---
title: "A light, inextensible string is wrapped around the outer rim of a uniform solid cylinder of mass \\(M\\) and radius \\(R\\) that is mounted on a fixed horizontal axle through its center. The free end of the string is attached to a block of mass \\(m\\). A constant frictional torque \\(\\tau_f\\) opposes the rotation of the cylinder about the axle. The block is released from rest and the string unwinds without slipping. What is the magnitude of the downward linear acceleration of the block in terms of \\(m\\), \\(M\\), \\(R\\), \\(\\tau_f\\), and fundamental constants?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/120835/"
date_modified: "2026-08-23T04:43:32+00:00"
---

# A light, inextensible string is wrapped around the outer rim of a uniform solid cylinder of mass \(M\) and radius \(R\) that is mounted on a fixed horizontal axle through its center. The free end of the string is attached to a block of mass \(m\). A constant frictional torque \(\tau_f\) opposes the rotation of the cylinder about the axle. The block is released from rest and the string unwinds without slipping. What is the magnitude of the downward linear acceleration of the block in terms of \(m\), \(M\), \(R\), \(\tau_f\), and fundamental constants?

A light, inextensible string is wrapped around the outer rim of a uniform solid cylinder of mass \(M\) and radius \(R\) that is mounted on a fixed horizontal axle through its center. The free end of the string is attached to a block of mass \(m\). A constant frictional torque \(\tau_f\) opposes the rotation of the cylinder about the axle. The block is released from rest and the string unwinds without slipping. What is the magnitude of the downward linear acceleration of the block in terms of \(m\), \(M\), \(R\), \(\tau_f\), and fundamental constants?

![A side-view diagram shows a circular disk of radius \(R\) and mass \(M\) mounted on a central horizontal axle. A vertical string extends downward tangentially from the rightmost edge of the disk to a hanging square block of mass \(m\). A curved arrow labeled \(\tau_f\) wraps counterclockwise around the central axle. An arrow labeled \(g\) points vertically downward next to the block. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460211-axDUoI.jpg)

- **A.** \(\dfrac{mg + \dfrac{\tau_f}{R}}{m + \dfrac{1}{2}M}\)
- **B.** \(\dfrac{mg - \dfrac{\tau_f}{R}}{m + M}\)
- **C.** \(\dfrac{mg - \dfrac{\tau_f}{R}}{m}\)
- **D.** \(\dfrac{mg - \dfrac{\tau_f}{R}}{m + \dfrac{1}{2}M}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120835/*