---
title: "Block 1 of mass \\(m_1\\) is placed on a rough horizontal tabletop with a coefficient of kinetic friction \\(\\mu_k\\). A light, unstretchable string is attached to Block 1, passes over a pulley, and is attached to hanging Block 2 of mass \\(m_2\\). The pulley has a rotational inertia \\(I\\) and radius \\(R\\), and it rotates on a frictionless axle. The string does not slip on the pulley. The system is released from rest and Block 2 accelerates downward."
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url: "https://nerd-notes.com/ubq/109532/"
date_modified: "2026-03-25T04:16:10+00:00"
---

# Block 1 of mass \(m_1\) is placed on a rough horizontal tabletop with a coefficient of kinetic friction \(\mu_k\). A light, unstretchable string is attached to Block 1, passes over a pulley, and is attached to hanging Block 2 of mass \(m_2\). The pulley has a rotational inertia \(I\) and radius \(R\), and it rotates on a frictionless axle. The string does not slip on the pulley. The system is released from rest and Block 2 accelerates downward.

Block 1 of mass \(m_1\) is placed on a rough horizontal tabletop with a coefficient of kinetic friction \(\mu_k\). A light, unstretchable string is attached to Block 1, passes over a pulley, and is attached to hanging Block 2 of mass \(m_2\). The pulley has a rotational inertia \(I\) and radius \(R\), and it rotates on a frictionless axle. The string does not slip on the pulley. The system is released from rest and Block 2 accelerates downward.

![A side-view diagram showing a horizontal table. Block 1, labeled 'm_1', rests on the left side of the table. A string is attached to the right side of Block 1, extends horizontally to the right, and passes over a solid disk pulley of radius R located at the right edge of the table. The string then goes straight down and attaches to Block 2, labeled 'm_2', which is hanging in the air.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774412169-LS3erG.jpg)

**Part a)** On the dots below, which represent Block 1, the pulley, and Block 2, **draw** and **label** the forces (not components) that are exerted on each object. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. *(3 points)*

**Part b)** **Derive** an expression for the magnitude of the linear acceleration \(a\) of the blocks. Express your answer in terms of \(m_1\), \(m_2\), \(I\), \(R\), \(\mu_k\), and physical constants, as appropriate. *(4 points)*

**Part c)** Let \(T_1\) be the tension in the horizontal section of the string and \(T_2\) be the tension in the vertical section of the string. *(3 points)*

**Part d)** The original pulley is removed and replaced with a new pulley that has the same mass \(M\) and radius \(R\), but its mass is concentrated near its outer rim (a hoop) rather than being distributed uniformly (a solid disk). The system is again released from rest. **Predict** whether the magnitude of the acceleration of the blocks will be greater than, less than, or equal to the acceleration of the blocks in the original setup. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109532/*