Hanging it Up

I was thinking again about the bent rail problem from a couple of days ago, and got to wondering what the height would be if it was an inverted catenary arc instead of a circular arc.  I was led to this idea by one of my students who also had suggested that it might NOT be circular....and I had no way to assure him it would be... I have no reason to assume it is a catenary either, but it was a curve I felt I needed to know more about so I used it as a self-teaching exercise.

At that point I realized I didn't know much about the catenary curve in general.  (what I mostly knew is about its history and can be found here).  So I set out to see if I could learn a little more and figure out how to solve the question of how high would the inverted catenary reach.  Below is what I have attempted to do... the answer is close enough to the circular amount of bowing that I wonder if it can be right, so if you are up on this stuff, please comment.

I knew that the catenary curve y= a cosh(x/a) has a minimum at (0,a).  For students who haven't seen "cosh" that's just a hyperbolic cosine function (much like the circular cosine function in many ways, and often pronounced to rhyme with "gosh"... see an explanation of hyperbolic functions here).  It can be replaced with an alternate form that may be more palatable to your eyes:

Of course it is easy to see that the vertex is at (0,a).  What I had never known, and find particularly wonderful about the curve, is that for any catenary, the area under the curve from x=c to x=d  divided by the constant a of the equation is the length of the curve.....A good calculus student can show this is true pretty easily with the exponential form since the derivative of cosh(x) is sinh(x).... and for hyperbolas we have cosh^2 - sinh^2 = 1 instead of the addition. That makes the integral for arc length the same as the integral for area between the curve and the x-axis.

So the length of the catenary from the vertex to any other value of x is given by a* Sinh(x/a)  (OK, you can guess that is the hyperbolic Sin, usually pronounced like "cinch" and it is the same as the hyperbolic cosine except the two exponentials are joined by subtraction rather than addition..)

I worked the problem out upside down because I wasn't smart enough to turn it rightside up... ... I assumed that the rod was hanging under a secant line.  For our hanging (and now very flexible) rod, the arc length is 5281 feet, and the secant (parallel to the x-axis) is 5280 feet.  We can position the end points so that they are symmetric around the y-axis so that our hanging rod will have it's vertex on the y-axis. 

So we know that the vertex is at (0,a) and we know that the length of the cable from the vertex to x=2640 is going to be 2640.5 which must equal a*sinh(x/a).    This seemed to give us enough to write an equation that we could solve for a, and my answer came out close to 82,809... which seems like a really big constant.  Buty the effect of a on the catenary is to adjust both the distance from the origin, but also how fast it turns up.  Here is a graph showing catenary curves for a=1 (red) and a=2(blue). 

So we want a very wide catenary, we need a really big number... maybe 82,809 will work.  So our equation for this curve is y = 82,809 * cosh(x/82809).   Now we can use the equation for the curve to find the height of the curve at the point where x= 2640.  This turns out to be just a smidge over 82,851 or almost exactly 42 feet above the y-intercept of 82,809.  So if we flipped it over and set it on the ground, the one mile wide arch would be only 42 feet high.   Very near to the 45 foot answer we got from the circular arc. 

Recently I wrote about Willima Whewell's role in the creation of the word scientist. I just found that there is also a Whewell equation for the catenary which gives the slope of the tangent of the catenary as a function of the distance from the origin, s. Tan(T)= s/a. That means in the catenary version of our inverted rail, the angle between the curve and the horizontal secant is just over .03 radians.... I think I did that correctly.

As a comparison, the St. Louis arch is 630 feet high, and has a width from side to side of 630 feet.  It is not a true catenary because it is adjusted for weight, called a "weighted" or "flattened" catenary. If we assume the difference is small (I don't know for sure) then the information above should allow you to find the constant "a" for the arch, and then find the length of the arch... Good luck... 

