Curve of hanging cable

Precision

The curve given by a hanging cable, for example an electrical cable between two poles, hangs in an equilibrium configuration such that its overall potential energy due to gravity is at a minimum.

Galileo is known to have mistakenly thought that the curve was a parabola.

However, the curve of a hanging cable is actually a curve known as a catenary.

Taken gravity to be acting downwards in the y-axis and
ignoring its location or scaling parameters in the x-axis or y-axis,
the equation of the catenary curve is,


y = e^(x) + e^(-x)


So, there's a bit of a challenge of a maths formula to derive.

As the olde saying goes, a problem shared is a problem halved!!

(Hmmm... maybe that saying is not quite as valid for mathematics problems.)

ecksor

This is a standard problem in the calculus of variations, which I'd explain if I'd revised it properly (although I suppose I should, I have an exam in it in a few weeks). There's a few examples of online notes in the subject however.

Precision

Thanks Ecksor,

I have not yet been able to derive this one myself yet. I am working on it though. I had been focusing too much on the potential energy of it. The maths started to get a bit messy though. I found what looks like would lead to a proof of the hanging cable formula on this link,

http://webpages.ursinus.edu/lriley/courses/p212/lectures/node14.html

It focuses more on the tension within the cable. I have not yet get around to going through it, just took a brief look. Plan to take a better look later.

Capt'n Midnight

For those with scientific calculators the function y = e^(x) + e^(-x)
is just the hyperbolic cosine of X / cosh(X)

Have a look at this for some clues on the maths - you have a series of triangles
http://web.njit.edu/~jcl7/pastimes/catenary/index.html

Capt'n Midnight

But a hanging cable is straight

Mellor

Capt'n Midnight wrote:

For those with scientific calculators the function y = e^(x) + e^(-x)
is just the hyperbolic cosine of X / cosh(X)

Have a look at this for some clues on the maths - you have a series of triangles
http://web.njit.edu/~jcl7/pastimes/catenary/index.html

Interesting link Capt'n.
There is a picture of three cables/strings hanginf to the same curve and the comment that density has no effect.
Surely its does, just that it is unnoticed in the cables as each curve is greater than the min radius of the cable.
If a longer catenary is attempted, it wont work with dense cables as the radius at the apex is less than the min radius capable of the cable.
So not all cables will produce catenarys

ZorbaTehZ

Disregarding whether or not a cable is physically capable of producing a catenary, would I be incorrect in saying that if the curve of a catenary was not independant of it's density (keeping the length between each end constant - and the length of the cable proper), then that would imply that the acceleration acting on each part of the cable was proportional to it's mass? Which clearly is incorrect. . .

Buffer

Just passing by and I thought I'd add an observation ...

Technically, a catenary curve is the shape of a free-hanging chain rather than cable, though with infinitesimally small link length. The reason for this distinction is that no bending stiffness along its length is assumed, so the equation does not take into account the possibility of a mininimum radius of curvature.

As with all idealised mathematical models, real-world entities don't exactly match the assumptions they make. A rope is a pretty good approximation but a cable is slightly less good because of its inherent bending stiffness; for example, a thick cable spanning a short distance might not bend visibly. A chain with just four large solid links also would not of course form a correct catenary.

The catenary equation describes a curve that is obviously independent of density because it contains no terms for density, mass or related quantities.