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Math Help - Calculating Total Length of Wraps Around A Cylinder

  1. #1
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    Calculating Total Length of Wraps Around A Cylinder

    So in this problem I am supposed to calculate the total length of a fabric wrapped around a cylinder.

    I am given the following:

    Thickness of the fabric: t
    Cylinder diameter: c
    Total diameter of all of the fabric and the cylinder together: d

    I originally solved this problem by summing the circumference of each layer of fabric, using an incremental increase in diameter with the value of the thickness. However, I am looking for a way to solve this without the use of a sum.

    I'm kind of lost as to where to start this though. I am assuming that the rate of change of thickness is a continuous function (as opposed to a step at the beginning of each layer). Any help in solving this problem would be greatly appreciated.
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  2. #2
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    The length depends on how many times it's wrapped around the cylinder.

    The arc length of a circular helix is given by:

    \frac{dx}{dt}=-a\cdot{sin(t)}, \;\  \frac{dy}{dt}=a\cdot{cos(t)}, \;\ \frac{dz}{dt}=c

    L=\int_{0}^{t_{0}}\sqrt{a^{2}sin^{2}(t)+a^{2}cos^{  2}(t)+c^{2}}dt=\int_{0}^{t_{0}}\sqrt{a^{2}+c^{2}}d  t=t_{0}\sqrt{a^{2}+c^{2}}

    Example:

    Let's say we have fabric of 1/2 inch dia. wrapped around a cylinder with dia. 12 inches. What length of tubing will make one complete turn in 20 inches measured along the axis of the cylinder.


    The helix is x=a\cdot{cos(t)}, \;\ y=a\cdot{sin(t)}, \;\ z=ct

    a=6.25

    c=\frac{10}{\pi}

    The radius of the helix is the distance from the axis of the cylinder to the center of the fabric, and the helix makes one turn in 20 inches., where t=2{\pi}.

    The length of the fabric in one turn is then 2{\pi}\sqrt{6.25^{2}+(\frac{10}{\pi})^{2}}\approx{  44.07} inches.


    Does that example help?.
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  3. #3
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    Here is a non-Calculus solution to the problem assuming I understand it correctly.

    The picture on the left if a Cylinder with a wrapped red string. Unwrape the cyclinder and you end up with a rectangle, whose width is 2\pi r and the height is d\cdot n where d is the distance between the strings and n is the number of sections. You can use the Pythagoren theorem to get, the length of each read string is \sqrt{4\pi^2 r^2 + d^2} and in total we have n\sqrt{4\pi^2 r^2 + d^2}.
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  4. #4
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    Thank you for your responses. However, I think I probably wasn't totally clear in my original description of the problem.

    The fabric is wrapped like toilet paper for example, where each layer is on top of the previous layer. And the 'thickness' is the gauge of the fabric, .01 inches for example. So if you take a cylinder with a 1" diameter and have 500 wraps, the final diameter will be 11". And I am trying to calculate the length of these 500 wraps.

    Is that more clear now?
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  5. #5
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    OK, I think I was making this more difficult than it really was. I think I've got it now.

    I put the equation in polar form, r=t/(2π)*θ + c

    And from here I can just use the arc length formula and solve.
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