---
title: "A block of mass \\(m\\) is released from rest on an inclined plane that makes an angle \\(\\theta\\) with the horizontal. The surface of the incline is rough, with a coefficient of kinetic friction \\(\\mu_k\\) and a coefficient of static friction \\(\\mu_s\\) between the block and the incline. An ideal spring of spring constant \\(k\\) is fixed at the bottom of the incline. The block is released from a distance \\(L\\), measured along the incline, from the uncompressed end of the spring."
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/112700/"
date_modified: "2026-04-30T20:51:32+00:00"
---

# A block of mass \(m\) is released from rest on an inclined plane that makes an angle \(\theta\) with the horizontal. The surface of the incline is rough, with a coefficient of kinetic friction \(\mu_k\) and a coefficient of static friction \(\mu_s\) between the block and the incline. An ideal spring of spring constant \(k\) is fixed at the bottom of the incline. The block is released from a distance \(L\), measured along the incline, from the uncompressed end of the spring.

A block of mass \(m\) is released from rest on an inclined plane that makes an angle \(\theta\) with the horizontal. The surface of the incline is rough, with a coefficient of kinetic friction \(\mu_k\) and a coefficient of static friction \(\mu_s\) between the block and the incline. An ideal spring of spring constant \(k\) is fixed at the bottom of the incline. The block is released from a distance \(L\), measured along the incline, from the uncompressed end of the spring.

![A side-view diagram of a physical setup. A right-angled triangular wedge is shown with its hypotenuse representing an inclined plane. The angle between the horizontal base and the incline is labeled \(\theta\). A rectangular block labeled 'm' rests on the upper part of the incline. At the bottom of the incline, a coiled spring is attached to a solid perpendicular stop. The distance from the bottom edge of the block to the top uncompressed edge of the spring is marked with a dashed dimension line labeled 'L'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1777582292-io6W2O.jpg)

**Part a)** **Derive** an expression for the magnitude of the acceleration \(a\) of the block as it slides down the incline before reaching the spring. Express your answer in terms of \(m\), \(\theta\), \(\mu_k\), \(g\), and fundamental constants. *(3 points)*

**Part b)** Suppose the experiment is repeated, but the angle \(\theta\) of the incline is decreased. The block is still able to slide down the incline. **Indicate** whether the magnitude of the block's acceleration down the incline will increase, decrease, or stay the same. - [ ] Increase - [ ] Decrease - [ ] Stay the same **Justify** your answer using physical principles. *(3 points)*

**Part c)** Returning to the original angle \(\theta\), **derive** an expression for the kinetic energy \(K\) of the block exactly at the moment it makes contact with the uncompressed spring. Express your answer in terms of \(m\), \(\theta\), \(\mu_k\), \(L\), \(g\), and fundamental constants. *(3 points)*

**Part d)** The block hits the spring, compresses it by a maximum distance \(d\), and momentarily comes to rest. Because the spring pushes back on the block, the block might rebound up the incline. **Derive** an expression for the minimum coefficient of static friction \(\mu_s\) required so that the block remains at rest at the position of maximum compression \(d\) and does not rebound. Express your answer in terms of \(m\), \(\theta\), \(k\), \(d\), \(g\), and fundamental constants. *(4 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/112700/*