Two rods, one of aluminium and the other of steel, having initial lengths ' $\mathrm{L}_1$ ' and '…

Two rods, one of aluminium and the other of steel, having initial lengths ' $\mathrm{L}_1$ ' and ' $\mathrm{L}_2$ ' are connected together to form a single rod of length $\left(\mathrm{L}_1+\mathrm{L}_2\right)$. The coefficients of linear expansion of aluminium and steel are ' $\alpha_1$ ' and ' $\alpha_2$ ' respectively. If the length of each rod increases by the same amount, when their temperatures are raised by ${ }^{\prime} t^{\circ} \mathrm{C}$, then the ratio $\frac{\mathrm{L}_1}{\mathrm{~L}_1+\mathrm{L}_2}$ will be
  1. $\frac{\alpha_2}{\alpha_1}$
  2. $\frac{\alpha_1}{\alpha_2}$
  3. $\frac{\alpha_2}{(\alpha_1+\alpha_2)}$
  4. $\frac{\alpha_1}{(\alpha_1+\alpha_2)}$

Solution

$\begin{array}{ll} & \text {Given } \Delta \mathrm{L}_1=\Delta \mathrm{L}_2 \Rightarrow \mathrm{~L}_1 \alpha_1 \mathrm{t}=\mathrm{L}_2 \alpha_2 \mathrm{t} \\ \therefore \quad & \frac{\mathrm{L}_1}{\mathrm{~L}_2}=\frac{\alpha_2}{\alpha_1} \Rightarrow \frac{\mathrm{~L}_1}{\mathrm{~L}_1+\mathrm{L}_2}=\frac{\alpha_2}{\alpha_1+\alpha_2}\end{array}$

Asked in: MHT CET 2024 (03 May Shift 1)

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