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title: "A uniform solid disk of mass \\(M\\) and radius \\(R\\) is horizontally suspended by a thin vertical wire attached to its center. When the disk is rotated through an angular displacement \\(\\theta\\) from its equilibrium orientation, the wire exerts a restoring torque given by \\(\\tau = -\\kappa \\theta\\), where \\(\\kappa\\) is a positive constant. The disk is rotated to an initial displacement \\(\\theta_0\\) and released from rest at time \\(t = 0\\). Assuming friction and air resistance are negligible, which of the following expressions represents the maximum angular speed \\(\\omega_{\\text{max}}\\) of the disk during its oscillations?"
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url: "https://nerd-notes.com/ubq/120990/"
date_modified: "2026-08-23T04:44:57+00:00"
---

# A uniform solid disk of mass \(M\) and radius \(R\) is horizontally suspended by a thin vertical wire attached to its center. When the disk is rotated through an angular displacement \(\theta\) from its equilibrium orientation, the wire exerts a restoring torque given by \(\tau = -\kappa \theta\), where \(\kappa\) is a positive constant. The disk is rotated to an initial displacement \(\theta_0\) and released from rest at time \(t = 0\). Assuming friction and air resistance are negligible, which of the following expressions represents the maximum angular speed \(\omega_{\text{max}}\) of the disk during its oscillations?

A uniform solid disk of mass \(M\) and radius \(R\) is horizontally suspended by a thin vertical wire attached to its center. When the disk is rotated through an angular displacement \(\theta\) from its equilibrium orientation, the wire exerts a restoring torque given by \(\tau = -\kappa \theta\), where \(\kappa\) is a positive constant. The disk is rotated to an initial displacement \(\theta_0\) and released from rest at time \(t = 0\). Assuming friction and air resistance are negligible, which of the following expressions represents the maximum angular speed \(\omega_{\text{max}}\) of the disk during its oscillations?

![A schematic diagram showing a horizontal ceiling at the top represented by a short horizontal line with hatching above it. A thin, single vertical line representing a suspension wire hangs downward from the center of the ceiling to the exact center of a flat horizontal disk. The disk is drawn in perspective as a flat ellipse with a light gray fill. A straight dashed radial line extends from the center of the disk to its rightmost outer edge, labeled with radius \(R\). The disk is labeled with mass \(M\). The vertical wire is labeled along its side with torsion constant \(\kappa\). A curved horizontal arrow wraps counterclockwise around the top surface of the disk, labeled with angular displacement \(\theta\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460297-zYWHUE.jpg)

- **A.** \(\theta_0 \sqrt{\dfrac{\kappa}{2MR^2}}\
- **B.** \(\theta_0 \sqrt{\dfrac{\kappa}{MR^2}}\
- **C.** \(\theta_0 \sqrt{\dfrac{3\kappa}{2MR^2}}\
- **D.** \(\theta_0 \sqrt{\dfrac{2\kappa}{MR^2}}\

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120990/*