Question

A copper rod (1.7×10-5°c)is 20cm longest than 2n aluminum rod (2.20×10-5°c)how long should the copper rod be if the difference in their length is to be independent of temperature

Answer 1

Let's write the thermal linear equations for copper and aluminum rods:

$$\Delta L_{01}=\alpha_1L_{01}\Delta T_1,$$ $$\Delta L_{02}=\alpha_2L_{02}\Delta T_2.$$

Since the difference in their length is to be independent of temperature, $\Delta L_{01}=\Delta L_{02}=\Delta L$ under the same temperature change ($\Delta T_1=\Delta T_2=\Delta T$). Therefore, we can equate both equations:

$$\alpha_1L_{01}\Delta T_1=\alpha_2L_{02}\Delta T_2,$$ $$\alpha_1L_{01}=\alpha_2(L_{01}-0.2\ m),$$ $$L_{01}=\dfrac{\alpha_2\cdot0.2\ m}{\alpha_2-\alpha_1},$$ $$L_{01}=\dfrac{2.20\cdot10^{-5}\ ^{\circ}C^{-1}\cdot0.2\ m}{2.20\cdot10^{-5}\ ^{\circ}C^{-1}-1.70\cdot10^{-5}\ ^{\circ}C^{-1}}=0.88\ m.$$