---
title: "A uniform solid flywheel of mass \\(M\\) and radius \\(R\\) is mounted on a fixed, vertical, frictionless axle passing through its center. The flywheel is initially spinning with a high angular speed \\(\\omega_0\\). At time \\(t = 0\\), the motor driving the flywheel is turned off, and the flywheel begins to slow down solely due to air resistance. The air resistance exerts a decelerating torque on the flywheel with a magnitude given by \\(\\tau_R = b\\omega\\), where \\(\\omega\\) is the angular speed of the flywheel and \\(b\\) is a positive constant. The rotational inertia of a uniform solid disk is \\(I = \\dfrac{1}{2}MR^2\\)."
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url: "https://nerd-notes.com/ubq/117751/"
date_modified: "2026-08-04T07:55:58+00:00"
---

# A uniform solid flywheel of mass \(M\) and radius \(R\) is mounted on a fixed, vertical, frictionless axle passing through its center. The flywheel is initially spinning with a high angular speed \(\omega_0\). At time \(t = 0\), the motor driving the flywheel is turned off, and the flywheel begins to slow down solely due to air resistance. The air resistance exerts a decelerating torque on the flywheel with a magnitude given by \(\tau_R = b\omega\), where \(\omega\) is the angular speed of the flywheel and \(b\) is a positive constant. The rotational inertia of a uniform solid disk is \(I = \dfrac{1}{2}MR^2\).

A uniform solid flywheel of mass \(M\) and radius \(R\) is mounted on a fixed, vertical, frictionless axle passing through its center. The flywheel is initially spinning with a high angular speed \(\omega_0\). At time \(t = 0\), the motor driving the flywheel is turned off, and the flywheel begins to slow down solely due to air resistance. The air resistance exerts a decelerating torque on the flywheel with a magnitude given by \(\tau_R = b\omega\), where \(\omega\) is the angular speed of the flywheel and \(b\) is a positive constant. The rotational inertia of a uniform solid disk is \(I = \dfrac{1}{2}MR^2\).

![A perspective view of a thick, horizontal solid cylinder representing the flywheel. A vertical dashed line passes exactly through its geometric center, indicating the axle. A curved arrow wraps around the axle above the disk, pointing in a counterclockwise direction, labeled \omega_0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830157-RbaSlu.jpg)

**Part a)** **Predict** whether the magnitude of the angular acceleration of the flywheel increases, decreases, or stays the same as time progresses. - [ ] Increases - [ ] Decreases - [ ] Stays the same **Justify** your prediction using qualitative physical reasoning (without referencing equations). *(3 points)*

**Part b)** Using calculus, **derive** an expression for the angular speed \(\omega(t)\) of the flywheel as a function of time \(t\). Express your answer in terms of \(M\), \(R\), \(b\), \(\omega_0\), and fundamental constants. *(4 points)*

**Part c)** A student claims that the flywheel will eventually come to a complete stop at a finite time \(t_f\). **Indicate** whether the student's claim is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your choice using evidence from your derivation in Part (b). *(2 points)*

**Part d)** A second identical flywheel (Flywheel 2) is given the same initial angular speed \(\omega_0\). Because Flywheel 2 is located in a denser atmosphere, it experiences a decelerating torque from air resistance with a constant \(2b\). On the axes provided, **sketch** a graph of the angular speed \(\omega\) as a function of time \(t\) for both flywheels. **Label** the curve for the original flywheel as "1" and the curve for the second flywheel as "2". *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117751/*