---
title: "A solid sphere of radius \\(R\\) and total mass \\(M\\) has a volume mass density that varies radially according to \\(\\rho(r) = \\rho_0 \\left(\\dfrac{r}{R}\\right)\\), where \\(\\rho_0\\) is a constant and \\(r\\) is the distance from the center of the sphere. Which of the following expressions represents the rotational inertia of the sphere about an axis passing through its center?"
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url: "https://nerd-notes.com/ubq/117605/"
date_modified: "2026-08-04T07:49:17+00:00"
---

# A solid sphere of radius \(R\) and total mass \(M\) has a volume mass density that varies radially according to \(\rho(r) = \rho_0 \left(\dfrac{r}{R}\right)\), where \(\rho_0\) is a constant and \(r\) is the distance from the center of the sphere. Which of the following expressions represents the rotational inertia of the sphere about an axis passing through its center?

A solid sphere of radius \(R\) and total mass \(M\) has a volume mass density that varies radially according to \(\rho(r) = \rho_0 \left(\dfrac{r}{R}\right)\), where \(\rho_0\) is a constant and \(r\) is the distance from the center of the sphere. Which of the following expressions represents the rotational inertia of the sphere about an axis passing through its center?

![A solid sphere of radius R centered at the origin, with a vertical dashed axis passing through its center. The interior of the sphere shows a radial shading gradient that becomes darker toward the outer edge at radius R, representing increasing mass density with radial distance r. An arrow labeled R extends from the center to the outer boundary. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829757-GH44nG.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117605/*