Thermodynamics, composite bar linear expansion

[RJWills]

Homework Statement

"A composit bar is made of two metals joined in series, with coefficients of linear thermal expansion 2.2E-5 and 0.9E-5 K^-1 respectively. The bar expands by 0.5% in length with a temperature rise of 500 K. Calculate the ratio of the initial lengths of the two metal sections."

Homework Equations

The main equation I think to be useful is ΔL=α L ΔT

The Attempt at a Solution

I am thinking that I cannot assume that the two bars will expand the same amount, meaning that the sum of ΔL1+ΔL2+L=1.005L...

500(ΔL1α1+ΔL2α2)+L=1.005L
500(ΔL1α1+ΔL2α2)/L=0.005

Whatever way I look at this I just see myself going round in circles :/

 

[TSny]

500(ΔL1α1+ΔL2α2)+L=1.005L

Are you sure you want the Δ's in this equation?

Also, can you express L in terms of the initial lengths of each section?

 

[tiny-tim]

Hi RJWills!

What does the question ask for?

The ratio L1/L2 …

so call that ratio "r", and put it into the equation! ​

 

[RJWills]

Okay so tried it the way you suggested, here's what I did:

500(L1α1+L2α2) +L1+L2 = 1.005(L1+L2)
1.011L1 + 1.0045L2 = 1.005(L1+L2)
Therefore L1/L2 = 1/12 (0.0833...) right?

 

[TSny]

Okay so tried it the way you suggested, here's what I did:

500(L1α1+L2α2) +L1+L2 = 1.005(L1+L2)
1.011L1 + 1.0045L2 = 1.005(L1+L2)
Therefore L1/L2 = 1/12 (0.0833...) right?

Looks good.

 

[RJWills]

Awesome thanks for the help guys :)