---
title: "Block A of mass \\(m_A\\) and Block B of mass \\(m_B\\) (where \\(m_B > m_A\\)) are connected by a string of negligible mass that passes over a uniform solid cylindrical pulley of mass \\(M\\) and radius \\(R\\). The string does not slip on the pulley, and the pulley rotates about a horizontal, frictionless axle through its center. The rotational inertia of a uniform solid cylinder is \\(I = \\dfrac{1}{2}MR^2\\). The system is released from rest."
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url: "https://nerd-notes.com/ubq/109528/"
date_modified: "2026-03-25T04:15:18+00:00"
---

# Block A of mass \(m_A\) and Block B of mass \(m_B\) (where \(m_B > m_A\)) are connected by a string of negligible mass that passes over a uniform solid cylindrical pulley of mass \(M\) and radius \(R\). The string does not slip on the pulley, and the pulley rotates about a horizontal, frictionless axle through its center. The rotational inertia of a uniform solid cylinder is \(I = \dfrac{1}{2}MR^2\). The system is released from rest.

Block A of mass \(m_A\) and Block B of mass \(m_B\) (where \(m_B > m_A\)) are connected by a string of negligible mass that passes over a uniform solid cylindrical pulley of mass \(M\) and radius \(R\). The string does not slip on the pulley, and the pulley rotates about a horizontal, frictionless axle through its center. The rotational inertia of a uniform solid cylinder is \(I = \dfrac{1}{2}MR^2\). The system is released from rest.

![A physical setup diagram showing an Atwood machine. A circular pulley of radius R is supported at its center by a bracket attached to a ceiling. A string hangs over the pulley. On the left side of the string, a rectangular box labeled 'Block A' is attached. On the right side of the string, a slightly larger rectangular box labeled 'Block B' is attached. Block B is positioned lower than Block A.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774412117-RDIhbp.jpg)

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

**Part b)** **Derive** an expression for the magnitude of the acceleration of Block B after the system is released. Express your answer in terms of \(m_A\), \(m_B\), \(M\), and physical constants, as appropriate. *(4 points)*

**Part c)** **Derive** an expression for the difference in the magnitudes of the string tensions, \((T_B - T_A)\), where \(T_B\) is the tension in the segment of string attached to Block B, and \(T_A\) is the tension in the segment attached to Block A. Express your answer in terms of \(m_A\), \(m_B\), \(M\), and physical constants, as appropriate. *(2 points)*

**Part d)** The solid cylindrical pulley is now replaced with a thin hoop of the same mass \(M\) and radius \(R\). The blocks are reset to their initial positions and released from rest. **Predict** whether the new downward acceleration of Block B is greater than, less than, or equal to the acceleration derived in part (b). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your reasoning. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109528/*