---
title: "A non-uniform thin rod of mass \\(M\\) and length \\(L\\) is pivoted about a fixed horizontal axis through one end at \\(x = 0\\), allowing it to oscillate freely in a vertical plane as a physical pendulum. The linear mass density of the rod as a function of distance \\(x\\) from the pivot is given by \\(\\lambda(x) = Cx\\), where \\(C\\) is a constant. Which of the following expressions represents the period \\(T\\) of small oscillations of the rod?"
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url: "https://nerd-notes.com/ubq/117645/"
date_modified: "2026-08-04T07:49:28+00:00"
---

# A non-uniform thin rod of mass \(M\) and length \(L\) is pivoted about a fixed horizontal axis through one end at \(x = 0\), allowing it to oscillate freely in a vertical plane as a physical pendulum. The linear mass density of the rod as a function of distance \(x\) from the pivot is given by \(\lambda(x) = Cx\), where \(C\) is a constant. Which of the following expressions represents the period \(T\) of small oscillations of the rod?

A non-uniform thin rod of mass \(M\) and length \(L\) is pivoted about a fixed horizontal axis through one end at \(x = 0\), allowing it to oscillate freely in a vertical plane as a physical pendulum. The linear mass density of the rod as a function of distance \(x\) from the pivot is given by \(\lambda(x) = Cx\), where \(C\) is a constant. Which of the following expressions represents the period \(T\) of small oscillations of the rod?

![A vertical thin rod suspended from a small circular pivot at its top end. A horizontal dashed pivot axis passes through the pivot, labeled x = 0. The rod extends downward a length L to its bottom tip, labeled x = L. A shaded gradient along the rod becomes darker toward the bottom tip to represent increasing linear mass density. An arrow pointing downward along the rod is labeled x. An angle arc at the pivot shows a small displacement angle \theta from the vertical dashed reference line. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829767-SH4O3e.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117645/*