---
title: "A solid cylinder of mass \\(M\\) and radius \\(R\\) rests on a rough horizontal table. A light string is attached to a frictionless axle at the center of the cylinder. The string passes over a light, frictionless pulley at the edge of the table and is attached to a hanging block of mass \\(m\\). The rotational inertia of a solid cylinder about its center is \\(I = \\dfrac{1}{2}MR^2\\). The system is released from rest, and the cylinder rolls without slipping along the table. Let \\(g\\) represent the acceleration due to gravity."
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url: "https://nerd-notes.com/ubq/110271/"
date_modified: "2026-04-01T02:44:28+00:00"
---

# A solid cylinder of mass \(M\) and radius \(R\) rests on a rough horizontal table. A light string is attached to a frictionless axle at the center of the cylinder. The string passes over a light, frictionless pulley at the edge of the table and is attached to a hanging block of mass \(m\). The rotational inertia of a solid cylinder about its center is \(I = \dfrac{1}{2}MR^2\). The system is released from rest, and the cylinder rolls without slipping along the table. Let \(g\) represent the acceleration due to gravity.

A solid cylinder of mass \(M\) and radius \(R\) rests on a rough horizontal table. A light string is attached to a frictionless axle at the center of the cylinder. The string passes over a light, frictionless pulley at the edge of the table and is attached to a hanging block of mass \(m\). The rotational inertia of a solid cylinder about its center is \(I = \dfrac{1}{2}MR^2\). The system is released from rest, and the cylinder rolls without slipping along the table. Let \(g\) represent the acceleration due to gravity.

![A horizontal table with a solid cylinder on top. A string connects to the exact center (axle) of the cylinder, extends horizontally to the right, goes over a small pulley at the right edge of the table, and drops vertically. A rectangular block is suspended from the vertical portion of the string. The table surface is shaded to indicate roughness. Labels 'M, R' point to the cylinder and 'm' points to the hanging block.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775011468-5wv3ig.jpg)

**Part a)** On the dots below, which represent the cylinder and the block, **draw** and **label** the forces (not components) exerted on each object. Each force must be represented by a distinct arrow starting on, and pointing away from, the appropriate dot. *(3 points)*

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

**Part c)** A student attempts to find the final speed \(v\) of the block after it has fallen a vertical distance \(d\) by applying the law of conservation of energy. The student writes the following equation: \[ mgd = \dfrac{1}{2}mv^2 + \dfrac{1}{2}Mv^2 \] *(3 points)*

**Part d)** Suppose the rough table is replaced with a completely frictionless table, but the system is otherwise identical. The cylinder now slides without rotating as it is pulled. **Indicate** whether the acceleration of the falling block on the frictionless table will be greater than, less than, or equal to the acceleration of the block when the table was rough. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using the mathematical expression you derived in part (b). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110271/*