Question

1.  At 0⁰C, a rod of aluminum and a rod of copper have the same length, 1.25 meters. To what common temperature must they be heated to differ in length by 0.0750 millimeter?

Answer 1

By the definition of thermal linear expansion, we have:

$$\dfrac{L_{Al}-L_{Al,0}}{L_{Al,0}}=\alpha_{Al}(t_{comm}-0^{\circ}C),$$ $$\dfrac{L_{Cu}-L_{Cu,0}}{L_{Cu,0}}=\alpha_{Cu}(t_{comm}-0^{\circ}C).$$

Then, taking into account that $L_{Al,0}=L_{Cu,0}=L_0$ we can rewrite our equations as follows:

$$L_{Al}-L_0=\alpha_{Al}L_0t_{comm},$$ $$L_{Cu}-L_0=\alpha_{Cu}L_0t_{comm}.$$

Subtracting second equation from the first one, we get:

$$L_{Al}-L_{Cu}=(\alpha_{Al}-\alpha_{Cu})L_0t_{comm},$$ $$t_{comm}=\dfrac{L_{Al}-L_{Cu}}{(\alpha_{Al}-\alpha_{Cu})L_0},$$ $$t_{comm}=\dfrac{7.5\cdot10^{-5}\ m}{(24\cdot10^{-6}\ \dfrac{1}{^{\circ}C}-17\cdot10^{-6}\ \dfrac{1}{^{\circ}C})\cdot1.25\ m}=8.57^{\circ}C.$$