Hello,
I have a strip of paper 20 feet long and 1/16 inches wide.
When I tape the ends together, I get a ring of paper, 20 feet in circumference(C).
Diameter(D) is 6.36 ft. Radius(R) is 3.18 ft.

Untape stripe.

Now I put one full twist or one full rotation of the strip and tape the ends together.
The diameter(d) of the twist is 1/16 ". The radius(r) is 1/32" The circumference(c) of the spiral is .19 inches. I believe that's right, check me.

This twist never changes the length of the stripe, but it made the circumference(C) to decrease.
What is the new C?

How does one express that decrease, as controlled by small r-the radius of the spiral?

This system is a rotating spring with one turn........ground state.

You can only add full twists to the spiral, 1 turn is the lowest.

As one adds turns to the spiral, C should decrease in steps. This size of steps should be controlled by the length of r.

Any thoughts?

 

Hi,

Yes, dont make a ring. Just make the spiral (helix), measure/calculate the length. More advanced, find the series that describes the decrease in length as turns are added.
To simplify, might use just two 'wires' spaced apart by a constant distance instead of a 'ribbon' of paper.

 

Of course, thank you kindly.

 

Hi again,

It looks like it should be even simpler, looking at the equations for a helix on Wikipedia.
Check that out, see what you think.

LATER:
Ok yes it looks pretty simple. I did this kinda fast but it looks right...
Parameterizing the helix as x=cos(w*t),y=sin(w*t), z=t
with w=2*pi*f and using the 3d arc length:
integral(sqrt((dx/dt)^2+(dy/dt)^2+(dz/dt)^2))dt
over z we get a very simple result:
L=sqrt(w^2+1)*z
where L is the length of the original piece of material.
So now solving for z is easy:
z=L/sqrt(w^2+1)

We got this simple result because of the choice of parameterization, which reduces the length along the edge to a simple algebraic form. For very long pieces we might even want to approximate with z=L/w. It would be easy to adapt this formula to a variable radius too.