---
title: "A uniform thin rod of length \\(L\\) and mass \\(M\\) is pivoted at one end and allowed to swing in a vertical plane. Let \\(\\theta\\) be the angle the rod makes with the downward vertical direction, as shown in the figure. The rod is released from rest at some initial angle \\(\\theta_0 > 90^\\circ\\). What is the ratio \\(\\dfrac{\\alpha_{45}}{\\alpha_{90}}\\) of the magnitude of the angular acceleration of the rod when it is at \\(\\theta = 45^\\circ\\) to the magnitude of its angular acceleration when it is at \\(\\theta = 90^\\circ\\)?"
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url: "https://nerd-notes.com/ubq/110564/"
date_modified: "2026-04-07T05:17:16+00:00"
---

# A uniform thin rod of length \(L\) and mass \(M\) is pivoted at one end and allowed to swing in a vertical plane. Let \(\theta\) be the angle the rod makes with the downward vertical direction, as shown in the figure. The rod is released from rest at some initial angle \(\theta_0 > 90^\circ\). What is the ratio \(\dfrac{\alpha_{45}}{\alpha_{90}}\) of the magnitude of the angular acceleration of the rod when it is at \(\theta = 45^\circ\) to the magnitude of its angular acceleration when it is at \(\theta = 90^\circ\)?

A uniform thin rod of length \(L\) and mass \(M\) is pivoted at one end and allowed to swing in a vertical plane. Let \(\theta\) be the angle the rod makes with the downward vertical direction, as shown in the figure. The rod is released from rest at some initial angle \(\theta_0 > 90^\circ\). What is the ratio \(\dfrac{\alpha_{45}}{\alpha_{90}}\) of the magnitude of the angular acceleration of the rod when it is at \(\theta = 45^\circ\) to the magnitude of its angular acceleration when it is at \(\theta = 90^\circ\)?

![A thin uniform rod is pivoted at its top end. A dashed vertical line passes through the pivot point P. The rod is positioned at an angle theta relative to the downward dashed vertical line. The center of mass is labeled CM at the midpoint of the rod. A curved arrow indicates the direction of swing toward the vertical line.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775539036-r61AZv.jpg)

- **A.** \(\dfrac{1}{2}\)
- **B.** \(\dfrac{\sqrt{2}}{2}\)
- **C.** \(1\)
- **D.** \(\sqrt{2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110564/*