A copper rod $\left(\alpha=1.70 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$ is $20 \mathrm{~cm}$ longer than an aluminum rod $\left(\alpha=2.20 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$. How long should the copper rod be if the difference in their lengths is to be independent of temperature? For their difference in lengths not to change with temperature, $\Delta L$ must be the same for both rods under the same temperature change. That is,where $L_{0}$ is the length of the copper rod, and $\Delta T$ is the same for both rods. Solving for the original length yields $L_{0}=0.88 \mathrm{~m}$.