---
title: "A thin, uniform wire of radius \\(R\\) and linear mass density \\(\\lambda\\) is bent into a semicircle. An axis of rotation lies in the plane of the wire and passes through both endpoints of the semicircle. The angle \\(\\theta\\) is measured from the axis of rotation to a segment of the wire, as shown. Which of the following integrals correctly represents the rotational inertia of the wire about this axis?"
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url: "https://nerd-notes.com/ubq/124227/"
date_modified: "2026-09-28T14:04:19+00:00"
---

# A thin, uniform wire of radius \(R\) and linear mass density \(\lambda\) is bent into a semicircle. An axis of rotation lies in the plane of the wire and passes through both endpoints of the semicircle. The angle \(\theta\) is measured from the axis of rotation to a segment of the wire, as shown. Which of the following integrals correctly represents the rotational inertia of the wire about this axis?

A thin, uniform wire of radius \(R\) and linear mass density \(\lambda\) is bent into a semicircle. An axis of rotation lies in the plane of the wire and passes through both endpoints of the semicircle. The angle \(\theta\) is measured from the axis of rotation to a segment of the wire, as shown. Which of the following integrals correctly represents the rotational inertia of the wire about this axis?

![A horizontal dashed line represents the axis of rotation. Resting on this horizontal line is a semicircular curved line of radius \(R\) centered at an origin point indicated by a small solid dot on the horizontal line. The semicircle curves upward into the upper half-plane, with its left endpoint and right endpoint touching the horizontal dashed line at distances \(R\) from the center dot. A straight solid line segment extends from the center dot to a point on the semicircle at an angle \(\theta\) above the right side of the horizontal line, with an arc labeled \(\theta\) indicating this angle. The straight line segment has length labeled \(R\). A small arc segment of the wire at this position is shaded dark gray to represent a mass element \(dm\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604259-oDCJue.jpg)

- **A.** \(\displaystyle \int_0^{\pi} \lambda R^3 \sin^2\theta \, d\theta\)
- **B.** \(\displaystyle \int_0^{\pi/2} \lambda R^3 \sin^2\theta \, d\theta\)
- **C.** \(\displaystyle \int_0^{\pi} \lambda R^3 \, d\theta\)
- **D.** \(\displaystyle \int_0^{2\pi} \lambda R^3 \sin^2\theta \, d\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124227/*