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[itex]\alpha[/itex]ΔT
ΔL/L_o = -F/AY = [itex]\alpha[/itex]ΔT

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[itex]\alpha[/itex]ΔT - L_o F_wire1/AY) + ( L_o[itex]\alpha[/itex]ΔT - 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.

 

[Nickie]

I would approach this problem by first considering the properties of thermal expansion and how it may affect the wires. Thermal expansion is the tendency for a substance to expand or contract in response to changes in temperature. In this case, the wires are being cooled, which will cause them to contract. The amount of expansion or contraction is determined by the coefficient of thermal expansion (α), the change in temperature (ΔT), and the original length of the wire (Lo).

Using the equation ΔL = LoαΔT, we can calculate the change in length of each wire due to the change in temperature. However, since the wires are arranged in a Y shape, the knot at the center will not move directly to the right or left, but rather at an angle determined by the geometry of the system. To determine the distance the knot moves to the right, we can use trigonometry to calculate the horizontal component of the expansion of each wire.

To find the final tension in each wire, we can use the equation ΔL/Lo = -F/AY, where F is the tension in the wire, A is the cross-sectional area of the wire, and Y is the Young's modulus of the wire. Since the initial tension is approximately zero, we can assume that the final tension will be equal to the force needed to keep the knot in place, which can be calculated using the horizontal component of the expansion and the cosine of the angle θ/2.

It is important to note that the assumption that θ does not change when the knot moves may not be entirely accurate. As the wires expand or contract, the angle θ may also change slightly, affecting the final tension in each wire. This can be taken into consideration by including a correction factor in the calculation.

Overall, the solution to this problem involves a combination of equations and geometric considerations. It is important to carefully consider the properties of thermal expansion and how they may affect the system in order to arrive at an accurate solution.