---
title: "A uniform solid disk of mass \\(M\\) and radius \\(R\\) rotates in a horizontal plane about a fixed vertical axle through its center with an initial angular speed \\(\\omega_0\\). The disk experiences a retarding torque due to fluid drag of magnitude \\(\\tau = \\gamma \\omega^2\\), where \\(\\gamma\\) is a positive constant and \\(\\omega\\) is the instantaneous angular speed. Which of the following expressions represents a correct integral setup for the angular displacement \\(\\Delta\\theta\\) as the disk slows to an angular speed \\(\\omega\\), and what does this relationship predict about the total angular displacement as the disk approaches rest (\\(\\omega \\to 0\\))?"
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url: "https://nerd-notes.com/ubq/124434/"
date_modified: "2026-09-28T14:05:30+00:00"
---

# A uniform solid disk of mass \(M\) and radius \(R\) rotates in a horizontal plane about a fixed vertical axle through its center with an initial angular speed \(\omega_0\). The disk experiences a retarding torque due to fluid drag of magnitude \(\tau = \gamma \omega^2\), where \(\gamma\) is a positive constant and \(\omega\) is the instantaneous angular speed. Which of the following expressions represents a correct integral setup for the angular displacement \(\Delta\theta\) as the disk slows to an angular speed \(\omega\), and what does this relationship predict about the total angular displacement as the disk approaches rest (\(\omega \to 0\))?

A uniform solid disk of mass \(M\) and radius \(R\) rotates in a horizontal plane about a fixed vertical axle through its center with an initial angular speed \(\omega_0\). The disk experiences a retarding torque due to fluid drag of magnitude \(\tau = \gamma \omega^2\), where \(\gamma\) is a positive constant and \(\omega\) is the instantaneous angular speed. Which of the following expressions represents a correct integral setup for the angular displacement \(\Delta\theta\) as the disk slows to an angular speed \(\omega\), and what does this relationship predict about the total angular displacement as the disk approaches rest (\(\omega \to 0\))?

![A thin solid circular disk of radius \(R\) and mass \(M\) drawn in perspective as a flat horizontal ellipse with a slight tilt. A vertical dashed line passes through the geometric center of the disk to represent a fixed axle of rotation, extending slightly above and below the disk. A straight dashed line segment is drawn from the center of the top face of the disk to the right edge, labeled with the variable \(R\). A curved arrow is drawn above the top surface of the disk, curving counterclockwise around the central vertical axis, labeled with the symbol \(\omega\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604330-QkZzwi.jpg)

- **A.** \(\Delta\theta = -\dfrac{M R^2}{\gamma} \int_{\omega_0}^{\omega} \dfrac{1}{\omega'}\,d\omega'\); the total angular displacement is finite as \(\omega \to 0\).
- **B.** \(\Delta\theta = \dfrac{M R^2}{2\gamma} \int_{\omega_0}^{\omega} \dfrac{1}{\omega'}\,d\omega'\); the total angular displacement is unbounded as \(\omega \to 0\).
- **C.** \(\Delta\theta = -\dfrac{M R^2}{2\gamma} \int_{\omega_0}^{\omega} \dfrac{1}{\omega'}\,d\omega'\); the total angular displacement is unbounded as \(\omega \to 0\).
- **D.** \(\Delta\theta = -\dfrac{2\gamma}{M R^2} \int_{\omega_0}^{\omega} \dfrac{1}{\omega'}\,d\omega'\); the total angular displacement is finite as \(\omega \to 0\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124434/*