---
title: "A thin, non-uniform rod of total mass \\(M\\) and length \\(L\\) lies along the positive \\(x\\)-axis between \\(x = 0\\) and \\(x = L\\). The linear mass density of the rod decreases exponentially from the left end according to the function \\(\\lambda(x) = \\lambda_0 e^{-x/L}\\), where \\(\\lambda_0\\) is a positive constant. Which of the following expressions correctly represents the center-of-mass coordinate \\(x_{\\text{cm}}\\) of the rod measured from \\(x = 0\\)?"
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url: "https://nerd-notes.com/ubq/124304/"
date_modified: "2026-09-28T14:04:42+00:00"
---

# A thin, non-uniform rod of total mass \(M\) and length \(L\) lies along the positive \(x\)-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod decreases exponentially from the left end according to the function \(\lambda(x) = \lambda_0 e^{-x/L}\), where \(\lambda_0\) is a positive constant. Which of the following expressions correctly represents the center-of-mass coordinate \(x_{\text{cm}}\) of the rod measured from \(x = 0\)?

A thin, non-uniform rod of total mass \(M\) and length \(L\) lies along the positive \(x\)-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod decreases exponentially from the left end according to the function \(\lambda(x) = \lambda_0 e^{-x/L}\), where \(\lambda_0\) is a positive constant. Which of the following expressions correctly represents the center-of-mass coordinate \(x_{\text{cm}}\) of the rod measured from \(x = 0\)?

![A horizontal coordinate axis labeled x extends to the right with an arrow. A vertical tick mark at the origin is labeled 0, and a second vertical tick mark to the right is labeled L. A horizontal rectangular bar representing a rod lies along the x-axis, spanning from x = 0 to x = L. The bar has a gradient shading that is darkest gray at the left end at x = 0 and smoothly transitions to very light gray at the right end at x = L. A thin vertical slice of width dx is marked within the rod at a horizontal distance x from the origin, labeled dm. A horizontal double-headed dimension arrow spans below the rod from 0 to L and is labeled L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604282-0PSEqR.jpg)

- **A.** \(x_{\text{cm}} = \dfrac{1}{L} \int_0^L x e^{-x/L}\,dx\)
- **B.** \(x_{\text{cm}} = \dfrac{e}{L} \int_0^L x e^{-x/L}\,dx\)
- **C.** \(x_{\text{cm}} = \dfrac{1}{L(1 - e^{-1})} \int_0^L x e^{-x/L}\,dx\)
- **D.** \(x_{\text{cm}} = \dfrac{1}{L(1 - e^{-1})} \int_0^L (L - x) e^{-x/L}\,dx\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124304/*