Finding the Length of a Coil

Date: 07/07/99 at 14:19:09
From: Robert Johnson
Subject: Finding the length of a coil

I am looking for a formula to figure the length of a coil or spring 
when given the diameter and pitch and number of turns.

Date: 07/07/99 at 17:32:09
From: Doctor Rick
Subject: Re: Finding the length of a coil

Hi, Robert.

Imagine that you take a paper triangle like this:

                                                              /|
                                                       /       |
                                                /              |
                                L        /                     |
                                 /                             | P*N
                          /                                    |
                   /|                                          |
           /       P|                                          |
  /_________pi*D____|__________________________________________|
                              pi*D*N

     L = length of coil
     D = diameter of coil
     N = number of turns
     P = pitch

Wrap it around a cylinder the diameter of the coil. It will wrap 
around exactly N times, since Pi*D is the circumference of the 
cylinder. At each turn, the diagonal edge (hypotenuse) of the triangle 
is a distance P farther along the cylinder than on the previous turn; 
this is the pitch. The hypotenuse follows the path of the coil, so its 
length L is the length of the coil. You can find it from the 
Pythagorean theorem:

     L^2 = (pi*D*N)^2 + (P*N)^2
         = N^2 * (pi*D)^2 + P^2)

       L = N*sqrt((pi*D)^2 + P^2)

where sqrt() means the square root of the expression in parentheses, 
and P^2 means P squared.

If you have any questions about this, feel free to ask.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/