Linear expansion metal rod problem

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Homework Help Overview

The problem involves three metal rods with different coefficients of linear expansion, focusing on how their lengths change with temperature. The original poster presents a scenario where two rods expand differently under the same temperature change, and a third rod is composed of segments of the first two. The goal is to determine the lengths of each segment in the third rod based on the given expansions.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss deriving the coefficients of linear expansion from the first two rods and setting up an equation for the third rod's length change. There are questions about the necessity of using the given expansion value for the third rod and how to express the fractions of the rod lengths in terms of a variable.

Discussion Status

The discussion includes attempts to clarify the setup of the problem and the relationships between the variables. Some participants have provided guidance on how to express the lengths of the materials in the third rod and how to approach solving for the variable representing the fraction of the first material. There is a lack of explicit consensus on the correctness of the calculated values, but participants are engaging with the problem constructively.

Contextual Notes

Participants are working within the constraints of the problem as presented, including the specific expansions for each rod and the requirement to find the lengths of the segments in the third rod. There is an acknowledgment of confusion regarding the interpretation of the variable used in the equations.

jsalapide
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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..
 
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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?
 
Yes, ∆L = 5.80 x 10-4 m, the amount by which the length of C changes.
 
dx said:
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?
 
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?
 
Ya that looks correct. I got β = 0.766.
 

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