---
title: "A uniform solid disk of mass \\(M\\) and radius \\(R\\) rotates about a fixed, frictionless axle through its center with an initial angular speed \\(\\omega_0\\). At time \\(t = 0\\), a magnetic brake is engaged, exerting a resistive torque on the disk of magnitude \\(\\tau = \\gamma \\sqrt{\\omega}\\), where \\(\\gamma\\) is a positive constant and \\(\\omega\\) is the instantaneous angular speed. Which of the following integral expressions correctly represents the total number of revolutions \\(N\\) the disk completes before coming to rest?"
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url: "https://nerd-notes.com/ubq/120839/"
date_modified: "2026-08-23T04:43:34+00:00"
---

# A uniform solid disk of mass \(M\) and radius \(R\) rotates about a fixed, frictionless axle through its center with an initial angular speed \(\omega_0\). At time \(t = 0\), a magnetic brake is engaged, exerting a resistive torque on the disk of magnitude \(\tau = \gamma \sqrt{\omega}\), where \(\gamma\) is a positive constant and \(\omega\) is the instantaneous angular speed. Which of the following integral expressions correctly represents the total number of revolutions \(N\) the disk completes before coming to rest?

A uniform solid disk of mass \(M\) and radius \(R\) rotates about a fixed, frictionless axle through its center with an initial angular speed \(\omega_0\). At time \(t = 0\), a magnetic brake is engaged, exerting a resistive torque on the disk of magnitude \(\tau = \gamma \sqrt{\omega}\), where \(\gamma\) is a positive constant and \(\omega\) is the instantaneous angular speed. Which of the following integral expressions correctly represents the total number of revolutions \(N\) the disk completes before coming to rest?

![A circular disk of radius \(R\) viewed at an oblique angle, mounted horizontally on a vertical central axle represented by a thin vertical line passing through the center of the disk. A curved arrow along the outer edge of the disk indicates an initial angular velocity labeled \(\omega_0\). A curved arrow in the opposite direction represents a resistive torque labeled \(\tau\). A straight dimension arrow extends from the center of the disk to its outer edge, labeled \(R\). The mass of the disk is labeled \(M\) near the top surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460213-3lE6xU.jpg)

- **A.** \(\dfrac{M R^2}{4\pi \gamma} \int_0^{\omega_0} \dfrac{1}{\sqrt{\omega}}\,d\omega\)
- **B.** \(\dfrac{M R^2}{2\pi \gamma} \int_0^{\omega_0} \sqrt{\omega}\,d\omega\)
- **C.** \(\dfrac{M R^2}{4\pi \gamma} \int_0^{\omega_0} \sqrt{\omega}\,d\omega\)
- **D.** \(\dfrac{M R^2}{2\gamma} \int_0^{\omega_0} \sqrt{\omega}\,d\omega\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120839/*