A wire is made by attaching two segments together, end to end. One segment is made of aluminum and the other is steel. The effective coefficient of linear expansion of the two-segment wire is 19.0 x 10-6 (C°)-1. What fraction of the length is aluminum?
To find the fraction of the length that is aluminum in a two-segment wire, we need to know the coefficients of linear expansion for both aluminum and steel, as well as the relative lengths of the two segments.
Let's assume that the length of the aluminum segment is L1 and the length of the steel segment is L2. The total length of the wire is then L = L1 + L2.
The change in length of the aluminum segment can be calculated using the formula ΔL1 = α1 * L1 * ΔT, where α1 is the coefficient of linear expansion for aluminum and ΔT is the change in temperature.
Similarly, the change in length of the steel segment can be calculated using the formula ΔL2 = α2 * L2 * ΔT, where α2 is the coefficient of linear expansion for steel.
Since the two segments are attached end to end, the total change in length of the wire should be equal to the sum of the individual changes: ΔL = ΔL1 + ΔL2.
Given that the effective coefficient of linear expansion for the two-segment wire is 19.0 x 10-6 (C°)-1, we can set up the following equation:
ΔL = ΔL1 + ΔL2
19.0 x 10-6 (C°)-1 * L * ΔT = α1 * L1 * ΔT + α2 * L2 * ΔT
Next, we need to find the fraction of the length that is aluminum, which can be expressed as the ratio L1 / L. To simplify the equation, we can replace L2 with L - L1:
19.0 x 10-6 (C°)-1 * L * ΔT = α1 * L1 * ΔT + α2 * (L - L1) * ΔT
Simplifying further, we can divide both sides by ΔT and cancel out the ΔT term:
19.0 x 10-6 (C°)-1 * L = α1 * L1 + α2 * (L - L1)
Expanding the equation:
19.0 x 10-6 (C°)-1 * L = α1 * L1 + α2 * L - α2 * L1
Now we can isolate L1 by moving the α1 * L1 term to the left side and the α2 * L term to the right side:
19.0 x 10-6 (C°)-1 * L - α1 * L1 = α2 * L - α2 * L1
Factoring out L1 from the left side and L from the right side:
(19.0 x 10-6 (C°)-1 - α2) * L1 = (α2 - 19.0 x 10-6 (C°)-1) * L
Finally, we can solve for the fraction of the length that is aluminum (L1 / L):
L1 / L = (α2 - 19.0 x 10-6 (C°)-1) / (19.0 x 10-6 (C°)-1 - α2)
Plugging in the respective coefficients of linear expansion for aluminum and steel will give you the answer to the fraction of the length that is aluminum in the two-segment wire. Note that the temperature change (ΔT) is not provided in the question, so you will need to know the change in temperature to calculate the fraction accurately.