1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: The spiral obtained when tape wound on a spool.
  1. Adhesive tape (Replies: 1)

  2. Rolling Spool (Replies: 3)

  3. Ticker tape (Replies: 5)

Loading...