# The spiral obtained when tape wound on a spool.

1. Apr 18, 2012

### suhasm

I was just thinking about a problem for fun where
n layers of tape of thickness t are wound on a spool of inner radius r
and one needs to find the the variation of angular speed of spool as a function of time such that tape is obtained at a constant time rate v.

But , my question is , what kind of a curve is the spiral?
To , me , at the first glance , it look like a discrete function.
Then some googling tells me it could be an archimedes spiral.
Few more suggest an involute circle.

Also , can anyone explain to me why its not a discrete function? I have trouble visualizing this.

Some inputs would be appreciated.

2. Apr 18, 2012

### haruspex

Since the tape has constant thickness it's an arithmetic spiral.
If the spool rotates at w = w(t), radius r(t) satisfies dr/dt = k.w(t), linear velocity v(t) = r(t).w(t). (k = tape thickness/2pi)
Setting v(t) = V, constant, we have k.w(t) = d(V/w)/dt = -(V/w^2).dw/dt.
k.dt = -V.dw/w^3
2k.t = V/w^2 - R.R/V, where R is radius at time 0.
w = V/sqrt(R.R + 2.k.V.t)

Looks reasonable.

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