Helix (spool and wire) geometry

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SUMMARY

The discussion focuses on the geometric problem of determining the shortest length of cable wound around a spool, as presented in Stewart's Calculus Early Transcendentals (4th ed). The key equation involves the term 2π(R + r), which represents the combined circumference of the spool and the cable. Participants seek clarification on the geometric representation of the problem, specifically the orientation of the spool and the significance of the vertical distance between coils. Understanding this geometric relationship is crucial for solving the problem accurately.

PREREQUISITES
  • Basic understanding of calculus concepts, particularly in geometry.
  • Familiarity with the properties of circles and circumferences.
  • Knowledge of geometric representations and their interpretations.
  • Experience with problem-solving in mathematical contexts.
NEXT STEPS
  • Study the derivation of the formula for the circumference of a circle, specifically 2πr.
  • Explore geometric representations of spirals and their applications in calculus.
  • Investigate the relationship between coil spacing and cable length on spools.
  • Review similar problems in Stewart's Calculus to reinforce understanding of geometric applications.
USEFUL FOR

Students studying calculus, particularly those focusing on geometric problems, as well as educators looking for examples of real-world applications of calculus concepts.

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Homework Statement


(From Stewart's Calculus Early Transcendentals (4th ed), Ch. 13 Problems Plus #8)

A cable has radius r and length L and is wound around a spool with radius R without overlapping. What is the shortest length along the spool that is covered by the cable?

Homework Equations



See image link below.

The Attempt at a Solution


I've read the solution from the solutions guide and it makes sense, aside from this geometric representation:
(h=vertical distance between coils)

http://img242.imagevenue.com/img.php?loc=loc164&image=th_34593_spool2_122_164lo.jpg"

I'm trying to understand the rationale for this geometric image. Specifically, where is the 2Pi(R+r) coming from? This is clearly the circumference of the spool/cable, but I don't understand its presence in the image. Could someone describe what the image is showing (where the spool is, how it's oriented, what the two parallel line are, etc)? Thanks!

http://img107.imagevenue.com/img.php?image=35205_spool_122_371lo.jpg"
 
Last edited by a moderator:
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I thought I would repost the image, as the link above has too many ads.
Any ideas?

20tqngh.jpg
 

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