---
title: "A smooth circular hoop of radius \\(R = 0.40 \\text{ m}\\) is oriented vertically and rotates at a constant angular speed \\(\\omega\\) about its vertical central diameter. A small bead is threaded onto the hoop and slides along it with negligible friction. When the hoop rotates, the bead maintains a constant position on the hoop at an angle \\(\\theta = 60^\\circ\\) measured from the lowest point of the hoop. What is the angular speed \\(\\omega\\) of the hoop?"
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url: "https://nerd-notes.com/ubq/120603/"
date_modified: "2026-08-23T04:41:54+00:00"
---

# A smooth circular hoop of radius \(R = 0.40 \text{ m}\) is oriented vertically and rotates at a constant angular speed \(\omega\) about its vertical central diameter. A small bead is threaded onto the hoop and slides along it with negligible friction. When the hoop rotates, the bead maintains a constant position on the hoop at an angle \(\theta = 60^\circ\) measured from the lowest point of the hoop. What is the angular speed \(\omega\) of the hoop?

A smooth circular hoop of radius \(R = 0.40 \text{ m}\) is oriented vertically and rotates at a constant angular speed \(\omega\) about its vertical central diameter. A small bead is threaded onto the hoop and slides along it with negligible friction. When the hoop rotates, the bead maintains a constant position on the hoop at an angle \(\theta = 60^\circ\) measured from the lowest point of the hoop. What is the angular speed \(\omega\) of the hoop?

![A vertical circle of radius \(R\) representing a wire hoop is centered on a vertical dashed line representing the axis of rotation. A small solid circular dot representing a bead is located on the lower-right section of the hoop. A dashed radial line connects the center of the hoop to the bead. A vertical dashed reference line extends downward from the center of the hoop. The angle between the downward vertical reference line and the radial line to the bead is labeled \(\theta = 60^\circ\). A curved arrow around the vertical axis at the top indicates rotation with angular speed \(\omega\). The radius from the center to the top edge of the hoop is labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460114-l31N75.jpg)

- **A.** \(3.5 \text{ rad/s}\)
- **B.** \(7.0 \text{ rad/s}\)
- **C.** \(9.9 \text{ rad/s}\)
- **D.** \(49 \text{ rad/s}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120603/*