---
title: "A thin, non-uniform rod of mass \\(M\\) and length \\(L\\) is pivoted at a frictionless axle at its top end (\\(x = 0\\)). The rod’s linear mass density varies along its length according to \\(\\lambda(x) = \\lambda_0 \\left(\\dfrac{x}{L}\\right)\\), where \\(\\lambda_0\\) is a constant and \\(x\\) is the distance from the pivot. The rod is held at rest displaced by an angle \\(\\theta\\) relative to the downward vertical and then released. What is the magnitude of the rod’s initial angular acceleration \\(\\alpha\\) immediately after it is released?"
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url: "https://nerd-notes.com/ubq/117610/"
date_modified: "2026-08-04T07:49:19+00:00"
---

# A thin, non-uniform rod of mass \(M\) and length \(L\) is pivoted at a frictionless axle at its top end (\(x = 0\)). The rod’s linear mass density varies along its length according to \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)\), where \(\lambda_0\) is a constant and \(x\) is the distance from the pivot. The rod is held at rest displaced by an angle \(\theta\) relative to the downward vertical and then released. What is the magnitude of the rod’s initial angular acceleration \(\alpha\) immediately after it is released?

A thin, non-uniform rod of mass \(M\) and length \(L\) is pivoted at a frictionless axle at its top end (\(x = 0\)). The rod's linear mass density varies along its length according to \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)\), where \(\lambda_0\) is a constant and \(x\) is the distance from the pivot. The rod is held at rest displaced by an angle \(\theta\) relative to the downward vertical and then released. What is the magnitude of the rod's initial angular acceleration \(\alpha\) immediately after it is released?

![A thin, straight rod of length L is pivoted at its upper end attached to a fixed horizontal ceiling. The rod hangs at an angle theta relative to a vertical dashed line extending downward from the pivot. The top end at the pivot is labeled x = 0, and the free lower end is labeled x = L. The rod is drawn slightly wider toward its lower end to visually indicate that its linear mass density lambda(x) increases with distance x. An arc labeled theta indicates the angle between the downward vertical dashed line and the rod. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829758-OQSAnj.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117610/*