---
title: "Block A of mass \\(m_A\\) rests on a rough horizontal table. It is attached to a vertical wall on the left by an ideal spring of spring constant \\(k\\). A light string attached to the right side of Block A passes over an ideal pulley at the edge of the table and is attached to Block B of mass \\(m_B\\), which hangs vertically. The horizontal surface exerts a constant kinetic friction force of magnitude \\(f_k\\) on Block A as it moves. The system is initially held at rest with the spring at its unstretched equilibrium length. The system is then released. Block B falls a maximum distance \\(D\\) before momentarily coming to rest."
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url: "https://nerd-notes.com/ubq/109229/"
date_modified: "2026-03-30T09:43:37+00:00"
---

# Block A of mass \(m_A\) rests on a rough horizontal table. It is attached to a vertical wall on the left by an ideal spring of spring constant \(k\). A light string attached to the right side of Block A passes over an ideal pulley at the edge of the table and is attached to Block B of mass \(m_B\), which hangs vertically. The horizontal surface exerts a constant kinetic friction force of magnitude \(f_k\) on Block A as it moves. The system is initially held at rest with the spring at its unstretched equilibrium length. The system is then released. Block B falls a maximum distance \(D\) before momentarily coming to rest.

Block A of mass \(m_A\) rests on a rough horizontal table. It is attached to a vertical wall on the left by an ideal spring of spring constant \(k\). A light string attached to the right side of Block A passes over an ideal pulley at the edge of the table and is attached to Block B of mass \(m_B\), which hangs vertically. The horizontal surface exerts a constant kinetic friction force of magnitude \(f_k\) on Block A as it moves. The system is initially held at rest with the spring at its unstretched equilibrium length. The system is then released. Block B falls a maximum distance \(D\) before momentarily coming to rest.

![A horizontal tabletop. On the table is a rectangular box labeled 'Block A' with mass m_A. A coiled spring labeled 'k' connects a vertical wall on the left to the left side of Block A. A string is attached to the right side of Block A, runs horizontally to the right over a circular pulley mounted at the right edge of the table, and goes straight down to a hanging rectangular box labeled 'Block B' with mass m_B. The table surface is hatched with diagonal lines to indicate a rough surface, with a label f_k near Block A.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774091729-gCg2uI.jpg)

**Part a)** **Complete** the energy bar chart below for the system consisting of Block A, Block B, the spring, and Earth, from the initial moment the system is released to the final moment Block B has fallen the maximum distance \(D\). Assume the gravitational potential energy of the system is zero when Block B is at its lowest position (distance \(D\) below its starting point). Let \(W_{nc}\) represent the work done by nonconservative forces. You may use letters, shaded bars, or check marks to indicate whether each quantity is zero, positive, or negative. | Initial Kinetic Energy \(K_i\) | Initial Grav. Potential \(U_{gi}\) | Initial Spring Potential \(U_{si}\) | Work Done \(W_{nc}\) | Final Kinetic Energy \(K_f\) | Final Grav. Potential \(U_{gf}\) | Final Spring Potential \(U_{sf}\) | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | | | | | | | | *(3 points)*

**Part b)** **Derive** an expression for the maximum distance \(D\) that Block B falls. Express your answer in terms of \(m_A\), \(m_B\), \(k\), \(f_k\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** The students want to determine the total kinetic energy of the two-block system at an arbitrary intermediate point during the fall. **Derive** an expression for the total kinetic energy \(K\) of the system when Block B has fallen a distance \(x\), where \(0 < x < D\). Express your answer in terms of \(m_A\), \(m_B\), \(k\), \(f_k\), \(x\), and fundamental constants, as appropriate. *(3 points)*

**Part d)** **Predict** whether the magnitude of the instantaneous rate at which mechanical energy is dissipated by friction is greater when Block B has fallen a distance \(x = D/4\), greater when it has fallen a distance \(x = 3D/4\), or the same at both positions. - [ ] Greater at \(x = D/4\) - [ ] Greater at \(x = 3D/4\) - [ ] The same at both positions **Justify** your answer using physical principles and your expression from part (c). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109229/*