There was an error in my note yesterday -- I left out the length of
paper in the final equation!  For "4 pi t", read "4 pi L t". I've been
asked to clarify it anyway, so here goes...

All you need to know to calculate acceleration in a roll is the length
and thickness of the paper that has been rolled onto the take-up spool,
and the original spool circumference. All are easy to measure.

It's hard to represent maths well in plain text, so I'll try harder
today!  Assuming that the wind motor can maintain a fixed rotational
speed of the take-up spool, the proportional paper speed acceleration as
the spool is thickened by paper build up is:

               New circumference
Acceleration = ----------------------
               Original Circumference

If the equation linking the circumferences (see yesterday) is:

(new circumference)squared = (orig. circumference)squared + 4 pi L t

Then, spelling it all out:

                             (     4 pi Length * thickness )
Acceleration  =  square root ( 1 + ----------------------- )
                             (     Original circumference  )

Original circumference, paper thickness and pi are all constants.
The acceleration is proportional to the square root of the length
of paper played (so adding 20' of paper does NOT give twice the
acceleration of 10').  Increasing the original circumference makes
the proportional acceleration drop, as does decreasing the thickness
of the paper -- both of which are intuitively correct.

Obviously you could redo the maths in turns of (say) time, or turns of
the spool (which are effectively the same thing), but this equation
shows acceleration per linear measure of roll, which is what you want
to know when marking out a roll on a sheet of paper or in the computer.

I hope this is clear now!

Julian Dyer