---
title: "A uniform solid disk of mass \\(M\\) and radius \\(R\\) (rotational inertia \\(I = \\dfrac{1}{2}MR^2\\)) rotates horizontally with angular speed \\(\\omega_0\\) about a fixed frictionless vertical axle through its center. A child of mass \\(m = \\dfrac{1}{2}M\\), modeled as a point mass, runs tangentially along the perimeter of the disk in the direction of rotation with linear speed \\(v_0 = 2\\omega_0 R\\) and steps onto the rim of the disk, sticking to it. Which of the following correctly identifies the final angular speed \\(\\omega_f\\) of the disk-child system and the ratio of the system’s final kinetic energy to its initial kinetic energy, \\(\\dfrac{K_f}{K_i}\\)?"
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url: "https://nerd-notes.com/ubq/120923/"
date_modified: "2026-08-23T04:44:20+00:00"
---

# A uniform solid disk of mass \(M\) and radius \(R\) (rotational inertia \(I = \dfrac{1}{2}MR^2\)) rotates horizontally with angular speed \(\omega_0\) about a fixed frictionless vertical axle through its center. A child of mass \(m = \dfrac{1}{2}M\), modeled as a point mass, runs tangentially along the perimeter of the disk in the direction of rotation with linear speed \(v_0 = 2\omega_0 R\) and steps onto the rim of the disk, sticking to it. Which of the following correctly identifies the final angular speed \(\omega_f\) of the disk-child system and the ratio of the system’s final kinetic energy to its initial kinetic energy, \(\dfrac{K_f}{K_i}\)?

A uniform solid disk of mass \(M\) and radius \(R\) (rotational inertia \(I = \dfrac{1}{2}MR^2\)) rotates horizontally with angular speed \(\omega_0\) about a fixed frictionless vertical axle through its center. A child of mass \(m = \dfrac{1}{2}M\), modeled as a point mass, runs tangentially along the perimeter of the disk in the direction of rotation with linear speed \(v_0 = 2\omega_0 R\) and steps onto the rim of the disk, sticking to it. Which of the following correctly identifies the final angular speed \(\omega_f\) of the disk-child system and the ratio of the system's final kinetic energy to its initial kinetic energy, \(\dfrac{K_f}{K_i}\)?

![A top-down view shows a circular platform of radius \(R\) centered on a small black dot representing a fixed vertical axle. A curved counterclockwise arrow labeled \(\omega_0\) indicates rotation of the platform. A small solid circle representing a child of mass \(m\) is located at the top edge of the platform, moving horizontally to the left along a path tangent to the platform's perimeter. A straight arrow labeled \(\vec{v}_0\) points horizontally to the left from the small circle. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460260-mmKz9n.jpg)

- **A.** \(\omega_f = \dfrac{1}{2}\omega_0\) and \(\dfrac{K_f}{K_i} = \dfrac{1}{10}\)
- **B.** \(\omega_f = \dfrac{4}{3}\omega_0\) and \(\dfrac{K_f}{K_i} = \dfrac{8}{9}\)
- **C.** \(\omega_f = \dfrac{3}{2}\omega_0\) and \(\dfrac{K_f}{K_i} = 1\)
- **D.** \(\omega_f = \dfrac{3}{2}\omega_0\) and \(\dfrac{K_f}{K_i} = \dfrac{9}{10}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120923/*