# Linear expansion metal rod problem

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
Homework Helper
Gold Member
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.

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

dx
Homework Helper
Gold Member
Yes, ∆L = 5.80 x 10-4 m, the amount by which the length of C changes.

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
Homework Helper
Gold Member
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.

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
Homework Helper
Gold Member
Ya that looks correct. I got β = 0.766.