---
title: "A uniform thin rod of length \\(L\\) and mass \\(M\\) is suspended from a frictionless horizontal pivot located a distance \\(x\\) from the rod’s center of mass, where \\(0 < x < L/2\\). The rod oscillates in a vertical plane with small angular amplitude. As \\(x \\to 0\\), the period of oscillation approaches infinity, and as \\(x\\) approaches \\(L/2\\), the period increases after passing through a minimum. In terms of \\(L\\), what distance \\(x\\) minimizes the period of oscillation?"
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url: "https://nerd-notes.com/ubq/124642/"
date_modified: "2026-09-28T14:08:55+00:00"
---

# A uniform thin rod of length \(L\) and mass \(M\) is suspended from a frictionless horizontal pivot located a distance \(x\) from the rod’s center of mass, where \(0 < x < L/2\). The rod oscillates in a vertical plane with small angular amplitude. As \(x \to 0\), the period of oscillation approaches infinity, and as \(x\) approaches \(L/2\), the period increases after passing through a minimum. In terms of \(L\), what distance \(x\) minimizes the period of oscillation?

A uniform thin rod of length \(L\) and mass \(M\) is suspended from a frictionless horizontal pivot located a distance \(x\) from the rod's center of mass, where \(0 < x < L/2\). The rod oscillates in a vertical plane with small angular amplitude. As \(x \to 0\), the period of oscillation approaches infinity, and as \(x\) approaches \(L/2\), the period increases after passing through a minimum. In terms of \(L\), what distance \(x\) minimizes the period of oscillation?

![A vertical thin rectangular rod of length L is suspended in a vertical plane. A small open circle labeled P represents a horizontal pivot located a distance x above the rod's midpoint. A solid black dot labeled CM marks the center of mass of the rod at its geometric center. A vertical double-headed arrow between P and CM is labeled x. A vertical bracket extending the full length of the rod is labeled L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604535-VDi5PO.jpg)

- **A.** \(\dfrac{L}{4}\)
- **B.** \(\dfrac{L}{2\sqrt{3}}\)
- **C.** \(\dfrac{L}{\sqrt{6}}\)
- **D.** \(\dfrac{L}{\sqrt{3}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124642/*