---
title: "Two thin, straight rods, Rod 1 and Rod 2, each of length \\(L\\), are aligned along the horizontal \\(x\\)-axis with their left ends at \\(x = 0\\) and right ends at \\(x = L\\). Rod 1 has a uniform linear mass density \\(\\lambda_1(x) = \\lambda_0\\), while Rod 2 has a non-uniform linear mass density given by \\(\\lambda_2(x) = Cx\\), where \\(C\\) is a positive constant. What is the ratio \\(\\dfrac{x_{\\text{cm}, 1}}{x_{\\text{cm}, 2}}\\) of the center-of-mass position of Rod 1 to that of Rod 2?"
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url: "https://nerd-notes.com/ubq/124233/"
date_modified: "2026-09-28T14:04:21+00:00"
---

# Two thin, straight rods, Rod 1 and Rod 2, each of length \(L\), are aligned along the horizontal \(x\)-axis with their left ends at \(x = 0\) and right ends at \(x = L\). Rod 1 has a uniform linear mass density \(\lambda_1(x) = \lambda_0\), while Rod 2 has a non-uniform linear mass density given by \(\lambda_2(x) = Cx\), where \(C\) is a positive constant. What is the ratio \(\dfrac{x_{\text{cm}, 1}}{x_{\text{cm}, 2}}\) of the center-of-mass position of Rod 1 to that of Rod 2?

Two thin, straight rods, Rod 1 and Rod 2, each of length \(L\), are aligned along the horizontal \(x\)-axis with their left ends at \(x = 0\) and right ends at \(x = L\). Rod 1 has a uniform linear mass density \(\lambda_1(x) = \lambda_0\), while Rod 2 has a non-uniform linear mass density given by \(\lambda_2(x) = Cx\), where \(C\) is a positive constant. What is the ratio \(\dfrac{x_{\text{cm}, 1}}{x_{\text{cm}, 2}}\) of the center-of-mass position of Rod 1 to that of Rod 2?

![A horizontal coordinate axis labeled x extends to the right, terminated by an arrowhead. Two vertical tick marks are drawn on the axis, labeled 0 and L. Positioned horizontally above the axis are two identical rectangular rods of length L, both spanning from x = 0 to x = L. The top rod, labeled Rod 1, has uniform light-gray filling across its entire length. The bottom rod, labeled Rod 2, features a smooth grayscale fill that gradually darkens from pure white at x = 0 to dark gray at x = L. Both rods are parallel to the x-axis and to each other. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604261-9U4mQ8.jpg)

- **A.** \(\dfrac{3}{4}\)
- **B.** \(1\)
- **C.** \(\dfrac{4}{3}\)
- **D.** \(\dfrac{3}{2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124233/*