---
title: "A non-uniform thin rod of length \\(L\\) and total mass \\(M\\) is pivoted about a frictionless horizontal axis at one end (\\(x = 0\\)). The rod has a linear mass density that increases linearly with distance from the pivot, given by \\(\\lambda(x) = bx\\), where \\(b\\) is a positive constant. The rod is displaced by a small angle from the vertical and released from rest. In terms of \\(g\\) and \\(L\\), what is the angular frequency \\(\\omega\\) of the resulting small-angle oscillations?"
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url: "https://nerd-notes.com/ubq/120847/"
date_modified: "2026-08-23T04:43:36+00:00"
---

# A non-uniform thin rod of length \(L\) and total mass \(M\) is pivoted about a frictionless horizontal axis at one end (\(x = 0\)). The rod has a linear mass density that increases linearly with distance from the pivot, given by \(\lambda(x) = bx\), where \(b\) is a positive constant. The rod is displaced by a small angle from the vertical and released from rest. In terms of \(g\) and \(L\), what is the angular frequency \(\omega\) of the resulting small-angle oscillations?

A non-uniform thin rod of length \(L\) and total mass \(M\) is pivoted about a frictionless horizontal axis at one end (\(x = 0\)). The rod has a linear mass density that increases linearly with distance from the pivot, given by \(\lambda(x) = bx\), where \(b\) is a positive constant. The rod is displaced by a small angle from the vertical and released from rest. In terms of \(g\) and \(L\), what is the angular frequency \(\omega\) of the resulting small-angle oscillations?

![A vertical dashed line extends downward from a small open circle representing a frictionless pivot at the origin. A straight rod of length \(L\) is attached at one end to the pivot and is tilted to the right of the dashed vertical line by an angle labeled \(\theta\). The rod is drawn with increasing thickness from its upper pivoted end at \(x = 0\) to its lower free end at \(x = L\) to indicate non-uniform linear mass density. A curved arc with an arrowhead indicates the angle \(\theta\) between the vertical dashed line and the centerline of the rod. A small coordinate axis along the length of the rod shows the pivot at \(x = 0\) and the bottom end at \(x = L\). A downward vertical arrow labeled \(g\) is located to the left of the pivot. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460216-oa78It.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120847/*