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title: "A uniform disk of mass \\(M\\) and radius \\(R\\) is mounted on a fixed, frictionless vertical axle passing through its center. An external horizontal force \\(\\vec{F}\\) of constant magnitude is applied to a point on the rim of the disk at a varying angle \\(\\phi\\) relative to the outward radial vector, where \\(0 < \\phi < \\pi\\) at all times. The axle exerts a horizontal contact force on the disk such that the center of mass of the disk remains stationary. Which of the following statements correctly describes the rotational kinetic energy of the disk as it rotates from rest through an angular displacement \\(\\Delta\\theta\\), and provides the correct justification?"
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url: "https://nerd-notes.com/ubq/124288/"
date_modified: "2026-09-28T14:04:39+00:00"
---

# A uniform disk of mass \(M\) and radius \(R\) is mounted on a fixed, frictionless vertical axle passing through its center. An external horizontal force \(\vec{F}\) of constant magnitude is applied to a point on the rim of the disk at a varying angle \(\phi\) relative to the outward radial vector, where \(0 < \phi < \pi\) at all times. The axle exerts a horizontal contact force on the disk such that the center of mass of the disk remains stationary. Which of the following statements correctly describes the rotational kinetic energy of the disk as it rotates from rest through an angular displacement \(\Delta\theta\), and provides the correct justification?

A uniform disk of mass \(M\) and radius \(R\) is mounted on a fixed, frictionless vertical axle passing through its center. An external horizontal force \(\vec{F}\) of constant magnitude is applied to a point on the rim of the disk at a varying angle \(\phi\) relative to the outward radial vector, where \(0 < \phi < \pi\) at all times. The axle exerts a horizontal contact force on the disk such that the center of mass of the disk remains stationary. Which of the following statements correctly describes the rotational kinetic energy of the disk as it rotates from rest through an angular displacement \(\Delta\theta\), and provides the correct justification?

![Top-down view of a circular disk of radius \(R\). At the center of the circle, a small solid black dot represents a fixed axle. A dashed line segment of length \(R\) connects the center dot to a point on the top edge of the circle's circumference. From that contact point on the circumference, an arrow representing \(\vec{F}\) points upward and toward the right. A dashed extension of the radial line segment continues outward beyond the rim, and a curved angle arc labeled \(\phi\) is marked between this outward dashed radial line and the force arrow \(\vec{F}\). Near the center of the disk, a curved arrow indicates a counterclockwise direction of rotation. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604279-pljtLt.jpg)

- **A.** The rotational kinetic energy of the disk increases because the work done on the disk is the line integral of \(\vec{F}\) along the displacement of its point of application, which equals \(\int (R F \sin\phi)\, d\theta\), while the force exerted by the axle does zero work.
- **B.** The rotational kinetic energy of the disk stays the same because the net external force acting on the disk is zero to keep the center of mass stationary, which requires the net work done on the disk to be zero.
- **C.** The rotational kinetic energy of the disk increases because the radial component of the force, \(F\cos\phi\), exerts an inward stress that is converted by the rigid structure of the disk into rotational kinetic energy.
- **D.** The rotational kinetic energy of the disk stays the same because the positive work done by the tangential component of \(\vec{F}\) is exactly balanced by the equal and opposite negative work done by the axle reaction force.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124288/*