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title: "A thin, uniform rod of mass \\(M\\) and length \\(L\\) is suspended from a horizontal frictionless axle perpendicular to the rod and allowed to oscillate with small angular amplitude. Three different pivot locations along the rod are tested, located at distances \\(d_1 = \\dfrac{L}{6}\\), \\(d_2 = \\dfrac{L}{3}\\), and \\(d_3 = \\dfrac{L}{2}\\) from the center of mass of the rod, resulting in oscillation periods \\(T_1\\), \\(T_2\\), and \\(T_3\\), respectively. Which of the following correctly ranks the periods of oscillation?"
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url: "https://nerd-notes.com/ubq/124254/"
date_modified: "2026-09-28T14:04:31+00:00"
---

# A thin, uniform rod of mass \(M\) and length \(L\) is suspended from a horizontal frictionless axle perpendicular to the rod and allowed to oscillate with small angular amplitude. Three different pivot locations along the rod are tested, located at distances \(d_1 = \dfrac{L}{6}\), \(d_2 = \dfrac{L}{3}\), and \(d_3 = \dfrac{L}{2}\) from the center of mass of the rod, resulting in oscillation periods \(T_1\), \(T_2\), and \(T_3\), respectively. Which of the following correctly ranks the periods of oscillation?

A thin, uniform rod of mass \(M\) and length \(L\) is suspended from a horizontal frictionless axle perpendicular to the rod and allowed to oscillate with small angular amplitude. Three different pivot locations along the rod are tested, located at distances \(d_1 = \dfrac{L}{6}\), \(d_2 = \dfrac{L}{3}\), and \(d_3 = \dfrac{L}{2}\) from the center of mass of the rod, resulting in oscillation periods \(T_1\), \(T_2\), and \(T_3\), respectively. Which of the following correctly ranks the periods of oscillation?

![A vertical rectangular bar representing a uniform rod of length \(L\) and mass \(M\). The center of mass is indicated by a black dot at the vertical midpoint of the bar, labeled \(\text{CM}\). Above the center of mass, three small circles along the centerline represent pivot points: the first circle is located at distance \(d_1\) above \(\text{CM}\), the second circle is located at distance \(d_2\) above \(\text{CM}\), and the third circle is at the top end of the bar at distance \(d_3\) above \(\text{CM}\). Vertical double-headed dimension arrows extend from the horizontal level of \(\text{CM}\) to each pivot point, labeled \(d_1 = \dfrac{L}{6}\), \(d_2 = \dfrac{L}{3}\), and \(d_3 = \dfrac{L}{2}\). A dashed vertical line marks the central longitudinal axis of the rod. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604271-o2meXJ.jpg)

- **A.** \(T_3 > T_2 > T_1\)
- **B.** \(T_1 > T_2 > T_3\)
- **C.** \(T_1 = T_3 > T_2\)
- **D.** \(T_2 > T_1 = T_3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124254/*