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MHT CET · Physics · Thermal Properties of Matter

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. A \(\frac{\alpha_i}{\alpha_c+\alpha_i}\)
  2. B \(\frac{\alpha_c}{\alpha_c+\alpha_i}\).
  3. C \(\frac{\alpha_i-\alpha_c}{\alpha_c+\alpha_i}\)
  4. D \(\frac{\alpha_c-\alpha_i}{\alpha_c+\alpha_i}\)
Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{\alpha_i-\alpha_c}{\alpha_c+\alpha_i}\)

Step-by-step Solution

Detailed explanation

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