---
title: "Four rigid, uniform objects each have the same total mass \\(M\\) and outer radius \\(R\\):  I. A thin cylindrical hoop II. A thin spherical shell III. A solid cylinder IV. A solid sphere  Each object is initially at rest and free to rotate about a frictionless fixed axis through its center of mass (the cylinders about their symmetry axes and the spheres about a diameter). A constant net torque \\(\\tau_0\\) is applied to each object for the same time interval \\(\\Delta t\\). Which of the following correctly ranks the rotational kinetic energy \\(K\\) of the four objects at the end of the time interval \\(\\Delta t\\)?"
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url: "https://nerd-notes.com/ubq/120875/"
date_modified: "2026-08-23T04:44:00+00:00"
---

# Four rigid, uniform objects each have the same total mass \(M\) and outer radius \(R\):

I. A thin cylindrical hoop
II. A thin spherical shell
III. A solid cylinder
IV. A solid sphere

Each object is initially at rest and free to rotate about a frictionless fixed axis through its center of mass (the cylinders about their symmetry axes and the spheres about a diameter). A constant net torque \(\tau_0\) is applied to each object for the same time interval \(\Delta t\). Which of the following correctly ranks the rotational kinetic energy \(K\) of the four objects at the end of the time interval \(\Delta t\)?

Four rigid, uniform objects each have the same total mass \(M\) and outer radius \(R\):

I. A thin cylindrical hoop
II. A thin spherical shell
III. A solid cylinder
IV. A solid sphere

Each object is initially at rest and free to rotate about a frictionless fixed axis through its center of mass (the cylinders about their symmetry axes and the spheres about a diameter). A constant net torque \(\tau_0\) is applied to each object for the same time interval \(\Delta t\). Which of the following correctly ranks the rotational kinetic energy \(K\) of the four objects at the end of the time interval \(\Delta t\)?

- **A.** \(K_{\text{hoop}} > K_{\text{spherical shell}} > K_{\text{solid cylinder}} > K_{\text{solid sphere}}\)
- **B.** \(K_{\text{hoop}} = K_{\text{spherical shell}} = K_{\text{solid cylinder}} = K_{\text{solid sphere}}\)
- **C.** \(K_{\text{solid sphere}} > K_{\text{solid cylinder}} > K_{\text{spherical shell}} > K_{\text{hoop}}\)
- **D.** \(K_{\text{solid sphere}} > K_{\text{spherical shell}} > K_{\text{solid cylinder}} > K_{\text{hoop}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120875/*