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title: "A composite cylindrical rod of length \\(2L\\) consists of two segments of equal length \\(L\\) and equal cross-sectional area \\(A\\), made of Metal X (\\(0 \\le x \\le L\\)) and Metal Y (\\(L \\le x \\le 2L\\)), joined end-to-end. The lateral surfaces of both segments are thermally insulated. The left end (\\(x = 0\\)) is maintained at \\(100\\text{ }^\\circ\\text{C}\\) and the right end (\\(x = 2L\\)) is maintained at \\(10\\text{ }^\\circ\\text{C}\\). The system reaches a steady state, and the temperature \\(T\\) along the rod as a function of position \\(x\\) is shown in the graph. Which of the following correctly compares the thermal conductivities \\(k_X\\) and \\(k_Y\\) of the two metals, and provides a valid justification?"
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url: "https://nerd-notes.com/ubq/117457/"
date_modified: "2026-08-04T06:52:13+00:00"
---

# A composite cylindrical rod of length \(2L\) consists of two segments of equal length \(L\) and equal cross-sectional area \(A\), made of Metal X (\(0 \le x \le L\)) and Metal Y (\(L \le x \le 2L\)), joined end-to-end. The lateral surfaces of both segments are thermally insulated. The left end (\(x = 0\)) is maintained at \(100\text{ }^\circ\text{C}\) and the right end (\(x = 2L\)) is maintained at \(10\text{ }^\circ\text{C}\). The system reaches a steady state, and the temperature \(T\) along the rod as a function of position \(x\) is shown in the graph. Which of the following correctly compares the thermal conductivities \(k_X\) and \(k_Y\) of the two metals, and provides a valid justification?

A composite cylindrical rod of length \(2L\) consists of two segments of equal length \(L\) and equal cross-sectional area \(A\), made of Metal X (\(0 \le x \le L\)) and Metal Y (\(L \le x \le 2L\)), joined end-to-end. The lateral surfaces of both segments are thermally insulated. The left end (\(x = 0\)) is maintained at \(100\text{ }^\circ\text{C}\) and the right end (\(x = 2L\)) is maintained at \(10\text{ }^\circ\text{C}\). The system reaches a steady state, and the temperature \(T\) along the rod as a function of position \(x\) is shown in the graph. Which of the following correctly compares the thermal conductivities \(k_X\) and \(k_Y\) of the two metals, and provides a valid justification?

![A line graph displaying temperature T in degrees Celsius on the vertical axis from 0 to 100, and distance x on the horizontal axis from 0 to 2L. A vertical dashed line at x = L divides the graph into region Metal X on the left and region Metal Y on the right. In region Metal X, from x = 0 to x = L, a straight line segment goes from (0, 100) down to (L, 40). In region Metal Y, from x = L to x = 2L, a straight line segment goes from (L, 40) down to (2L, 10). The line segment in Metal X is visibly steeper than the line segment in Metal Y. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785826333-bqRost.jpg)

- **A.** \(k_X > k_Y\) because Metal X has a steeper temperature gradient, indicating that heat is conducted at a greater rate through Metal X than through Metal Y.
- **B.** \(k_X > k_Y\) because the average temperature of Metal X is higher than the average temperature of Metal Y.
- **C.** \(k_X < k_Y\) because both metals conduct heat at the same rate in steady state, so the metal with the steeper temperature gradient must have a lower thermal conductivity.
- **D.** \(k_X < k_Y\) because heat flows at a greater rate through Metal Y than through Metal X to reach the cold reservoir.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117457/*