---
title: "Spring and Pulley System"
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url: "https://nerd-notes.com/ubq/109086/"
date_modified: "2026-03-16T10:46:23+00:00"
---

# Spring and Pulley System

A block of mass \(m_1\) rests on a rough horizontal table with a coefficient of kinetic friction \(\mu_k\). The left side of the block is attached to a wall by an ideal spring of spring constant \(k\). The right side of the block is attached to a lightweight string that passes over an ideal, frictionless pulley at the edge of the table and connects to a hanging block of mass \(m_2\). The system is held at rest with the spring at its natural, unstretched length. At time \(t = 0\), the system is released and the blocks begin to move. Assume that \(m_2 g > \mu_k m_1 g\) so that the system moves upon release.

**Part a)** **Derive** an expression for the magnitude of the initial acceleration of the blocks immediately after they are released. Express your answer in terms of \(m_1\), \(m_2\), \(\mu_k\), \(k\), and physical constants, as appropriate. *(3 points)*

**Part b)** As the blocks move, they reach a maximum speed before eventually slowing down. **Derive** an expression for the distance \(D\) that block \(m_1\) has moved from its initial position at the instant the system reaches its maximum speed. Express your answer in terms of \(m_1\), \(m_2\), \(\mu_k\), \(k\), and physical constants, as appropriate. *(2 points)*

**Part c)** The blocks eventually momentarily come to rest. **Derive** an expression for the maximum distance \(x_{max}\) that the spring is stretched. Express your answer in terms of \(m_1\), \(m_2\), \(\mu_k\), \(k\), and physical constants, as appropriate. *(3 points)*

**Part d)** A student replaces the rough table with a frictionless table and repeats the experiment, releasing the system from rest with the spring unstretched. **Indicate** whether the new maximum distance the spring stretches, \(x_{new}\), is greater than, less than, or equal to \(x_{max}\). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using physical principles. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109086/*