---
title: "A solid sphere of total mass \\(M\\) and radius \\(R\\) has a volume mass density that varies with distance \\(r\\) from its center according to \\(\\rho(r) = Cr\\), where \\(C\\) is a constant. The sphere rotates about a fixed axis passing through its center with a constant angular speed \\(\\omega\\). In terms of \\(M\\), \\(R\\), and \\(\\omega\\), what is the rotational kinetic energy of the sphere?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/117583/"
date_modified: "2026-08-04T07:49:12+00:00"
---

# A solid sphere of total mass \(M\) and radius \(R\) has a volume mass density that varies with distance \(r\) from its center according to \(\rho(r) = Cr\), where \(C\) is a constant. The sphere rotates about a fixed axis passing through its center with a constant angular speed \(\omega\). In terms of \(M\), \(R\), and \(\omega\), what is the rotational kinetic energy of the sphere?

A solid sphere of total mass \(M\) and radius \(R\) has a volume mass density that varies with distance \(r\) from its center according to \(\rho(r) = Cr\), where \(C\) is a constant. The sphere rotates about a fixed axis passing through its center with a constant angular speed \(\omega\). In terms of \(M\), \(R\), and \(\omega\), what is the rotational kinetic energy of the sphere?

- **A.** \(\dfrac{2}{9} M R^2 \omega^2\)
- **B.** \(\dfrac{1}{5} M R^2 \omega^2\)
- **C.** \(\dfrac{1}{3} M R^2 \omega^2\)
- **D.** \(\dfrac{4}{9} M R^2 \omega^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117583/*