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Find the angle that lets you make a bendy pipe

Mathematics Asked by John Porter on December 13, 2020

I have a pipe with $n$ segments of equal length. I move the farthest end nearer than the combined length to make the pipe bend. How do I find the angle between each segment with only the combined length (straight length), the distance between each end, and the number of segments?

  1. $theta$ is the angle to solve for
  2. $l$ is segment length
  3. $n$ is the number of segments
  4. $dx$ is how the distance between ends

With only two segments I would use the law of cosines:$dx^2=2*l^2 – 2*l^2*cos theta $

One Answer

You have to assume all the angles between the segments are the same or the problem is underdetermined. In that case all the bends lie on the arc of a circle. A figure with four segments is below. $BG$ is your $dx$, $n=4$, and $l approx 3.6$ as drawn. All the triangles are isosceles. If angle $DAB=theta, GAB=4theta, GB=2rsin(2theta),l=BD=2rsin(theta/2)$. You need to use $l, GB$ to get $r, theta$

enter image description here

Answered by Ross Millikan on December 13, 2020

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