Arrangements for 10-tune nickelodeon rolls must deal with the
constantly increasing paper speed, as the paper builds up on
the constant-speed take-up spool.  This paper presents the
mathematics involved.

Units:

  distance: inches
  time: seconds

Parameters:

  p  = paper thickness (inches)
  r0 = initial radius
  pi = 3.14
 2pi = 6.28

  w  = revolutions per second
  r  = radius (inches)
  t  = time (seconds)
  v  = paper velocity (inches per second)
 v0  = initial paper velocity
  a  = acceleration (inches/sec/sec)

Each revolution increases the radius by one paper thickness:

   r = r0 + w*t*p             [1]
 w*t = accumulated revolutions

Paper velocity:

   v = 2pi*w*r
     = 2pi*w*(r0 + w*t*p)
     = 2pi*w*r0 + 2pi*w^2*p*t   [2]

This is form v = v0 + at, where

  v0 = 2pi*w*r0
   a = 2pi*w^2*p

A practical example.   Assume

  ten tunes, each tune duration = 180 seconds (3 minutes)
  take-up spool initial radius r0 = 1.813 inch (A-roll core)
  paper thickness p = 0.0028 inch (Typical of modern paper)
  initial velocity v0 = 1.2 inch/sec (Tempo 60, 6.0 feet/minute)

  e.t.  = elapsed time (seconds) at beginning of tune
  speed = paper speed (ips, inches per second) at beginning of tune
  Tempo = paper speed, 10*feet/minute, = 50 * speed
  radius = radius of paper on take-up spool (Inches)

tune   e.t. speed  Tempo  radius
       sec   ips          inches

  1      0  1.200  60.000  1.813   (A-roll core, 3.625 inch diameter)
  2    180  1.235  61.758  1.866
  3    360  1.270  63.516  1.919
  4    540  1.305  65.274  1.972
  5    720  1.341  67.032  2.025
  6    900  1.376  68.790  2.078
  7   1080  1.411  70.548  2.132
  8   1260  1.446  72.306  2.185
  9   1440  1.481  74.064  2.238
 10   1620  1.516  75.822  2.291

 11   1800  1.552  77.580  2.344  (4.69 inch diameter)

A spread-sheet could perform the equations for songs of different
durations.

Robbie Rhodes