---
title: "A thin hoop of radius \\(R\\) lies in the \\(xy\\)-plane with its center at the origin. The linear mass density \\(\\lambda\\) of the hoop varies with angular position \\(\\theta\\) measured from the positive \\(x\\)-axis according to \\(\\lambda(\\theta) = \\lambda_0 (1 + \\cos^2\\theta)\\), where \\(\\lambda_0\\) is a positive constant. Which of the following expressions represents the rotational inertia of the hoop about a central axis perpendicular to the plane of the hoop?"
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url: "https://nerd-notes.com/ubq/117622/"
date_modified: "2026-08-04T07:49:21+00:00"
---

# A thin hoop of radius \(R\) lies in the \(xy\)-plane with its center at the origin. The linear mass density \(\lambda\) of the hoop varies with angular position \(\theta\) measured from the positive \(x\)-axis according to \(\lambda(\theta) = \lambda_0 (1 + \cos^2\theta)\), where \(\lambda_0\) is a positive constant. Which of the following expressions represents the rotational inertia of the hoop about a central axis perpendicular to the plane of the hoop?

A thin hoop of radius \(R\) lies in the \(xy\)-plane with its center at the origin. The linear mass density \(\lambda\) of the hoop varies with angular position \(\theta\) measured from the positive \(x\)-axis according to \(\lambda(\theta) = \lambda_0 (1 + \cos^2\theta)\), where \(\lambda_0\) is a positive constant. Which of the following expressions represents the rotational inertia of the hoop about a central axis perpendicular to the plane of the hoop?

![A circular ring of radius \(R\) is centered at the origin of a two-dimensional coordinate system with a horizontal x-axis and vertical y-axis. A dashed z-axis extends perpendicular to the plane of the ring through the origin. A narrow angular segment of the ring is highlighted at an angle \(\theta\) measured counterclockwise from the positive x-axis. A radial line segment of length \(R\) extends from the origin to this highlighted segment. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829761-U5b5gW.jpg)

- **A.** \(2\pi \lambda_0 R^3\)
- **B.** \(3\pi \lambda_0 R^3\)
- **C.** \(4\pi \lambda_0 R^3\)
- **D.** \(6\pi \lambda_0 R^3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117622/*