

Spirals... Circles... Radius
Name: Name
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
Question: Is there any mathematical formula for determining the unrolled
length of a spiral? For instance, if I had a roll of ribbon that had a radius
of two inches, and the ribbon was one sixteenth of an inch thick, how long
would the ribbon be? If there is not a formula for it, what would be the best
way to determine this?
Replies:
A very close approximation for this kind of spiral is to pretend
that the roll of ribbon is a series of concentric circular loops of ribbon.
Then, the length of each loop is 2 * pi * r, where r = the loop's radius and
pi = 3.1415926...so the answer for your example would be: 2 * pi * (1/16 +
2/16 + 3/16 + ... + 32/16) = 2 * pi * 528/16. If the ribbon was on a spindle
of radius, say, 1 inch, the sum would be (16/16 + 17/16 + ... + 32/16). An
interesting related problem is find (and prove) a formula for ( 1 + 2 + 3 +
... + N )
John Hawley
Here is another approach. If we assume that the ribbon is tightly
wound up (i.e., no gaps between the layers of ribbon), then the volume
occupied by the 'disk' of woundup ribbon equals the volume of the unwound
strip (a very flat and long 'box'). Let R denote the radius of the 'disk', T
the thickness of the ribbon, L its unwound length (the quantity we want to
compute), and W its width. The woundup volume is pi*R^2*W ("R^2 means "R
squared"), and the unwound volume is L*W*T. Setting these equal and solving
for L, we get L = pi*R^2/T. For the values given (R=2 in., T=1/16 in.), we
get L = 64*pi (about 201) inches. By the way, your idea of using a spiral
would work. One could write an equation (polar coordinates would be the
easiest choice) for a curve that would lie within the ribbonsay, at mid
interior. There is a formula for length of a curve, using integral calculus.
As it happens, the answer you get is the same as that obtained above. If you
would like more details on this calculus approach, please ask (here or via e
mail).
Ron Winther
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Update: June 2012

