---
title: "A uniform solid disk of mass \\(M\\) and radius \\(R\\) is mounted on a fixed, frictionless axle passing through its center of mass. The disk is attached to a non-linear torsion spring that exerts a restoring torque \\(\\tau(\\theta) = -\\kappa \\theta^3\\), where \\(\\kappa\\) is a positive constant and \\(\\theta\\) is the angular displacement from equilibrium. The disk is released from rest at an initial angular displacement \\(\\theta = \\theta_0 > 0\\). Which of the following integral expressions correctly represents the angular speed \\(\\omega\\) of the disk as it passes through the equilibrium position \\(\\theta = 0\\)?"
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url: "https://nerd-notes.com/ubq/124331/"
date_modified: "2026-09-28T14:04:53+00:00"
---

# A uniform solid disk of mass \(M\) and radius \(R\) is mounted on a fixed, frictionless axle passing through its center of mass. The disk is attached to a non-linear torsion spring that exerts a restoring torque \(\tau(\theta) = -\kappa \theta^3\), where \(\kappa\) is a positive constant and \(\theta\) is the angular displacement from equilibrium. The disk is released from rest at an initial angular displacement \(\theta = \theta_0 > 0\). Which of the following integral expressions correctly represents the angular speed \(\omega\) of the disk as it passes through the equilibrium position \(\theta = 0\)?

A uniform solid disk of mass \(M\) and radius \(R\) is mounted on a fixed, frictionless axle passing through its center of mass. The disk is attached to a non-linear torsion spring that exerts a restoring torque \(\tau(\theta) = -\kappa \theta^3\), where \(\kappa\) is a positive constant and \(\theta\) is the angular displacement from equilibrium. The disk is released from rest at an initial angular displacement \(\theta = \theta_0 > 0\). Which of the following integral expressions correctly represents the angular speed \(\omega\) of the disk as it passes through the equilibrium position \(\theta = 0\)?

![A circular disk of radius R shown in perspective, mounted on a central horizontal axle aligned with its symmetry axis. The disk has mass M indicated by a label M near the outer edge. A coiled spiral torsion spring is wrapped coaxially around the axle at the center of the disk, with one end attached to a fixed vertical support and the other attached to the disk. A curved reference arrow indicates the angular displacement \theta from a vertical reference dashed line labeled \theta = 0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604292-ANEhXl.jpg)

- **A.** \(\sqrt{\dfrac{4\kappa}{MR^2}\int_0^{\theta_0} \theta^3\,d\theta}\)
- **B.** \(\sqrt{\dfrac{2\kappa}{MR^2}\int_0^{\theta_0} \theta^3\,d\theta}\)
- **C.** \(\sqrt{\dfrac{4\kappa}{MR^2}\int_{\theta_0}^0 \theta^3\,d\theta}\)
- **D.** \(\sqrt{\dfrac{\kappa}{MR^2}\int_0^{\theta_0} \theta^3\,d\theta}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124331/*