Linear expantion and elastic modulus- brass vs. glass

[mmoadi]

Homework Statement

Onto a thick brass rod we attach equally long glass thread. At what temperature change will the glass thread break if the temperature coefficient of linear expansion for brass is
α1= 20 x 10 ^-6 K^-1, and for glass is α2= 7 x 10^-6 K^-1? Young’s (elastic) modulus for glass is E1= 7 x 10^10 N/m², and shear modulus for glass is σ1= 7 x 10^7 N/m².

Homework Equations

Linear thermal expansion: α(L)= ΔL/ L(0)ΔT
Young’s (elastic) modulus: E= FL(0)/ AΔL

The Attempt at a Solution

Thank you mgb_phys for hint- it helped a lot:

Qute: "You don't need a relation between them as such.
You use the modulus for glass to work out at what strain the glass would break.

Then you use the relative expansion of brass and glass to work out at what temperature the glass would have been stretched that amount.

hint. remember the glass is also expanding as the brass does" (mgb_phys)

So I calculated strain:
Strain: ε= ΔL/ L
Stress: σ= F/A

E= FL(0)/ AΔL → E= stress/strain= σ/ ε → ε= σ/ E
ε = 7 x 10^7 N/m²/ 7 x 10^10 N/m²
ε= 0.001

But now, I don't know how to combine brass and glass to find the final answer.
I tried this, but I doubt is correct:

α(L-glass)= [ΔL/ L(0)]/ ΔT= ε/ ΔT → ΔT= ε/ α(L-glass)
ΔT= 0.001/ 7 x 10^-6 K^-1
ΔT=0.14 x 10^3 K= 140 K

Is this correct (big doubt?!, too low temperature)?

Any hints?
Thank you for helping!

 

[anathema]

Your approach is on the right track, but there are a few things to consider:

1. The strain you calculated is for the shear modulus, but you need to use the Young's modulus for this problem. So you should use E=FL(0)/AΔL instead.

2. The strain you calculated is for the glass thread, but you also need to consider the expansion of the brass rod. So your equation should include both materials, like this: ΔT= ε/(α1L-brass + α2L-glass)

3. Make sure you convert the units properly. The strain should be in meters per meter, so you may need to convert the dimensions to meters before calculating the strain.

4. Your final answer is in Kelvin, which is the correct unit for temperature change. But it might be helpful to also convert this to Celsius, which is a more intuitive unit for temperature change.

With these adjustments, you should be able to calculate the correct temperature change at which the glass thread will break. Good luck!

 

[kerimek]

I would suggest looking at the problem from a different angle. Instead of trying to combine the properties of brass and glass, it may be more useful to consider the properties of glass alone.

Using the given values, we can calculate the maximum strain that the glass thread can withstand before breaking:

σ = Eε → ε = σ/E = (7 x 10^7 N/m²)/(7 x 10^10 N/m²) = 0.001

This means that the glass thread can withstand a maximum strain of 0.001 before breaking. Now, we can use the coefficient of linear expansion for glass to determine the temperature change that would cause this strain:

α(L-glass) = ΔL/ L(0)ΔT → ΔT = ΔL/(L(0)α(L-glass)) = (0.001)/(7 x 10^-6 K^-1) = 143 K

Therefore, the glass thread will break when the temperature changes by 143 K. Since the coefficient of linear expansion for brass is larger than that for glass, the brass rod will expand more than the glass thread and cause it to break.