---
title: "Block 1 of mass \\(m_1\\) is released from rest at the top of a rough inclined plane of length \\(L\\) that makes an angle \\(\\theta\\) with the horizontal. The coefficient of kinetic friction between Block 1 and the incline is \\(\\mu_k\\). At the bottom of the incline, Block 1 smoothly transitions to a horizontal, frictionless surface.   Block 1 then collides with and sticks to Block 2 of mass \\(m_2\\), which is initially at rest and attached to an ideal, uncompressed spring of spring constant \\(k\\). The other end of the spring is attached to a rigid wall."
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url: "https://nerd-notes.com/ubq/109538/"
date_modified: "2026-04-01T08:13:19+00:00"
---

# Block 1 of mass \(m_1\) is released from rest at the top of a rough inclined plane of length \(L\) that makes an angle \(\theta\) with the horizontal. The coefficient of kinetic friction between Block 1 and the incline is \(\mu_k\). At the bottom of the incline, Block 1 smoothly transitions to a horizontal, frictionless surface. 

Block 1 then collides with and sticks to Block 2 of mass \(m_2\), which is initially at rest and attached to an ideal, uncompressed spring of spring constant \(k\). The other end of the spring is attached to a rigid wall.

Block 1 of mass \(m_1\) is released from rest at the top of a rough inclined plane of length \(L\) that makes an angle \(\theta\) with the horizontal. The coefficient of kinetic friction between Block 1 and the incline is \(\mu_k\). At the bottom of the incline, Block 1 smoothly transitions to a horizontal, frictionless surface. 

Block 1 then collides with and sticks to Block 2 of mass \(m_2\), which is initially at rest and attached to an ideal, uncompressed spring of spring constant \(k\). The other end of the spring is attached to a rigid wall.

![A block labeled '\(m_1\)' is at the top of an inclined plane of length '\(L\)' and angle '\(\theta\)' above the horizontal. The incline is shaded to indicate a rough surface. The bottom of the incline connects smoothly to a horizontal surface. On the horizontal surface, a block labeled '\(m_2\)' is attached to a coiled spring with spring constant '\(k\)'. The other end of the spring is attached to a rigid vertical wall on the right. The horizontal surface has a smooth line indicating it is frictionless.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774412295-ly6DcL.jpg)

**Part a)** **Draw** and label the forces (not components) that act on Block 1 while it is sliding down the incline. Draw each force as a distinct arrow starting on, and pointing away from, the dot. *(3 points)*

**Part b)** **Derive** an expression for the speed \(v_1\) of Block 1 at the bottom of the incline, just before it collides with Block 2. Express your answer in terms of \(m_1\), \(L\), \(\theta\), \(\mu_k\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** **Derive** an expression for the maximum compression \(x_{\text{max}}\) of the spring after the collision. Express your answer in terms of \(v_1\), \(m_1\), \(m_2\), \(k\), and fundamental constants, as appropriate. *(3 points)*

**Part d)** Suppose the experiment is repeated, but the mass of Block 2 is changed to \(2m_2\). Block 1 is again released from rest at the top of the incline. **Indicate** whether the new maximum compression of the spring will be greater than, less than, or equal to the original maximum compression \(x_{\text{max}}\). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using proportional reasoning and physical principles, without deriving a new, complete algebraic expression. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109538/*