Spirals

[THEROB]

This is a strange question – that I can’t figure out. I have a measuring tape – one of those that has a spring to wind it in automatically. When the tape is inside it is in a spiral shape. To make it easier we can say that the tape has a thickness of 1 unit. It also spirals from the center.

How would I find the Radius and Angle of a particular measurement? Also given the Radius and Angle – how would I find the measurement?

Thanks

Rob

[Rassis]

Consider a coil formed of (n) spirals each having a thickness of (n), resulting in (L) units long. The coil is winded in such a way that its inner radius is (Ri) and the outer radius is (Re).

Assuming that the material is not elastic, the number of spirals (n) is related to the other measures by n = (Re – Ri)/e.

And the length (L), for any given angle (a) in radians, is calculated by the series:

L = a.[Ri + (Ri + e) + (Ri + 2e) + (Ri + 3e) +...+ (Ri + n.e)]
L = a.[(n + 1).Ri + n.e]
L = a.{[(Re – Ri)/e + 1].Ri + [(Re – Ri)/e].e}
L = a.{[(Re – Ri)/e + 1].Ri + Re – Ri}

If a = 0, you get L = 0;
If a = 2(Pi) and e ~= 0, then Ri = Re = R and you get L = 2(Pi).R

Now a numerical example. Let:

Ri = 10 mm
Re = 40 mm
e = 1 mm
a = 2(Pi)

And you get:

L = 2(Pi)*{[(40 – 10)/1 + 1]*10 + 40 – 10} = 2,136 mm or 2.136 meters

Rui

[THEROB]

Thanks - thats great.

Rob