Hρπd²
Tension at top of cable = 0.25Hρgπd²
Stress at top of cable = Hρg
H_max = σ_frac/ρg = 32934.3954896 ≈ 32.9km

There is an absolute difference of 138.43493 ≈ 138m here, which converts to a surprisingly sizeable percentage error of 0.419%!

Now it has been established how important the non-uniformity of the Earth's gravitational field is in such a matter, one must now wonder how the shape of the cable could maximise the value of H_max - my hypothesis is that having a slight linear taper on the cable such that it is fatter at the top than at the bottom, with the cross-sectional area scale factor per metre climbed matching the scale-factor for decrease in gravitational field strength per metre climbed, would be optimal. This is because having less mass at the bottom is essential for marginal gains, where the gravitational pull is strongest, and a taper to this specification should account logically for the inverse-square nature of Newtonian gravity.

Steel was a fairly old-school example for testing the value of H_max: a huge focus of the material science field is the applications of carbon nanotubes, tailor-made materials based on graphene and its chemical derivatives. Wikipedia informs me that one particular type, Armchair Single-Walled NanoTubes, has a breaking stress of 126.2GPa, nearly 50 times the strength of 2800 maraging steel. Ignoring the considerable strain produced by such a breaking stress (0.231), an oversight it seems, the H_max values can be calculated with the calculus and simplistic methods, given an approximate density value for Armchair SWNT of 1660kg per cubic metre (from here):

Calculus H_max = -35976055.5613
Simplistic H_max = 7749653.04644

This is pretty stunning as a result. The nanotubes are so strong that the calculus formula breaks down to form a negative result - this implies the cable can stretch infinitely out into space (assuming it is not affected by the gravitational fields of other bodies, which isn't technically true of course), effectively escaping the effects of the Earth's pull, without fracture. I suppose the combination of the cable being nearly 5 times less dense and 50 times stronger means there is an effective relative increase in potential height of 250 times.