Two rods, one of copper $(. \mathrm{Cu})$ and the other of iron ( Fe ) having initial lengths $\mathrm{L}_1$…

Two rods, one of copper $(. \mathrm{Cu})$ and the other of iron ( Fe ) having initial lengths $\mathrm{L}_1$ and $\mathrm{L}_2$ respectively are connected together to form a single rod of length $L_1+L_2$. The coefficient of linear expansion of Cu and Fe are $\alpha_c$ and $\alpha_i$ respectively. If the length of each rod increases by the same amount when their temperatures are raised by $t^{\circ} \mathrm{C}$, then ratio of $\frac{L_1-L_2}{L_1+L_2}$ will be
  1. $\frac{\alpha_i}{\alpha_c+\alpha_i}$
  2. $\frac{\alpha_c}{\alpha_c+\alpha_i}$.
  3. $\frac{\alpha_i-\alpha_c}{\alpha_c+\alpha_i}$
  4. $\frac{\alpha_c-\alpha_i}{\alpha_c+\alpha_i}$

Solution

$\begin{aligned} & \text { Given } \Delta L_1=\Delta L_2 \\ & L_1 \alpha_c t=L_2 \alpha_i t \\ \therefore \quad & L_1=\frac{\alpha_i}{\alpha_c} L_2 \\ \therefore \quad & \frac{L_1-L_2}{L_1+L_2}=\frac{\frac{\alpha_i}{\alpha_c} L_2-L_2}{\frac{\alpha_i}{\alpha_c} L_2+L_2}=\frac{\frac{\alpha_i}{\alpha_c}-1}{\frac{\alpha_i}{\alpha_c}+1} \\ & \\ & =\frac{\alpha_i-\alpha_c}{\alpha_c+\alpha_i}\end{aligned}$

Asked in: MHT CET 2024 (03 May Shift 2)

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