The Catenary: Definition and Derivation
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- čas přidán 30. 09. 2019
- A catenary is the mathematical curve formed by a uniform cable hanging under its own weight. Here, we give a complete derivation of the equation describing a catenary, using intro calculus and some basic physics.
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Excellent, I was looking for a solution without using variational calculus, thank you
Excellent video in terms of your commentary. The math is what is important to me, and you nailed it, no gaps, clear as a bell. Thanks for that especially because I feel that that is lacking when it comes to other videos that attempt to derive the equation.
Could you upload a video of the background music? It gave a certain warmth that I really enjoyed and I'd love to listen to it on its own. Loved the proof btw!
Hi Mason, send me an email at scotthersh42 [at] gmail.com and I'd be happy to send you the audio recording of the music. I'm glad you enjoy it!
This is an elegant proof of the catenary function.
As an engineer I admire the purity of the physics.
A minor quibble - the use of the secant function.
Trigonometry IMO suffers from having too many functions in use. All that is needed is one. Let's say we pick sine : Then cosine is equivalent to sqrt( 1 - sin(x)^2 ), tangent equivalent to sin(x)/(1-sqrt(1- sin(x)^2)) and secant, cosecant and cotangent are unnecessary luxuries.
One could add arcsin(x) as a very useful function, especially if you are not bothered by irrational quantities.
The same remarks could be applied to the hyperbolic functions, where sinh(x) and arcsinh(x) are sufficient to carry the day.
I agree that the proliferation of trig functions isn't necessary, and that perhaps we'd be better off without sec, csc, and cot - however, I think there is substantial value in using at least sine and cosine instead of just sine. These two show up so often together (parametrizing curves, polar form of complex numbers, etc), and putting everything in terms of sine (whether by the pythagorean identity or by shifting the argument by pi/2) would risk making things appear needlessly complicated.
@@on_the_cusp I cannot disagree that cosine is a convenient function and seems to work hand-in-glove so to speak with sine, and is probably well worth keeping. Actually it only occurred to me since writing the comment, that the probable reason that this apparent plethora of trigonometric functions were invented, is that not too many decades ago, most calculations were manual, which necessitated the frequent use of tables, which much simplified the process so that tan and the reciprocal functions were more useful then than they might be today.
The function arctan is also pretty useful in the real world also.
I can also comment that if one is not too concerned about periodicity, and is more concerned with rational functions then sine can be expressed as the ratio ( 2*u )/( 1 + u^2 ) where 0 < u
@@crustyoldfart Good point - ease of manual calculation was a major motivating factor for the way trig and a lot of other mathematics were developed.
Daaaaaamn, my maths is rusty...! It's a skill you need to practice constantly, I think...
Plz make more videos
Great video Scott! I'll leave some feedback here, instead of in a message, because algorithms. (smash that like and subscribe button, leave a comment, all that jazz)
My first impression from this video is that it is a tutorial for students of physics. Most of the time was spent doing math, which, as someone without a heavy math background, I didn't really follow too well. If I was in a physics class and wanted to find an explanation for catenaries, however, I think I would have found this video both helpful and pleasing to watch. The music really does make a good vibe, I think (loved the little flourish when you finished btw).
If you want your target audience to be people with a general interest in science, rather than strictly physics students, then perhaps a slight change you could consider would be spending more time talking about what cool things catenaries can do or are used for, or where they are found in nature - something that would capture the interest of those without much background in physics/maths, but would still be fun for someone with a physics background to watch. You don't want to straddle the line too much, and it's ok to be in a niche, but you should consider how big of an audience you want to reach. Are you a physics tutor, or are you Neil deGrasse Tyson?
In terms of production quality, I was surprised that it was this good, especially for your first video. The math portion was very legible and clearly explained, and there were some nice pictures to complement (the music, to me, also matched the "mysterious physics stuff" mood).
In case you're interested, I'll leave a few links to some more academically-inclined/educational channels that I like to watch, in no particular order. Maybe they'll provide some new ideas for you.
- Wisecrack (philosophy in film) - czcams.com/channels/6-ymYjG0SU0jUWnWh9ZzEQ.html
- Philosophy Tube (philosophy in contemporary issues) - czcams.com/channels/2PA-AKmVpU6NKCGtZq_rKQ.html
- ContraPoints (philosophy of social issues, dialogue/skit-style) - czcams.com/channels/NvsIonJdJ5E4EXMa65VYpA.html
- Academy of Ideas (quotes books by "thinkers" over nice paintings, makes you feel big-brain) - czcams.com/channels/iRiQGCHGjDLT9FQXFW0I3A.html
- The Armchair Historian (military history) - czcams.com/channels/eUJFQ0D9qs6aVNyUt9fkeQ.html
- Adam Neely (I'm guessing you're already subbed) - czcams.com/channels/nkp4xDOwqqJD7sSM3xdUiQ.html
- Sideways (film composition) - czcams.com/channels/i7l9chXMljpUft67vw78qw.html
- Imaginary Ambition ("how to [popular musician]", memey and informative) - czcams.com/channels/P5bYRGZUJMG93AVoMekz9g.html
- Valuetainment (business/entrepreneurship) - czcams.com/channels/IHdDJ0tjn_3j-FS7s_X1kQ.html
- The Plain Bagel (finance) - czcams.com/channels/FCEuCsyWP0YkP3CZ3Mr01Q.html
- Hello Future Me (fiction writing) - czcams.com/channels/FQMO-YL87u-6Rt8hIVsRjA.html
Thanks Dillon for your thoughts! You have a good point that this video is pretty heavy on the math and light on the applications. I was originally going to add more about where catenaries show up in the world, but with the video already almost 10 minutes long I decided that that would be better left for another video (which I might make in the future). My thinking at this point is that some of my videos will be heavy on the math, and others less so--this just happened to be the topic that I picked for the first one.
