a) It means that the relative change of length of the copper, [tex]\Delta L/L[/tex], increases by 0.000017 for each increase in temperature of 1 degrees
b) The safety gap must be 0.33 mm
Explanation:
a)
The linear expansivity of a material is a measure of how much the length of a sample of that material expands per unit increase of temperature.
Mathematically, the linear expansivity can be written as
[tex]\alpha = \frac{\Delta L/L}{\Delta T}[/tex]
where
[tex]\Delta L[/tex] is the change in length of the sample
L is the original length of the sample
[tex]\Delta T[/tex] is the change in temperature of the sample
For instance, in this problem, we are told that the linear expansivity of copper is
[tex]\alpha = 0.000017/K[/tex]
Which means that the relative change of length of the copper, [tex]\Delta L/L[/tex], increases by 0.000017 for each increase in temperature of 1 degrees.
b)
We can re-write the formula written in part a as
[tex]\Delta L = \alpha L \Delta T[/tex]
where, applied to this problem, each variable means:
[tex]L[/tex] is the length of the steel bar
[tex]\alpha[/tex] is the linear expansivity of steel
[tex]\Delta L[/tex] is the change in length of the steel bar when the temperature increases by [tex]\Delta T[/tex]
In this problem, we have:
L = 3 m
[tex]\alpha = 1.0\cdot 10^{-5}/K[/tex]
[tex]\Delta T = 40^{\circ}C-29^{\circ}C=11^{\circ}C[/tex]
Therefore, the change in length of a steel bar is
[tex]\Delta L = (1.0\cdot 10^{-5})(3)(11)=0.00033 m = 0.33 mm[/tex]
Which means that the safety gap between two successive bars must be 0.33 mm.
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