---
title: "A uniform disk of rotational inertia \\(I\\) is initially at rest and driven by a motor that delivers a constant mechanical power \\(P_0\\). The rotation of the disk is opposed by a viscous damping torque of magnitude \\(\\tau = b\\omega\\), where \\(b\\) is a positive constant and \\(\\omega\\) is the instantaneous angular speed. The angular speed of the disk as a function of time \\(t\\) is given by \\(\\omega(t) = \\sqrt{\\dfrac{P_0}{b}\\left(1 – e^{-2bt/I}\\right)}\\). For a fixed time \\(t > 0\\), which of the following expressions represents the angular speed of the disk in the limit as the damping coefficient \\(b\\) approaches zero?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/124095/"
date_modified: "2026-09-28T13:30:24+00:00"
---

# A uniform disk of rotational inertia \(I\) is initially at rest and driven by a motor that delivers a constant mechanical power \(P_0\). The rotation of the disk is opposed by a viscous damping torque of magnitude \(\tau = b\omega\), where \(b\) is a positive constant and \(\omega\) is the instantaneous angular speed. The angular speed of the disk as a function of time \(t\) is given by \(\omega(t) = \sqrt{\dfrac{P_0}{b}\left(1 – e^{-2bt/I}\right)}\). For a fixed time \(t > 0\), which of the following expressions represents the angular speed of the disk in the limit as the damping coefficient \(b\) approaches zero?

A uniform disk of rotational inertia \(I\) is initially at rest and driven by a motor that delivers a constant mechanical power \(P_0\). The rotation of the disk is opposed by a viscous damping torque of magnitude \(\tau = b\omega\), where \(b\) is a positive constant and \(\omega\) is the instantaneous angular speed. The angular speed of the disk as a function of time \(t\) is given by \(\omega(t) = \sqrt{\dfrac{P_0}{b}\left(1 - e^{-2bt/I}\right)}\). For a fixed time \(t > 0\), which of the following expressions represents the angular speed of the disk in the limit as the damping coefficient \(b\) approaches zero?

- **A.** \(\sqrt{\dfrac{P_0 t}{2I}}\)
- **B.** \(\sqrt{\dfrac{P_0 t}{I}}\)
- **C.** \(\sqrt{\dfrac{2P_0 t}{I}}\)
- **D.** \(2\sqrt{\dfrac{P_0 t}{I}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124095/*