How to find length of thread given the characteristic of a spool?

Question

I would like to know the equation that perfectly model the correlation between the length of a thread given the inner and outer diameter of a spool and its height, and the diameter of the thread. This model is largely used in ou everyday life, but wikipedia and google are giving me the electric equation, while I want the physical model of a spool and the thread...

I would like to know the equation that perfectly PERFECTLY!!! model the correlation between the length of a thread given the inner and outer diameter of a spool and its height, and the diameter of the thread. This model is largely used in ou everyday life, but wikipedia and google are giving me the electric equation, while I want the physical model of a spool and the thread...
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Answer 1

This problem relies on three assumed constant conditions. First, that the thread is uniformly wound around the spool. Second, that the thread in all layers is uniform in width (perpendicular to the spool surface) under the pressure of being wound. Finally, it assumes normal thread, being cylindrical in nature, and not irregular, like a ribbon.

First off, you need to figure the length of thread on the outer surface of the spool, assuming that the thread covers the entire surface, from top to bottom. Area of cylinder divided by width of thread.

Second, do the same for the inner surface.

These provide the values you will need for simple sigma notation. The last step is to figure out how many layers are in the spool, which is the difference between the outer diameter and the inner diameter divided by the width of the thread. Without getting into the complex math of Sigma notation, you add the lengths of the outer surface to the length of the inner surface, and multiply that total by half the number of layers of thread on the spool.

[L(outer) + Length(inner)] * (# of layers/2)

Answer 2

The volume of the space in which the thread is wound is given by the formula,

V = Pi (R^2- r^2) h. R- outer radius and r inner radius and H is he height of the spool.


Let L be the total length of the thread. The total volume of thread is given by L x Pi a^2 where ‘a’ is radius of the thread.

Therefore, L x Pi a^2 = Pi (R^2- r^2) h

Or L = h (R^2- r^2) / a^2.

This gives the maximum length possible. In practice the length will be smaller than this value.