---
title: "A non-uniform ladder of length \\(L\\) and total mass \\(M\\) leans against a frictionless vertical wall, making an angle \\(\\theta\\) with a rough horizontal floor. The ladder has a linear mass density that increases linearly from the base according to \\(\\lambda(x) = \\dfrac{2M}{L^2}x\\), where \\(x\\) is the distance measured along the ladder from the base. The coefficient of static friction between the base of the ladder and the floor is \\(\\mu_s\\). In terms of the given quantities and fundamental constants, what is the maximum distance \\(d\\) along the ladder that a painter of mass \\(m\\) can climb from the base before the ladder begins to slip?"
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/124420/"
date_modified: "2026-09-28T14:05:18+00:00"
---

# A non-uniform ladder of length \(L\) and total mass \(M\) leans against a frictionless vertical wall, making an angle \(\theta\) with a rough horizontal floor. The ladder has a linear mass density that increases linearly from the base according to \(\lambda(x) = \dfrac{2M}{L^2}x\), where \(x\) is the distance measured along the ladder from the base. The coefficient of static friction between the base of the ladder and the floor is \(\mu_s\). In terms of the given quantities and fundamental constants, what is the maximum distance \(d\) along the ladder that a painter of mass \(m\) can climb from the base before the ladder begins to slip?

A non-uniform ladder of length \(L\) and total mass \(M\) leans against a frictionless vertical wall, making an angle \(\theta\) with a rough horizontal floor. The ladder has a linear mass density that increases linearly from the base according to \(\lambda(x) = \dfrac{2M}{L^2}x\), where \(x\) is the distance measured along the ladder from the base. The coefficient of static friction between the base of the ladder and the floor is \(\mu_s\). In terms of the given quantities and fundamental constants, what is the maximum distance \(d\) along the ladder that a painter of mass \(m\) can climb from the base before the ladder begins to slip?

![A side-view diagram shows a straight thick line segment representing a ladder of length \(L\) leaning against a vertical wall on the right and resting on a horizontal floor at the bottom left. The floor and wall meet at a perpendicular corner. The base of the ladder contacts the floor at an angle \(\theta\) above the horizontal floor. The top of the ladder contacts the vertical wall at height \(L\sin\theta\). A coordinate axis labeled \(x\) runs parallel to the ladder, pointing upward from the base where \(x = 0\) toward the top where \(x = L\). A shaded circle representing a person of mass \(m\) is positioned on the ladder at a distance \(d\) along the ladder from the base. A double-headed dimension arrow parallel to the ladder indicates the distance \(d\) from the floor contact point to the circle. No other labels, lines, text, or forces appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604318-edQC0h.jpg)

- **A.** \(\left[ \mu_s\left(1 + \dfrac{M}{m}\right)\tan\theta - \dfrac{M}{2m}\right] L\)
- **B.** \(\left[ \mu_s\left(1 + \dfrac{M}{m}\right)\tan\theta - \dfrac{2M}{3m}\right] L\)
- **C.** \(\left[ \mu_s\left(1 + \dfrac{M}{m}\right)\tan\theta - \dfrac{M}{3m}\right] L\)
- **D.** \(\left[ \mu_s\left(1 + \dfrac{M}{m}\right)\cot\theta - \dfrac{2M}{3m}\right] L\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124420/*