Linear expansion metal rod problem

[jsalapide]

Metal rod A is 0.300 m long expands by 6.50x10^-4 m when its temperature is increased by 100 degrees Celsius. Another rod made of different metal B and of the same length expands by 3.50x10^-4 m for the same increase in temperature. A third rod C, also 0.300 m long, is made up of pieces of the first two metals placed end-to-end and expands 5.80x10^4 m for the same increase in temperature. Find the length of each portion of the third bar.

i have no idea to solve this.. help..

 

[dx]

From the first part of the question, you can get the coefficients of linear expansion of both materials α₁ and α₂. Now for the third rod C, let β be the fraction of 0.3 m which is made of material one. Then the fraction of 0.3 m made of material two will be (1 - β). The change in length of C is ∆L = ∆L₁ + ∆L₂, where ∆L₁ is the change in the part made of material one and ∆L₂ is the change in length of the part made of material two. Write out this equation and solve for β.

 

[jsalapide]

do i have to use the 5.80x10^-4 m given?

 

[dx]

Yes, ∆L = 5.80 x 10^-4 m, the amount by which the length of C changes.

 

[jsalapide]

Now for the third rod C, let β be the fraction of 0.3 m which is made of material one. Then the fraction of 0.3 m made of material two will be (1 - β).

how do i solve this part?

 

[dx]

There's nothing to solve in that part. You're just saying that the fraction of 0.3 m that is made of material one is β. For example, if β = ½, then the length of material one in C will be ½ 0.3 = 0.15 m.

β is a variable and you don't know what it is yet. That's what you have to find.

Do this first: Expand out this equation: ∆L = ∆L₁ + ∆L₂ using the formula for linear expansion.

Hint: ∆L₁ = α₁β(0.3)∆T.

I have to go now so I may not reply for a while.

 

[jsalapide]

ok i was just confused... my β is equal to 0.763

i solve for the length of each material and the first part is 0.229 m while the second part is 0.071 m..

is that correct?

 

[dx]

Ya that looks correct. I got β = 0.766.