---
title: "A uniform solid horizontal disk of mass \\(M\\) and radius \\(R\\) is mounted on a frictionless vertical axle through its center. The disk is initially rotating freely with a constant angular speed \\(\\omega_0\\). A small robot of mass \\(m\\) is initially at rest relative to the disk at the center of the disk (\\(r = 0\\)). At time \\(t = 0\\), the robot begins to crawl outward along a radial line on the disk. The robot maintains a constant radial speed \\(v_0\\) relative to the disk. The robot can be treated as a point mass, and its motion is analyzed for the time interval \\(0 \\le t \\le R/v_0\\) before it reaches the edge of the disk."
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url: "https://nerd-notes.com/ubq/117766/"
date_modified: "2026-08-04T07:56:10+00:00"
---

# A uniform solid horizontal disk of mass \(M\) and radius \(R\) is mounted on a frictionless vertical axle through its center. The disk is initially rotating freely with a constant angular speed \(\omega_0\). A small robot of mass \(m\) is initially at rest relative to the disk at the center of the disk (\(r = 0\)). At time \(t = 0\), the robot begins to crawl outward along a radial line on the disk. The robot maintains a constant radial speed \(v_0\) relative to the disk. The robot can be treated as a point mass, and its motion is analyzed for the time interval \(0 \le t \le R/v_0\) before it reaches the edge of the disk.

A uniform solid horizontal disk of mass \(M\) and radius \(R\) is mounted on a frictionless vertical axle through its center. The disk is initially rotating freely with a constant angular speed \(\omega_0\). A small robot of mass \(m\) is initially at rest relative to the disk at the center of the disk (\(r = 0\)). At time \(t = 0\), the robot begins to crawl outward along a radial line on the disk. The robot maintains a constant radial speed \(v_0\) relative to the disk. The robot can be treated as a point mass, and its motion is analyzed for the time interval \(0 \le t \le R/v_0\) before it reaches the edge of the disk.

![A top-down view of a large solid circle representing the horizontal disk. A central dot marks the axle. A straight dashed line representing the robot's radial path extends from the center toward the right edge. On this dashed line, at roughly half the radius, is a small shaded square representing the robot. An arrow labeled v_0 starts at the robot and points radially outward along the dashed line. A curved arrow outside the edge of the disk points counterclockwise and is labeled omega_0. The radius of the disk is marked with a line labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830170-7SrtVf.jpg)

**Part a)** **Derive** an expression for the rotational inertia \(I(t)\) of the disk-robot system about the axle as a function of time \(t\). Express your answer in terms of \(M\), \(R\), \(m\), \(v_0\), \(t\), and physical constants, as appropriate. *(2 points)*

**Part b)** **Derive** an expression for the angular speed \(\omega(t)\) of the system as a function of time \(t\). Express your answer in terms of \(M\), \(R\), \(m\), \(\omega_0\), \(v_0\), \(t\), and physical constants, as appropriate. *(2 points)*

**Part c)** The robot's movement alters the energy of the system. *(4 points)*

**Part d)** Consider a new scenario where a motor is attached to the axle. The motor exerts a continuous external torque on the disk to keep the angular speed constant at \(\omega_0\) while the robot crawls outward with the same constant radial speed \(v_0\). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117766/*