Hyperbolic Functions: Asymmetric Catenaries

In this video I go over asymmetric catenaries which are just catenaries but hung at different heights as opposed. I was recently asked whether the curve formed by hanging a cable across two different heights was the same as that for when the heights are identical, and so I looked into this question further to realize that yes, yes it is. The reason for this is that in my earlier derivation video on catenaries I set up the coordinate system so that the lowest point on the catenary is located at the y-intercept, and then set up a free body diagram of the a segment of the rope or cable. The free body diagram illustrates the tension in the rope at the lowest point and the linear weight density as the only physical constants that need to be obtained, and thus the derivation does not include the actual heights at which the catenary is hung. Thus the resulting hyperbolic catenary function y = a*cosh(x/a) + c is the for both asymmetric and symmetric catenaries! The only difference is that the location of the lowest point of the curve, and tension at that point may change at different heights.

Also in this video I go over a couple of applications of the catenary functions, including catenary mooring line systems and the Sea-To-Sky Gondola transportation service for mountain climbers. This is a very interesting video illustrating varying types of catenaries and how we can always look to the original derivation of to see how the resulting shapes will be like, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvs1IuM3BFpCy2sDQQ

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-hyperbolic-functions-asymmetric-catenaries

Related Videos:

Hyperbolic Functions: Catenary: Example 4: Arc Length: https://youtu.be/mnBLG_D1nHg
Hyperbolic Functions: Catenary: Example 3: Telephone Lines: https://youtu.be/GbDGUYTrHQ0
Hyperbolic Functions: Catenary: Example 2: Graphing Catenaries: https://youtu.be/FlqcdaJn1NU
Hyperbolic Functions: Catenary: Example 1: Reverse Proof: https://youtu.be/KK4FoanPHzA
Hyperbolic Functions: Catenary: Formula and Proof: https://youtu.be/EYb1p9r1fnM
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc .

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