Meet Math in St. Louis

Happy Birthday, St. Louis (the city, not the guy)... It was on this day, Feb 15, 1764 that the city of St. Louis was founded.... I know, Wikipedia said so. Two hundred one years later, the Gateway Arch opened to the public in the Jefferson National Expansion Memorial. But for the math person, it is a big catenary arch... note to students, it is NOT a parabola... (OK, as you will see below, it is not an exact catenary)

Here is a video I found out about from Dave Richeson's Division by Zero blog that he posted a little while back about the math involved.


A little history for those, like me, who enjoy that sort of thing. It may be so perfectly fitting that the Jefferson National Expansion Memorial should have a catenary arch, since President Jefferson may have invented that exact English term for the shape. The shape is also sometimes called a chainette and a funicular curve from the Latin "funiculus" for a cord or rope.

For students of American History, it may be interesting that the first use of "catenary", rather than the longer, more formal "catenaria", may have been in a letter from Thomas Jefferson to Thomas Paine. Jeff Miller's wonderful web-site on the first use of mathematical words has

In a letter to Thomas Jefferson dated Sept. 15, 1788, Thomas Paine, discussing the design of a bridge, used the term catenarian arch:

Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypothenuse,(I believe that was the spelling in the letter, not my typo) the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypothenuse and right angle will be always given.

In a reply to Paine dated Dec. 23, 1788, Thomas Jefferson used the word catenary:

You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abbe Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'

The earliest citation for catenary in the OED2 is from the above letter.
============================================
Oh, and is anyone else surprised that Thomas Paine is involved in Bridge Design? I found another very brief comment from the same period that says: "1788: Mr. Paine resided at Rotherham in Yorkshire during part of the year, where an iron bridge upon the principle he had invented and laid before the Academy of Science was cast and erected."

In another article about the Walker Brothers Ironworks in Rotheram I found, "The Walkers were also known for their work with Thomas Paine from 1788 to 1791 in the construction of prototypical iron bridges. Thomas Paine, in addition to writing such works as The Rights of Man and Common Sense, was also one of the very earliest designers of single span iron bridges. In 1785 Paine tried unsuccessfully to gain support to erect an iron bridge over the Schuylkill River in Philadelphia. Unable to find investors in America, at Benjamin Franklin's suggestion Paine built a model of his design and traveled to England to show it to the Royal Society. Though his model was well received, still no backers were forthcoming, prompting Paine to seek out an ironworks which would built him a full scale prototype.
The Walker Ironworks in Rotherham agreed to do just that in the winter of 1788-1789. The bridge, with a span of 110 feet, was completed in 1790, but instead of being erected across the Thames River, as Paine would have liked, was instead set up in a field called Lisson Green, where other spectacles and attractions were located. Curious passers by could walk over the bridge or simply marvel at its construction. This endeavor, intended to attract possible investors in the building of pre-fabricated iron bridges, failed miserably. The rusting Lisson Green bridge was taken back to Rotherham by the Walkers in October of 1791. Thomas Paine, though disappointed, traveled to France where his writings about the French Revolution would land him in prison and nearly cost him his life.) The Walkers later built some of the earliest iron bridges ever made, including the bridge over the River Wear in Sunderland."

Now I wonder, was it (is it) circular or catenary?
And now I know...."The arch had a catenary shape and weighed 36½ tons, of which 23 tons was wrought iron and the remainder cast. It had a span of 110 feet and a rise of 5 feet. It remained on display for a year, attracting some fee-paying visitors but no offers to finance a Thames crossing. Eventually the abutments gave way—always a danger with a shallow arch—whereupon the bridge was dismantled and sent back to Rotherham. By then Paine had once again flung himself into revolutionary activities, this time in France, and his career as a bridge designer and promoter began a decade-long hiatus." From American Heritage Magazine

I should point out that this was not a singular stroll into science and engineering for the President. He also designed an improved moldboard (from Old English molde "earth, sand, dust, soil; land, country, world,") plow for which the French Society of Agriculture awarded Jefferson its gold medal and membership as a foreign associate.