Thermal expansion with set of 3 wires

[awerter]

Homework Statement

Three identical wires L_o, 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.)

Homework Equations

ΔL = L_o\alphaΔT
ΔL/L_o = -F/AY = \alphaΔT

The Attempt at a Solution

Tensions in wires: F_wire1 = 2 * F_wire2 * cos(θ/2)
Here is where I'm stuck. I think that the total expansion is zero, so the equation is something like this

ΔL_total = ΔL_wire1 + ΔL_wire2 cos(θ/2)
= (L_o\alphaΔT - L_o F_wire1/AY) + ( L_o\alphaΔT - L_o F_wire2/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.

 

[Sunil Simha]

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.

 

[CWatters]

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.

 

[awerter]

Thank you for all your help! I can do it now.