Question

Two rods made of metals A and B, each of length \(20\,\text{cm}\), expand by \(0.075\,\text{cm}\) and \(0.045\,\text{cm}\) respectively, when heated from \(0^\circ\text{C}\) to \(100^\circ\text{C}\). A composite rod of the same length is made with a portion of metal A and the remaining portion with B. The composite rod expands by \(0.060\,\text{cm}\) for the same rise in temperature. Then the portion of the composite rod made of A has initial length

• A. \(8\,\text{cm}\)
• B. \(10\,\text{cm}\)
• C. \(15\,\text{cm}\)
• D. \(18\,\text{cm}\) \bigskip

Hint:

For a composite rod, \[ \Delta L_{\text{total}} = \Delta L_1+\Delta L_2. \] Always find the expansion coefficients first using \[ \alpha=\frac{\Delta L}{L\Delta T}. \] Then apply the expansion formula separately to each segment.

Answer 1

The Correct Option is B

Concept: Find \(\alpha_A, \alpha_B\) from individual expansions: \(\alpha_A = 0.075/(20\times100) = 3.75\times10^{-5}\), \(\alpha_B = 2.25\times10^{-5}\). For composite: \(0.060 = \alpha_A x(100) + \alpha_B(20-x)(100) \Rightarrow 0.060 = 3.75\times10^{-3}x + 0.045 - 2.25\times10^{-3}x \Rightarrow 0.015 = 1.5\times10^{-3}x \Rightarrow x=10\) cm.

Step 1: Write the final answer. \(\boxed{x=10\,\text{cm}}\)