---
title: "A small sphere of mass \\(m\\) is attached to a string of length \\(L\\). The other end of the string is attached to the center of a horizontal ceiling. A smooth, rigid vertical cylindrical wall of radius \\(R\\) (where \\(R < L\\)) is centered on the attachment point of the string. The sphere is set into a horizontal circular path so that it revolves along the inside of the cylindrical wall with a constant angular speed \\(\\omega\\). The string makes an angle \\(\\theta\\) with the vertical, as shown in Figure 1. Friction between the sphere and the wall is negligible."
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url: "https://nerd-notes.com/ubq/109535/"
date_modified: "2026-04-01T08:27:41+00:00"
---

# A small sphere of mass \(m\) is attached to a string of length \(L\). The other end of the string is attached to the center of a horizontal ceiling. A smooth, rigid vertical cylindrical wall of radius \(R\) (where \(R < L\)) is centered on the attachment point of the string. The sphere is set into a horizontal circular path so that it revolves along the inside of the cylindrical wall with a constant angular speed \(\omega\). The string makes an angle \(\theta\) with the vertical, as shown in Figure 1. Friction between the sphere and the wall is negligible.

A small sphere of mass \(m\) is attached to a string of length \(L\). The other end of the string is attached to the center of a horizontal ceiling. A smooth, rigid vertical cylindrical wall of radius \(R\) (where \(R < L\)) is centered on the attachment point of the string. The sphere is set into a horizontal circular path so that it revolves along the inside of the cylindrical wall with a constant angular speed \(\omega\). The string makes an angle \(\theta\) with the vertical, as shown in Figure 1. Friction between the sphere and the wall is negligible.

![A 2D cross-sectional side view showing a horizontal line representing a ceiling. A vertical dashed line extends downward from the center of the ceiling. A solid vertical line representing a wall is parallel to the dashed line, located a distance R to the right. A string of length L is attached to the ceiling where the dashed line begins, extending diagonally down and to the right, ending at a small sphere of mass m. The sphere is touching the vertical wall. An angle theta is marked between the vertical dashed line and the string. A circular arrow around the dashed line indicates the sphere rotates around the central axis.](https://nerd-notes.com/wp-content/uploads/ubq-diagrams/ubq-frq-generatedstem-fig-1-1775032045-WYnUl1.jpg)

**Part a)** On the dot below, which represents the sphere, **draw** and **label** the forces (not components) that are exerted on the sphere while it rotates at angular speed \(\omega\). Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. *(3 points)*

**Part b)** **Derive** an expression for the minimum angular speed \(\omega_{min}\) the sphere must have to maintain its horizontal circular path of radius \(R\) without dropping away from the wall. Express your answer in terms of \(m\), \(L\), \(R\), \(\theta\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** The sphere now moves with an angular speed \(\omega\) that is greater than \(\omega_{min}\). **Derive** an expression for the magnitude of the normal force \(F_N\) exerted on the sphere by the wall. Express your answer in terms of \(m\), \(L\), \(R\), \(\theta\), \(\omega\), and fundamental constants, as appropriate. *(2 points)*

**Part d)** The original sphere is removed and replaced with a new sphere of mass \(2m\). The new sphere is set into horizontal circular motion at the same angular speed \(\omega\) as in part (c). **Indicate** whether the magnitude of the normal force exerted by the wall on the new sphere is greater than, less than, or equal to the normal force exerted on the original sphere. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109535/*