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The spiral obtained when tape wound on a spool.

  1. Apr 18, 2012 #1
    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. jcsd
  3. Apr 18, 2012 #2

    haruspex

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    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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