---
title: "A uniform solid cylinder of mass \\(M\\) and radius \\(R\\) has a light string wrapped multiple times around its circumference. The free end of the string is attached to a fixed horizontal ceiling. The cylinder is held at rest and then released, falling vertically as the string unwinds without slipping. The rotational inertia of a solid cylinder about its center of mass is \\(I = \\dfrac{1}{2}MR^2\\). After the center of mass of the cylinder has fallen a vertical distance \\(h\\), what is the speed of the center of mass of the cylinder?"
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url: "https://nerd-notes.com/ubq/112315/"
date_modified: "2026-04-23T01:42:30+00:00"
---

# A uniform solid cylinder of mass \(M\) and radius \(R\) has a light string wrapped multiple times around its circumference. The free end of the string is attached to a fixed horizontal ceiling. The cylinder is held at rest and then released, falling vertically as the string unwinds without slipping. The rotational inertia of a solid cylinder about its center of mass is \(I = \dfrac{1}{2}MR^2\). After the center of mass of the cylinder has fallen a vertical distance \(h\), what is the speed of the center of mass of the cylinder?

A uniform solid cylinder of mass \(M\) and radius \(R\) has a light string wrapped multiple times around its circumference. The free end of the string is attached to a fixed horizontal ceiling. The cylinder is held at rest and then released, falling vertically as the string unwinds without slipping. The rotational inertia of a solid cylinder about its center of mass is \(I = \dfrac{1}{2}MR^2\). After the center of mass of the cylinder has fallen a vertical distance \(h\), what is the speed of the center of mass of the cylinder?

![A vertical view of a cylinder hanging from a ceiling. A thin line representing a string is fixed to the ceiling and extends downward, wrapping around the left side of a gray solid cylinder. The cylinder is shown at two positions: an initial top position and a lower position labeled with a downward arrow of length h pointing to the center of the cylinder. A radius R is drawn from the center of the cylinder to its edge.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1776908550-qC65PV.jpg)

- **A.** \(\sqrt{gh}\)
- **B.** \(\sqrt{\dfrac{4gh}{3}}\)
- **C.** \(\sqrt{\dfrac{3gh}{2}}\)
- **D.** \(\sqrt{2gh}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/112315/*