Linear Expansion - Finding temperature

[JoeyBob]

Homework Statement:

See attached

Relevant Equations:

change length= a*original length * change in temp

So if I am understanding the question correctly, I need to find the change in temperature that causes one rights outer radii to be the same as another rings inner radii.

Now what I tried is two equations

change length_1= a_1*original length_inner * change in temp

change length_2= a_2*original length_outer * change in temp

But the problem is that there is one too many unknowns here - i don't know the temp, change in L_1 or change in L-2. Either there's another equation I am missing or I am using the wrong approach.

 

[JoeyBob]

Okay I think I found the equation I was missing.

L_1+change in L_2 = L_2 + change in L_2

The answer I got was only a bit off from the right answer, probably from rounding or something.

 

[haruspex]

Okay I think I found the equation I was missing.

L_1+change in L_2 = L_2 + change in L_2

The answer I got was only a bit off from the right answer, probably from rounding or something.

That equation makes no sense since you have "change in L_2" on both sides. What did you mean, in terms of the given variables r_i etc? Use α₁, α₂ for the coefficients.
Please post your answer and working so that we can check whether it is just rounding error.

 

[Chestermiller]

That equation makes no sense since you have "change in L_2" on both sides. What did you mean, in terms of the given variables r_i etc? Use α₁, α₂ for the coefficients.
Please post your answer and working so that we can check whether it is just rounding error.

It looks like a typo.

 

[JoeyBob]

That equation makes no sense since you have "change in L_2" on both sides. What did you mean, in terms of the given variables r_i etc? Use α₁, α₂ for the coefficients.
Please post your answer and working so that we can check whether it is just rounding error.

Its a typo. change in L_1.

The 3 equations are correct?

 

[haruspex]

The 3 equations are correct?

Impossible to say without definitions of the variables.
This is certainly correct:
find the change in temperature that causes one ring's outer radius to be the same as the other ring's inner radius.
But you have not posted an equation that expresses that in terms of r₁, r₂, α₁, α₂ and Δθ.