---
title: "A uniform solid cylinder of mass \\(M\\) and radius \\(R\\) is placed on a rough horizontal surface. A light string is attached to a frictionless axle that passes through the center of mass of the cylinder. The string is pulled horizontally to the right, passes over a light, frictionless pulley at the edge of the surface, and is attached to a hanging block of mass \\(m\\), as shown in Figure 1. The rotational inertia of a uniform solid cylinder about its center is \\(I = \\dfrac{1}{2}MR^2\\). The system is released from rest, and the cylinder rolls to the right without slipping."
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url: "https://nerd-notes.com/ubq/110139/"
date_modified: "2026-04-01T07:59:06+00:00"
---

# A uniform solid cylinder of mass \(M\) and radius \(R\) is placed on a rough horizontal surface. A light string is attached to a frictionless axle that passes through the center of mass of the cylinder. The string is pulled horizontally to the right, passes over a light, frictionless pulley at the edge of the surface, and is attached to a hanging block of mass \(m\), as shown in Figure 1. The rotational inertia of a uniform solid cylinder about its center is \(I = \dfrac{1}{2}MR^2\). The system is released from rest, and the cylinder rolls to the right without slipping.

A uniform solid cylinder of mass \(M\) and radius \(R\) is placed on a rough horizontal surface. A light string is attached to a frictionless axle that passes through the center of mass of the cylinder. The string is pulled horizontally to the right, passes over a light, frictionless pulley at the edge of the surface, and is attached to a hanging block of mass \(m\), as shown in Figure 1. The rotational inertia of a uniform solid cylinder about its center is \(I = \dfrac{1}{2}MR^2\). The system is released from rest, and the cylinder rolls to the right without slipping.

![A side-view diagram of a physical setup. A solid cylinder is on a horizontal surface representing a table. A string is attached to the exact center (axis) of the cylinder, extending horizontally to the right. The string goes over a small circular pulley mounted at the right edge of the table. From the pulley, the string extends vertically downward and is attached to a square block hanging below the table edge. The cylinder is labeled 'M', its radius is labeled 'R', and the hanging block is labeled 'm'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774589552-TjN0lF.jpg)

**Part a)** The circle below represents the solid cylinder. **Draw** and **label** the forces (not components) that are exerted on the cylinder as it rolls. Each force must be represented by a distinct arrow starting on, and pointing away from, the specific point on the cylinder where the force is exerted. *(3 points)*

**Part b)** **Derive** an expression for the magnitude of the linear acceleration of the block. Express your answer in terms of \(M\), \(m\), and physical constants, as appropriate. *(4 points)*

**Part c)** The solid cylinder is now replaced with a thin hollow hoop of the same mass \(M\) and radius \(R\). The experiment is repeated. **Indicate** whether the magnitude of the acceleration of the block is greater than, less than, or equal to the acceleration determined in part (b). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your reasoning using physical principles. *(3 points)*

**Part d)** Return to the original setup with the solid cylinder. The system is once again released from rest. **Derive** an expression for the final speed \(v\) of the block after it has fallen a vertical distance \(h\), using conservation of energy principles. Express your answer in terms of \(M\), \(m\), \(h\), and fundamental constants. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110139/*