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Homework Statement
"A round plug made of brass has a diameter of 87.53 mm at 20 C. The plug is to be fitted to a steel ring of inside diameter 87.43 mm. To what common temperature must they be brought in order to fit?"
The linear coefficient of expansion for brass:
[tex]\alpha_{b} = 18.7 \cdot 10^{-6} {\frac{\texttt{m}}{\texttt{m} \cdot \texttt{K}}[/tex]
and steel:
[tex]\alpha_{s} = 13 \cdot 10^{-6} {\frac{\texttt{m}}{\texttt{m} \cdot \texttt{K}}[/tex]
Unknowns: Delta T, L sub F (final L) for both brass and steel.
Homework Equations
As we are seeking a diameter, I assumed that the linear expansion equation would be sufficient:
[tex]\Delta L = \alpha \cdot \Delta T \cdot L_{0}[/tex]
The Attempt at a Solution
I interpret this as a system of linear equations. Delta T is what is sought; L final is also unknown. So we have two equations with two unknowns.
Taking the original linear expansion equation for brass:
[tex]L_{b} - L_{f} = \alpha_{b} \cdot \Delta T \cdot L_{b}[/tex]
I transform it algebraically to make L final the subject:
[tex]L_{f} = L_{b} - \alpha_{b} \cdot \Delta T \cdot L_{b}[/tex]
I do the same for the steel ring and make the two equations equal each other, then I solve for delta T:
[tex]\Delta T = \frac{L_{s}-L_{b}}{\alpha_{s} \cdot L_{s} - \alpha_{b} \cdot L_{b}}[/tex]
I substitute in my values, converting everything to base units (metres from millimetres), then cancelling units, and my result is
[tex]\Delta T = 199.91 \texttt{K} [/tex]
Now, here is the rub - the answer provided in the assignment is -123 C. If I take 20 C and subtract 199, I get -179 C. These answers vary widely, but I cannot see what I have done wrong here. The coefficients were not the ones provided with the assignment, but came out of a table http://www.engineeringtoolbox.com/linear-expansion-coefficients-d_95.html" , which the instructor has used as a source in the past.
What have I done wrong here? If the work looks good, how likely is that this discrepancy would be caused by the linear expansion coefficients used?
PS: The LaTeX renderer in the post previewer is broken, so forgive me if the first example of the linear expansion equation is actually the linear expansion coefficient for steel. That is not what is in my source code.
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