---
title: "A thin, non-uniform rod of total mass \\(M\\) and length \\(L\\) has a linear mass density given by \\(\\lambda(x) = \\dfrac{2M}{L^2}x\\), where \\(x\\) is the distance from the rod’s lighter end. The rod is attached to a frictionless horizontal pivot at the end \\(x = 0\\) and held in a horizontal orientation. Immediately after the rod is released from rest, what is the magnitude of its angular acceleration about the pivot?"
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url: "https://nerd-notes.com/ubq/120841/"
date_modified: "2026-08-23T04:43:34+00:00"
---

# A thin, non-uniform rod of total mass \(M\) and length \(L\) has a linear mass density given by \(\lambda(x) = \dfrac{2M}{L^2}x\), where \(x\) is the distance from the rod’s lighter end. The rod is attached to a frictionless horizontal pivot at the end \(x = 0\) and held in a horizontal orientation. Immediately after the rod is released from rest, what is the magnitude of its angular acceleration about the pivot?

A thin, non-uniform rod of total mass \(M\) and length \(L\) has a linear mass density given by \(\lambda(x) = \dfrac{2M}{L^2}x\), where \(x\) is the distance from the rod's lighter end. The rod is attached to a frictionless horizontal pivot at the end \(x = 0\) and held in a horizontal orientation. Immediately after the rod is released from rest, what is the magnitude of its angular acceleration about the pivot?

![A schematic of a horizontal rod of length \(L\) supported by a pivot at its left end. The rod is drawn as an elongated horizontal rectangle. The left end of the rod is mounted on a small circular pivot supported by a triangular base, labeled \(x = 0\). The right end of the rod is free and labeled \(x = L\). The interior of the rod has grayscale gradient shading that is light at the left end and becomes progressively darker toward the right end, representing increasing mass density. A horizontal dimension line with arrows at both ends spans below the full length of the rod and is labeled \(L\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460214-DuwCKN.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120841/*