Re: What is the length of cloth rolled on a spool?

The formula is approximately given by

        L  = pi [ (d2 - d1)/t ] [ (d1 + d2)/2 ]    + pi (d2 - d1) / 2

Where d2 is the outer diameter, d1 is the inner diameter, and t is the
thickness.

The derivation is as follows:

The first      layer has circumference pi (d1 + t)
      second layer has circumference pi (d1 + 2t)
      third     layer has circumference pi (d1 + 3 t)

      last      layer has circumference pi (d1 + nt)  where n is the number
of layers.

      Since we don't want to count the layers, we estimate n by setting
 d1 + nt = d2.

                         nt = d2 - d1

                         n = (d2 - d1)/t

So the total length is   pi [  (d1+t) + (d1+2t) + (d1+3t) + ... + (d1 + nt) 
]

Since there are n layers, the d1's add up to n d1.
Since the average of 1,2,3,....n   is (n+1)/2

the t + 2t + 3t + ....  nt   add up to t n (n+1)/2

So the total length is pi ( n d1 + t n (n+1)/2)  = pi n (d1 + t (n+1)/2 )


But since n = (d2 - d1)/t

   total length L = pi [ (d2 - d1)/t ]  [  d1 + t (    [d2 - d1]/t  +1 )/2
                      = pi [ (d2 - d1)/t ] [ d1 + (    d2 - d1) + t)/2

                       = pi [ (d2 - d1)/ t  ] [  d1 + (d2 - d1)/2  + t/2 ]

                       = pi [ (d2 - d1)/t ] (d2 + d1 + t)/2

                               The above form is sufficiently easy for
calculation.

                        = pi [ (d2 - d1) /t ]  [ (d2 + d1)/2    + t/2 ]

                        = pi ( d2 - d1) (d2 + d1) /( 2t)  +  pi (d2 - d1) t 
/
(2t)

                        = pi (d2 ^2 - d1^2) /(2t)  + pi (d2-d1)/2

                        = pi[ (d2 - d1)/t ] [(d2+d1)/2 ] + pi (d2-d1)/2

This final form is for conceptual understanding.

              The length is pi times the number of layers ( = [ d2 - d1]/t 
)
times the average diameter ( =  [d2 + d1]/2 )  plus a correction term  to
compensate for the fact that each layer has non-zero thickness.

Kermit
kermit@polaris.net

ADMIN NOTE:
A simpler way to approach this question might be to assume that the
volume of the cloth does not change when it is rolled onto the spool.  In
this case, as long as the width of the cloth isn't changed, the surface
area of the edge of the cloth (L x t) must be constant whether the cloth
is rolled or straight.  this means that the length of the cloth is equal
to the surface area of the cloth on the spool divided by the thickness,
or:

L = pi[(d2/2)^2 - (d1/2)^2)]/t

If you don't like to square things, this can be rewritten using "the
difference of two squares" as:

L = pi[(d2/2 + d1/2) x (d2/2 - d1/2)]/t

Or:

L = pi(d2 + d1)(d2 - d1)/4t