(Solved): GOAL Apply the concept of linear expansion and relate it to stress. PROBLEM (a) A steel railroad t ...

GOAL Apply the concept of linear expansion and relate it to stress. PROBLEM (a) A steel railroad track has a length of \( 30.000 \mathrm{~m} \) when the temperature is \( 0^{\circ} \mathrm{C} \). What is its length on a hot day when the temperature is \( 40.0^{\circ} \mathrm{C} \) ? (b) Suppose the track is nailed down so that it cant expand. What stress results in the track due to the temperature change? STRATEGY (a) Apply the linear expansion equation. (b) A track that cannot expand by \( \triangle L \) due to external constraints is equivalent to compressing the track by \( \Delta L \), creating a stress in the track. Using the equation relating tensile stress to tensile strain together with the linear expansion equation, the amount of (compressional) stress can be calculated using the proper equation. SOLUTION (A) Find the length of the track at \( 40.0^{\circ} \mathrm{C} \). Substitute given quantities into the \( \quad \Delta L=\alpha L_{0} \Delta T=\left[1.1 \times 10^{-5}\left({ }^{\circ} \mathrm{C}\right)^{-1}\right](30.000 \mathrm{~m})\left(40.0{ }^{\circ} \mathrm{C}\right)= \) equation to right, finding the change in \( 0.013 \mathrm{~m} \) length. Add the change to the original length \( \quad L=L_{0}+\Delta L=30.013 \mathrm{~m} \) to find the final length. (B) Find the stress if the track cannot expand. Substitute into the equation to right to find the stress. \[ \frac{F}{A}=Y \frac{\Delta L}{L}=\left(2.00 \times 10^{11} \mathrm{~Pa}\right)\left(\frac{0.013 \mathrm{~m}}{30.0 \mathrm{~m}}\right)=8.7 \times 10^{7} \mathrm{~Pa} \] EXERCISE HINTS: GETTING STARTED I I'M STUCK! Use the values from PRACTICE IT to help you work this exercise. What is the length of the same railroad track on a cold winter day when the temperature is \( 4^{\circ} \mathrm{F} \) ? \( \mathrm{m} \)