Question

Even though steel has a relatively low linear expansion coefficient \(\left(\alpha_{\text {steel }}=13 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right),\) the expansion of steel railroad tracks can potentially create significant problems on very hot summer days. To accommodate for the thermal expansion, a gap is left between consecutive sections of the track. If each section is \(25.0 \mathrm{~m}\) long at \(20.0{ }^{\circ} \mathrm{C}\) and the gap between sections is \(10.0 \mathrm{~mm}\) wide, what is the highest temperature the tracks can take before the expansion creates compressive forces between sections?

Short answer

Answer: The highest temperature the tracks can take is approximately 50.77°C.

Step-by-step solution

Identifying the values given in the exercise

We are given: - Linear expansion coefficient of steel: \(\alpha_{\text {steel }}=13 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\) - Length of each track section, \(L_0 = 25.0 \mathrm{~m}\) - Gap width between sections, \(\Delta L = 10.0 \mathrm{~mm}\) - Initial temperature, \(T_0 = 20.0 { }^{\circ} \mathrm{C}\) Our task is to find the highest temperature, \(T_{max}\), before compressive forces between sections occur.

Understanding the linear expansion formula and rearrange to find highest temperature

The formula for linear expansion is: $$\Delta L = \alpha L_0 \Delta T$$ Where \(\Delta L\) is the change in length, and \(\Delta T\) is the change in temperature. We need to rearrange this formula to find the highest temperature (\(T_{max}\)). First, let's find \(\Delta T\) in terms of given values: $$\Delta T = \frac{\Delta L}{\alpha L_0}$$ And since the change in temperature is the difference between the highest temperature and initial temperature, we can write: $$T_{max} = T_0 + \Delta T$$ Now, substitute the expression for \(\Delta T\) we derived earlier: $$T_{max} = T_0 + \frac{\Delta L}{\alpha L_0}$$

Plug in the given values and calculate the highest temperature

Using the values given in Step 1, we can plug them into the rearranged formula and find the highest temperature: $$T_{max} = 20.0 + \frac{10.0 \times 10^{-3}}{13 \cdot 10^{-6} \times 25.0}$$ Calculate the expression: $$T_{max} \approx 20.0 + 30.77 = 50.77 { }^{\circ} \mathrm{C}$$ So the highest temperature that the tracks can take before the expansion creates compressive forces between sections is approximately \(50.77 { }^{\circ} \mathrm{C}\).