---
title: "A uniform ladder with mass \\( m_2 \\) and length \\( L \\) rests against a smooth wall. A do-it-yourself enthusiast of mass \\( m_1 \\) stands on the ladder a distance \\( d \\) from the bottom (measured along the ladder). The ladder makes an angle \\( \\theta \\) with the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude \\( f \\) between the floor and the ladder.\\( N_1 \\) is the magnitude of the normal force exerted by the wall on the ladder, and \\( N_2 \\) is the magnitude of the normal force exerted by the ground on the ladder. Throughout the problem, consider counterclockwise torques to be positive."
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url: "https://nerd-notes.com/ubq/47231/"
date_modified: "2025-10-25T01:30:33+00:00"
---

# A uniform ladder with mass \( m_2 \) and length \( L \) rests against a smooth wall. A do-it-yourself enthusiast of mass \( m_1 \) stands on the ladder a distance \( d \) from the bottom (measured along the ladder). The ladder makes an angle \( \theta \) with the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude \( f \) between the floor and the ladder.\( N_1 \) is the magnitude of the normal force exerted by the wall on the ladder, and \( N_2 \) is the magnitude of the normal force exerted by the ground on the ladder. Throughout the problem, consider counterclockwise torques to be positive.

A uniform ladder with mass \( m_2 \) and length \( L \) rests against a smooth wall. A do-it-yourself enthusiast of mass \( m_1 \) stands on the ladder a distance \( d \) from the bottom (measured along the ladder). The ladder makes an angle \( \theta \) with the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude \( f \) between the floor and the ladder. \( N_1 \) is the magnitude of the normal force exerted by the wall on the ladder, and \( N_2 \) is the magnitude of the normal force exerted by the ground on the ladder. Throughout the problem, consider counterclockwise torques to be positive.

**Part a)** What is the minimum coefficient of static friction \( \mu_{\text{min}} \) required between the ladder and the ground so that the ladder does not slip? Express \( \mu_{\text{min}} \) in terms of \( m_1 \), \( m_2 \), \( d \), \( L \), and \( \theta \). *(3 points)*

**Part b)** Suppose that the actual coefficient of friction is one and a half times as large as the value of \( \mu_{\text{min}} \). That is, \( \mu_s = \frac{3}{2} \mu_{\text{min}} \). Under these circumstances, what is the magnitude of the force of friction \( f \) that the floor applies to the ladder? Express your answer in terms of \( m_1 \), \( m_2 \), \( d \), \( L \), \( g \), and \( \theta \). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/47231/*