Question

5. What is length of a steel bar in Arizona (37oC) if its length in North Dakota is 200 m? (Temperature in North Dakota is -7oC). 5a. What is length of a steel bar in North Dakota ( -7oC) if in Arizona it was 200 m (temperature in AZ is 37oC )?

Answer 1

ANSWERS

5 - Length of the bar in Arizona = 200.1056 m

5.a - Length of the bar in North Dakota = 199.8944 m

Step-by-step explanation

The linear thermal expansion follows the formula:

`\Delta L=\alpha\cdot L_i\cdot\Delta T`

where α is the linear expansion coefficient, Li is the initial length and ΔT is the temperature difference.

In this problem we have that the initial length Li is 200m (in North Dakota), so the initial temperature is -7°C and the final temperature is 37°C. The linear expansion coefficient of steel is 12 x 10⁻⁶ 1/°C:

`\begin{gathered} \Delta L=12*10^(-6)\cdot200\cdot(37-(-7)) \\ \Delta L=12*10^(-6)\cdot200\cdot47 \\ \Delta L=0.1056m \end{gathered}`

This is how much the steel bar expanded. It's length in Arizona is the length the bar had in North Dakota plus the expansion:

`\begin{gathered} L_{\text{Arizona}}=L_{\text{NorthDakota}}+\Delta L \\ L_{\text{Arizona}}=200m+0.1056m \\ L_{\text{Arizona}}=200.1056m \end{gathered}`

Item 5.a is the opposite. Since we know the length of the bar in Arizona, where it's hotter, the bar will contract when taken to North Dakota. Note that the contraction will be the same as the expansion, but negative, because the difference of temperature, as well as the initial length of the bar, is the same:

`\begin{gathered} L_{\text{NorthDakota}}=L_{\text{Arizona}}-\Delta L \\ L_{\text{NorthDakota}}=200m-0.1056m \\ L_{\text{NorthDakota}}=199.8944m \end{gathered}`