What is the Relationship Between Radius and Length of Coiled Tape?

[r4nd0m]

Imagine you have a tape wrapped around a coil (e.g. an audio magnetic tape or adhesive tape). The thickness of the tape is T and the radius of the coil is R.
The task is to determine the dependence of the length of the coiled tape and the radius of the whole (coil + tape).
(e.g. - if I know that I wrapped 50 meters of tape around the coil - then what is the radius of the whole? I hope you know what I mean)

 

[Hammie]

Hint: think about the surface area.

 

[r4nd0m]

so what is the surface area? It is certainly not 2*pi*R, because it's a spiral.

 

[Hammie]

No, it is not. but by using pi*r^2 and subtracting the inner core, you have the surface area of the tape. Also consider, length times thickness is also the same surface area.

A good strategy to this problem may be to find what is in common between the two different ways to find it.. and that is surface area.

 

[r4nd0m]

pi*r^2 is the area of a disk. But this is not a disk - only if the tape was very thin, we could regard it as a disk.

 

[Hammie]

I would say that the nature of tape itself would suggest that it is quite thin.

If you wrap the hub with precisely one layer of tape, the shape is not fundamentally changed. It is still circular, and the only thing changed is its radius, and slightly, at that.

Add a "whole bunch" of precise layers, and it is still circular.

 

[Hammie]

so what is the surface area? It is certainly not 2*pi*R, because it's a spiral.

I would suggest that it is not. The idea here is that we are wrapping single layers on a surface that is cylindrical to begin with.

If you started at the center with tape only, (no hub) I would agree that it would be a spiral of sorts, at least up to a point in the wrapping process.

 

[berkeman]

Next hint -- use the volume of the tape instead of the surface area...