---
title: "A thin, non-uniform rod of length \\(L\\) lies along the \\(x\\)-axis between \\(x = 0\\) and \\(x = L\\). The linear mass density of the rod is given by \\(\\lambda(x) = \\alpha x\\), where \\(\\alpha\\) is a positive constant. Which of the following expressions gives the position \\(x_{\\text{cm}}\\) of the center of mass of the rod?"
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url: "https://nerd-notes.com/ubq/117554/"
date_modified: "2026-08-04T07:49:02+00:00"
---

# A thin, non-uniform rod of length \(L\) lies along the \(x\)-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod is given by \(\lambda(x) = \alpha x\), where \(\alpha\) is a positive constant. Which of the following expressions gives the position \(x_{\text{cm}}\) of the center of mass of the rod?

A thin, non-uniform rod of length \(L\) lies along the \(x\)-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod is given by \(\lambda(x) = \alpha x\), where \(\alpha\) is a positive constant. Which of the following expressions gives the position \(x_{\text{cm}}\) of the center of mass of the rod?

![A horizontal axis labeled x with vertical tick marks at 0 and L. A thin horizontal rectangular rod lies along the x-axis from x = 0 to x = L. The rod has a shading gradient, starting light gray at x = 0 and becoming solid dark gray at x = L. An origin tick mark is labeled 0 at the left end of the rod. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829742-eKuVEH.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117554/*