---
title: "A thin, non-uniform rod of length \\(L\\) is pivoted about a frictionless horizontal axle through one end. The rod has a linear mass density given by \\(\\lambda(x) = k x^3\\), where \\(x\\) is the distance from the pivot and \\(k\\) is a positive constant. The rod is held in a horizontal orientation and released from rest. Immediately after release, what is the magnitude of the angular acceleration of the rod about the pivot?"
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url: "https://nerd-notes.com/ubq/124236/"
date_modified: "2026-09-28T14:04:21+00:00"
---

# A thin, non-uniform rod of length \(L\) is pivoted about a frictionless horizontal axle through one end. The rod has a linear mass density given by \(\lambda(x) = k x^3\), where \(x\) is the distance from the pivot and \(k\) is a positive constant. The rod is held in a horizontal orientation and released from rest. Immediately after release, what is the magnitude of the angular acceleration of the rod about the pivot?

A thin, non-uniform rod of length \(L\) is pivoted about a frictionless horizontal axle through one end. The rod has a linear mass density given by \(\lambda(x) = k x^3\), where \(x\) is the distance from the pivot and \(k\) is a positive constant. The rod is held in a horizontal orientation and released from rest. Immediately after release, what is the magnitude of the angular acceleration of the rod about the pivot?

![A horizontal rod of length \(L\) lies along the horizontal axis, supported at its left end by a circular pivot pin attached to a fixed vertical mount. The left end of the rod is labeled \(x = 0\) and the right free end is labeled \(x = L\). The rod is shaded with a smooth grayscale gradient that is light gray near \(x = 0\) and transitions to dark gray near \(x = L\) to indicate non-uniform mass density. A downward vertical arrow near the pivot indicates the direction of gravitational acceleration, labeled \(g\). A horizontal double-headed dimension arrow spans below the rod from \(x = 0\) to \(x = L\) and is labeled \(L\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604261-h8HFPt.jpg)

- **A.** \(\dfrac{3}{4}\dfrac{g}{L}\)
- **B.** \(\dfrac{4}{5}\dfrac{g}{L}\)
- **C.** \(\dfrac{5}{6}\dfrac{g}{L}\)
- **D.** \(\dfrac{6}{5}\dfrac{g}{L}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124236/*