---
title: "A block of mass \\(m\\) is placed on a horizontal track and pushed against an ideal spring of spring constant \\(k\\), compressing it a distance \\(D\\). The block is released from rest. The track is frictionless except for a rough section of length \\(L\\) with a coefficient of kinetic friction \\(\\mu_k\\). After passing the rough section, the block slides up a frictionless incline of angle \\(\\theta\\). The block remains on the track at all times, and you may assume the block’s size is negligible. The block completely passes through the rough section and reaches a maximum vertical height \\(H\\) on the incline."
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url: "https://nerd-notes.com/ubq/110127/"
date_modified: "2026-03-27T05:29:30+00:00"
---

# A block of mass \(m\) is placed on a horizontal track and pushed against an ideal spring of spring constant \(k\), compressing it a distance \(D\). The block is released from rest. The track is frictionless except for a rough section of length \(L\) with a coefficient of kinetic friction \(\mu_k\). After passing the rough section, the block slides up a frictionless incline of angle \(\theta\). The block remains on the track at all times, and you may assume the block’s size is negligible. The block completely passes through the rough section and reaches a maximum vertical height \(H\) on the incline.

A block of mass \(m\) is placed on a horizontal track and pushed against an ideal spring of spring constant \(k\), compressing it a distance \(D\). The block is released from rest. The track is frictionless except for a rough section of length \(L\) with a coefficient of kinetic friction \(\mu_k\). After passing the rough section, the block slides up a frictionless incline of angle \(\theta\). The block remains on the track at all times, and you may assume the block's size is negligible. The block completely passes through the rough section and reaches a maximum vertical height \(H\) on the incline.

![A horizontal track that transitions into a rightward upward-sloping incline. On the far left of the horizontal track, a vertical wall has a spring attached to it. The right end of the spring is touching a rectangular block labeled 'm'. A horizontal bracket labeled 'D' indicates the compression of the spring. Further to the right on the horizontal track is a shaded rectangular region labeled 'Rough patch'. A horizontal bracket above the rough patch is labeled 'L'. To the right of the rough patch, the track slants upward. An angle arc labeled '\theta' is shown between the horizontal dashed line and the slanted incline. A dashed outline of the block is shown at a vertical height 'H' on the incline to indicate its maximum position.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774589369-KlQj5m.jpg)

**Part a)** **Derive** an expression for the speed \(v\) of the block immediately after it loses contact with the spring, but before it reaches the rough patch. Express your answer in terms of \(m\), \(k\), \(D\), and fundamental constants, as appropriate. *(2 points)*

**Part b)** **Derive** an expression for the maximum vertical height \(H\) the block reaches on the incline. Express your answer in terms of \(m\), \(k\), \(D\), \(L\), \(\mu_k\), \(\theta\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** Consider the system consisting only of the block and Earth. **Derive** an expression for the net external work done on this system from the instant the block is released to the instant it reaches its maximum height. Express your answer in terms of \(m\), \(k\), \(D\), \(L\), \(\mu_k\), and fundamental constants, as appropriate. *(2 points)*

**Part d)** The original block is removed and replaced by a block of mass \(2m\). The spring is again compressed by a distance \(D\) and the heavier block is released. Assume the heavier block has enough energy to completely pass through the rough patch. *(5 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110127/*