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
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
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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.
Updated On: Jul 9, 2026
- \(8\,\text{cm}\)
- \(10\,\text{cm}\)
- \(15\,\text{cm}\)
- \(18\,\text{cm}\) \bigskip
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The Correct Option is B
Solution and Explanation
Concept:
Linear expansion is given by
\[
\Delta L=\alpha L\Delta T.
\]
First determine the coefficients of linear expansion of metals A and B.
Step 1: Calculate the coefficient of linear expansion of metal A. For metal A, \[ 0.075=\alpha_A(20)(100). \] \[ \alpha_A=\frac{0.075}{2000}. \] \[ \alpha_A=3.75\times10^{-5}\,^\circ\text{C}^{-1}. \]
Step 2: Calculate the coefficient of linear expansion of metal B. For metal B, \[ 0.045=\alpha_B(20)(100). \] \[ \alpha_B=\frac{0.045}{2000}. \] \[ \alpha_B=2.25\times10^{-5}\,^\circ\text{C}^{-1}. \]
Step 3: Form the equation for the composite rod. Let the initial length of metal A in the composite rod be \[ x\,\text{cm}. \] Then the length of metal B is \[ 20-x. \] Total expansion is \(0.060\,\text{cm}\). Hence, \[ 0.060 = \alpha_A x(100) + \alpha_B(20-x)(100). \] Substituting the values, \[ 0.060 = 3.75\times10^{-3}x + 2.25\times10^{-3}(20-x). \]
Step 4: Solve for \(x\). \[ 0.060 = 3.75\times10^{-3}x + 0.045 - 2.25\times10^{-3}x. \] \[ 0.015 = 1.5\times10^{-3}x. \] \[ x=10. \]
Step 5: Write the final answer. \[ \boxed{x=10\,\text{cm}} \] \[ \boxed{\text{Answer = (B)}} \]
Step 1: Calculate the coefficient of linear expansion of metal A. For metal A, \[ 0.075=\alpha_A(20)(100). \] \[ \alpha_A=\frac{0.075}{2000}. \] \[ \alpha_A=3.75\times10^{-5}\,^\circ\text{C}^{-1}. \]
Step 2: Calculate the coefficient of linear expansion of metal B. For metal B, \[ 0.045=\alpha_B(20)(100). \] \[ \alpha_B=\frac{0.045}{2000}. \] \[ \alpha_B=2.25\times10^{-5}\,^\circ\text{C}^{-1}. \]
Step 3: Form the equation for the composite rod. Let the initial length of metal A in the composite rod be \[ x\,\text{cm}. \] Then the length of metal B is \[ 20-x. \] Total expansion is \(0.060\,\text{cm}\). Hence, \[ 0.060 = \alpha_A x(100) + \alpha_B(20-x)(100). \] Substituting the values, \[ 0.060 = 3.75\times10^{-3}x + 2.25\times10^{-3}(20-x). \]
Step 4: Solve for \(x\). \[ 0.060 = 3.75\times10^{-3}x + 0.045 - 2.25\times10^{-3}x. \] \[ 0.015 = 1.5\times10^{-3}x. \] \[ x=10. \]
Step 5: Write the final answer. \[ \boxed{x=10\,\text{cm}} \] \[ \boxed{\text{Answer = (B)}} \]
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