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Thermal expansion with set of 3 wires

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data
    Three identical wires Lo, diameter d, are arranged like a Y letter (please see attachment)

    Each end of the wires is secured to a wall. Initial tension is approximately zero. If the wires are cooled ΔT, find the distance the knot moves to the right and the final tension in each wire. (assume θ does not change when the knot moves.)


    2. Relevant equations
    ΔL = Lo[itex]\alpha[/itex]ΔT
    ΔL/Lo = -F/AY = [itex]\alpha[/itex]ΔT

    3. The attempt at a solution
    Tensions in wires: Fwire1 = 2 * Fwire2 * cos(θ/2)
    Here is where I'm stuck. I think that the total expansion is zero, so the equation is something like this

    ΔLtotal = ΔLwire1 + ΔLwire2 cos(θ/2)
    = (Lo[itex]\alpha[/itex]ΔT - Lo Fwire1/AY) + ( Lo[itex]\alpha[/itex]ΔT - Lo Fwire2/AY) cos(θ/2) = 0

    But I got the wrong answers. It is hard for me to visualize how the system changes with the assumption that θ is still the same. It doesn't make sense. Please help me.
    Thank you very much.
     

    Attached Files:

  2. jcsd
  3. Mar 31, 2013 #2
    This question can be solved by assuming a Young's modulus (Y) for the material. It helps to draw the diagram of individual wires before cooling, after cooling (assuming absence of other wires) and the real scenario after cooling. I have attached these in this reply. So just check it out and see whether it works. Here x is the required extension.
     

    Attached Files:

    Last edited: Mar 31, 2013
  4. Mar 31, 2013 #3

    CWatters

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    θ would change but the change could be small so they are telling you to assume it's constant.

    I reckon for some angles Δx could be -ve, 0 or +ve.
     
  5. Mar 31, 2013 #4
    Thank you for all your help! I can do it now.
     
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