---
title: "A thin, non-uniform circular disk of mass \\(M\\) and radius \\(R\\) lies in the \\(xy\\)-plane. The surface mass density \\(\\sigma(r)\\) of the disk varies with radial distance \\(r\\) from the central axis according to \\(\\sigma(r) = \\sigma_0 \\left(1 – \\dfrac{r}{R}\\right)\\), where \\(\\sigma_0\\) is a positive constant. Which of the following expressions correctly gives the rotational inertia \\(I\\) of the disk about an axis perpendicular to the disk passing through its center?"
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url: "https://nerd-notes.com/ubq/117636/"
date_modified: "2026-08-04T07:49:25+00:00"
---

# A thin, non-uniform circular disk of mass \(M\) and radius \(R\) lies in the \(xy\)-plane. The surface mass density \(\sigma(r)\) of the disk varies with radial distance \(r\) from the central axis according to \(\sigma(r) = \sigma_0 \left(1 – \dfrac{r}{R}\right)\), where \(\sigma_0\) is a positive constant. Which of the following expressions correctly gives the rotational inertia \(I\) of the disk about an axis perpendicular to the disk passing through its center?

A thin, non-uniform circular disk of mass \(M\) and radius \(R\) lies in the \(xy\)-plane. The surface mass density \(\sigma(r)\) of the disk varies with radial distance \(r\) from the central axis according to \(\sigma(r) = \sigma_0 \left(1 - \dfrac{r}{R}\right)\), where \(\sigma_0\) is a positive constant. Which of the following expressions correctly gives the rotational inertia \(I\) of the disk about an axis perpendicular to the disk passing through its center?

![A circular disk lies horizontally with its center at the origin. A vertical dashed line passes through the center of the disk, representing the rotational axis perpendicular to the surface. An arrow labeled R extends from the central axis to the outer edge of the disk. Shading on the disk is darker near the center and gradually fades toward the outer edge to depict decreasing surface density. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829765-pVNOZv.jpg)

- **A.** \(\dfrac{3}{10} M R^2\)
- **B.** \(\dfrac{1}{6} M R^2\)
- **C.** \(\dfrac{1}{2} M R^2\)
- **D.** \(\dfrac{3}{5} M R^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117636/*