Matt Springer is a graduate student of physics at Texas A&M university. He is also an occasional writer and tinkerer, and he is probably too curious for his own good.
when you order from Amazon, and a fraction of the purchase price will be sent to me at zero cost to you. Much obliged, and thanks for your patronage.I trust you're having a relaxing Sunday? Mathematical physics can be relaxing too, especially when you just look at it. We're just going to look at this one. In fact, this is a literal mathematical instantiation of Sunday relaxation.
If you fix a wire or a rope at two points and let it hang naturally, it forms a shape called a catenary.
It looks a lot like a parabola, and it turns out that in fact as long as the suspension points aren't too close together compared to the length of the rope, it's a very good approximation. For this particular graph the error from a purely parabolic approximation would be nowhere greater than 4%.
What's physically interesting about this curve is that the sum total of the potential energy of each little part of this rope is a minimum. There's no possible way to arrange the curve to have a lower potential energy. Nature does it automatically.
Physical Science
Comments
What does the h stands for?
Posted by: IBY | August 10, 2008 12:50 PM
The function is not Cosine, it is the hyperbolic cosine, abbreviated "cosh".
cosh x = (e^x + e^-x)/2
Is it possible to get latex enable in comments?
Posted by: Palmer | August 10, 2008 1:14 PM
Unfortunately I don't have enough access to the site internal code to put in latex myself, but I'll start agitating for the powers-that-be to work on it.
Posted by: Matt Springer | August 10, 2008 1:57 PM
Wow! Now that brings back some old memories of static’s class back in college, I always loved how nature tends to do things with such mathematical precision, in the midst of chaotic environment.
Posted by: Ken Clark | August 10, 2008 2:45 PM
That's why The St. Louis Arch closely approximates an inverted catenary.
For details, see:
Weisstein, Eric W., "Catenary." From MathWorld--A Wolfram Web Resource.
Even more interesting, google: "brachistochrone" ...
Posted by: Jonathan Vos Post | August 11, 2008 12:29 AM
A looong time ago I was given this wonderful book 'A Book of Curves' by E. H. Lockwood, Cambridge University Press, 1961. I was going to copy you from the 'Tractrix and Catenary' section, but a little Googling found it available online in several formats at the Internet Archive 'http://www.archive.org/details/bookofcurves006299mbp'. So there you have it all, for free no less. Not the same as having the original hard-cover in hand, but nonetheless a fascinating read for this audience.
The emphasis is on actually drawing these curves, not just the mathematics. For me, a play-along read as a teenager with too much time on his hands. So a challenge to program, or to implement in Mathlab, ...
Posted by: GrayGaffer | August 11, 2008 1:46 AM
Man, trigonometry makes every simple thing seem complicated. ^_^ Thanks for the response!
Posted by: IBY | August 11, 2008 3:59 AM
"Is it possible to get latex enable in comments?"
I'm all for latex...oh! Wait a minute - wrong blog....
Seriously, what's the difference between a catenary and an arc?!
Posted by: Ian | August 11, 2008 7:53 AM
"What's physically interesting about this curve is that the sum total of the potential energy of each little part of this rope is a minimum."
Are there any static systems that _don't_ settle at minimum potential energy?
Posted by: Tom Jackson | August 11, 2008 11:05 AM
Set a block of lead on a block of wood, and the minimum potential would of course be for the blocks to be the other way around. It's still a local minimum in the potential though. Thermodynamics can also complicate things, as for instance if a heavy gas is mixed in a somewhat lighter gas.
Posted by: Matt Springer | August 11, 2008 6:50 PM
Unless i'm taking the wrong approach, finding this function f involves solving for f by minimizing a finite integral, whose integrand involves f and f'. i can't think of an analytical way to solve it right now.
Posted by: venkat | August 12, 2008 9:05 AM
Calculus of Variations! ::Fanfare plays::
I remember this problem from my classical dynamics class.
Posted by: Chris Hertlein | August 12, 2008 5:22 PM
Just FYI, the differential equation involved is 1+(y')^2=yy". One needs to integrate the gravitational potential energy along the rope and subject the integrand to Euler-Lagrange equation to find the minimal value for the integral (potential). It's a good physics problem.
Posted by: Anonymous | March 20, 2009 12:15 PM