I do think that there is an unfulfilled niche for videos aimed at people with more of a technical background in math/science, and so I do want to make some videos like that, but I want the channel in general to be accesible and appealing to a broader audience, so that is something I will keep in mind as I plan future videos.
Thanks for sharing so many channels that I can learn from! I will definitely check them out. (You are right, of course, that I am already subscribed to Adam Neely)
It looks like the catenary would be the same shape no matter the force of gravity.
Catenary and parabola are related or not???
You call a a constant, but it really is a function of lambda, because it contains T naught. That’s helpful when you take the derivative of s with respect to lambda, but I don’t see how you can treat a as a constant. In fact, a must be a function of something else, or it would never change, which it does, giving us different curves that are all still catenaries. So, if you have two functions multiplied together, you have to use the product rule to find the derivative, here, of s. But that would screw up the whole proof, which is otherwise unquestionable. Would love it if you would address this. I’m not saying you’re wrong, just that I don’t understand. Thank you.
Hi James, for a given catenary T naught is a constant because the center point of the curve is uniquely defined and there is a certain tension at that point, which we call T naught. For a different catenary, this value will be different, but once we have fixed a particular catenary, that tension at the center point is a constant. Lambda is also a constant, again because we are considering a particular string/cable, and we assume that the mass density is uniform. (If lambda is a function of x or s the problem can still be solved, but it's much more complicated and in most real-life situations the assumption that lambda=constant is reasonable.) Thus, a is also a constant. Hope this helps, let me know if you're still confused!
@@on_the_cusp So, effectively, we are happy to choose a particular catenary, by thinking of T naught as a constant. Even after we choose one specific catenary, there is still the task of defining its position in the Cartesian plane. What I say about that, as a way of bridging the gap between x and y and lambda, is: As the cable gets longer, the angle of the tension vector increases, indicating a relationship between the angle and length of the cable, its shape, and, ultimately, the function that gives the curve’s location in the Cartesian plane. Shape is a function of length, thus, x and y are both rightly thought of parametrically, as functions of lambda. By the way, I’m a 58 year old guy, dad of James, who swims and is in 9th grade, and is already taking Calculus.
@@212ntruesdale lambda, mass per unit length, is an inherent property of the cable and does not depend on the shape--whether you put the cable in a circle, or a straight line, or let it hang as a catenary, the mass per unit length is unchanged. It also does not depend on the length of the cable--just the material from which it is made. For example 2 inches of copper wire has twice the mass of 1 inch of copper wire, but the same mass *per unit length*.
You are correct that x and y can be thought of parametrically in terms of the shape, but here that shape is described by arc length s, not lambda.
Congratulations to James on taking calc so early and I hope he keeps learning as much math as he can!
@@on_the_cusp Oh, dear! I realize I meant phi, not lambda!. Hard to keep them all straight. Still thinking about why it’s okay to treat Tcosphi as a constant. Then I thought about how different cables still make the same curve. As cables of different density, they obviously have different tension vectors. The difference is in magnitude for a given length/catenary. Thus, magnitude can’t be viewed as a function of phi. So even though Tcosphi has phi in it, because it isn’t a vector, it can be thought of as a constant. Do you follow/like this analysis? Be honest! Add, subtract to it as necessary. However, I would like to do something with the observation that a heavy chain and piece of string, one much heavier than the other, but of the same length, trace the same path nonetheless.
@@212ntruesdale Ah ok, no worries! Tcosphi is not inherently a constant--since both T and phi depend on arc length s, it's conceivable that the combination Tcosphi would also depend on s. However, the physical constraints of the setup (setting the sum of the forces equal to zero) leads to the equation Tcosphi=T naught, where T naught is a constant independent of s. This fact does not strictly speaking result from the fact that Tcosphi is not a vector--rather, it results from the physics of the situation putting a constraint on the components of the vector tension.
As for a heavy chain and a piece of string tracing the same path, the analysis I did in this video isn't sufficient to show that that is the case. What happens is that "a" contains the ratio T naught / lambda, and a heavy chain will have both a larger lambda and a larger T naught than the piece of string. These effects cancel each other out, giving the same "a" value for both curves.
e is everywhere
Yes, it certainly is!
horrible background music, i am trying to learn, not playing minecraft,