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title: "A flywheel with rotational inertia \\(I\\) is mounted on a fixed frictionless axle and rotates freely with an initial angular speed \\(\\omega_0\\). At time \\(t = 0\\), a magnetic brake is engaged, exerting a retarding torque of magnitude \\(\\tau(t) = \\tau_0 e^{-t/T}\\) that opposes the rotation, where \\(\\tau_0\\) and \\(T\\) are positive constants satisfying \\(\\tau_0 T < I \\omega_0\\). Which of the following statements correctly describes the angular speed \\(\\omega(t)\\) and total angular displacement \\(\\Delta\\theta(t)\\) of the flywheel in the limit as \\(t \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/124406/"
date_modified: "2026-09-28T14:05:12+00:00"
---

# A flywheel with rotational inertia \(I\) is mounted on a fixed frictionless axle and rotates freely with an initial angular speed \(\omega_0\). At time \(t = 0\), a magnetic brake is engaged, exerting a retarding torque of magnitude \(\tau(t) = \tau_0 e^{-t/T}\) that opposes the rotation, where \(\tau_0\) and \(T\) are positive constants satisfying \(\tau_0 T < I \omega_0\). Which of the following statements correctly describes the angular speed \(\omega(t)\) and total angular displacement \(\Delta\theta(t)\) of the flywheel in the limit as \(t \to \infty\)?

A flywheel with rotational inertia \(I\) is mounted on a fixed frictionless axle and rotates freely with an initial angular speed \(\omega_0\). At time \(t = 0\), a magnetic brake is engaged, exerting a retarding torque of magnitude \(\tau(t) = \tau_0 e^{-t/T}\) that opposes the rotation, where \(\tau_0\) and \(T\) are positive constants satisfying \(\tau_0 T < I \omega_0\). Which of the following statements correctly describes the angular speed \(\omega(t)\) and total angular displacement \(\Delta\theta(t)\) of the flywheel in the limit as \(t \to \infty\)?

![A perspective diagram showing a circular flywheel viewed at an angle as a thin shaded gray ellipse centered on a horizontal axle. The axle is represented by a solid horizontal line passing through the center of the ellipse. A curved arrow labeled \(\omega_0\) loops over the top edge of the ellipse pointing in the direction of rotation. Adjacent to the upper rim of the ellipse is a small shaded rectangular block labeled Brake. A small straight arrow labeled \(\tau(t)\) points along the rim opposite to the direction of \(\omega_0\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604312-7gNhjy.jpg)

- **A.** The angular speed approaches zero asymptotically as \(t \to \infty\), and the flywheel reaches a finite maximum angular displacement of \(\Delta\theta_{\max} = \omega_0 T\).
- **B.** The angular speed approaches a constant non-zero value of \(\omega_0 - \dfrac{\tau_0 T}{I}\), and the total angular displacement increases without bound.
- **C.** The angular speed approaches a constant non-zero value of \(\omega_0 - \dfrac{\tau_0 T}{I}\), and the flywheel reaches a finite maximum angular displacement of \(\dfrac{\tau_0 T^2}{I}\).
- **D.** The angular speed reaches zero at a finite time \(t = \dfrac{I\omega_0}{\tau_0}\), after which the total angular displacement remains constant.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124406/*