---
title: "A hollow vertical cylinder of radius \\(R\\) rotates about its central vertical axis. A small block of mass \\(m\\) is placed against the rough inner wall of the cylinder. The coefficient of static friction between the block and the wall is \\(\\mu_s\\). The cylinder rotates at a constant angular speed \\(\\omega\\), and the block moves with the cylinder without sliding down the wall."
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url: "https://nerd-notes.com/ubq/110305/"
date_modified: "2026-04-01T05:56:30+00:00"
---

# A hollow vertical cylinder of radius \(R\) rotates about its central vertical axis. A small block of mass \(m\) is placed against the rough inner wall of the cylinder. The coefficient of static friction between the block and the wall is \(\mu_s\). The cylinder rotates at a constant angular speed \(\omega\), and the block moves with the cylinder without sliding down the wall.

A hollow vertical cylinder of radius \(R\) rotates about its central vertical axis. A small block of mass \(m\) is placed against the rough inner wall of the cylinder. The coefficient of static friction between the block and the wall is \(\mu_s\). The cylinder rotates at a constant angular speed \(\omega\), and the block moves with the cylinder without sliding down the wall.

![A 3D-style line drawing of a hollow open-topped vertical cylinder. A dashed vertical line runs perfectly down the center of the cylinder, representing the axis of rotation. A curved arrow labeled '\(\omega\)' loops around the top of this dashed central axis to indicate rotation. On the right inner wall of the cylinder, halfway down, a small rectangular block labeled '\(m\)' is resting against the inner surface. A horizontal dashed line connects the central axis to the block, with the label '\(R\)' placed above the dashed line to indicate the radius.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775022989-zsZFh6.jpg)

**Part a)** On the dot below, which represents the block, **draw** and **label** the forces (not components) that are exerted on the block. 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 cylinder must have such that the block does not slide down the wall. Express your answer in terms of \(m\), \(R\), \(\mu_s\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** The cylinder is now rotated at a new constant angular speed \(\omega_2\) such that \(\omega_2 = 2\omega_{min}\). **Indicate** how the magnitude of the actual frictional force \(f_s\) exerted on the block at this new speed compares to the weight of the block \(mg\). - [ ] \(f_s < mg\) - [ ] \(f_s = mg\) - [ ] \(f_s > mg\) **Justify** your answer using physical principles. *(2 points)*

**Part d)** The original block is removed and replaced by a second block of mass \(2m\). The coefficient of static friction between the second block and the wall is still \(\mu_s\). **Indicate** how the new minimum angular speed \(\omega_{new}\) required to keep the second block from sliding down compares to the original minimum angular speed \(\omega_{min}\). - [ ] \(\omega_{new} > \omega_{min}\) - [ ] \(\omega_{new} < \omega_{min}\) - [ ] \(\omega_{new} = \omega_{min}\) **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/110305/*