---
title: "A crate of mass \\(m\\) rests on a level horizontal surface where the coefficient of static friction between the crate and the surface is \\(\\mu_s\\). In Scenario 1, a force of magnitude \\(F_1\\) is applied to pull the crate at an angle \\(\\theta\\) above the horizontal. In Scenario 2, a force of magnitude \\(F_2\\) is applied to push the crate at the same angle \\(\\theta\\) below the horizontal. If both forces represent the maximum magnitudes that can be exerted without causing the crate to slip, and \\(\\mu_s \\tan\\theta < 1\\), what is the ratio \\(\\dfrac{F_1}{F_2}\\)?"
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url: "https://nerd-notes.com/ubq/124177/"
date_modified: "2026-09-28T13:30:57+00:00"
---

# A crate of mass \(m\) rests on a level horizontal surface where the coefficient of static friction between the crate and the surface is \(\mu_s\). In Scenario 1, a force of magnitude \(F_1\) is applied to pull the crate at an angle \(\theta\) above the horizontal. In Scenario 2, a force of magnitude \(F_2\) is applied to push the crate at the same angle \(\theta\) below the horizontal. If both forces represent the maximum magnitudes that can be exerted without causing the crate to slip, and \(\mu_s \tan\theta < 1\), what is the ratio \(\dfrac{F_1}{F_2}\)?

A crate of mass \(m\) rests on a level horizontal surface where the coefficient of static friction between the crate and the surface is \(\mu_s\). In Scenario 1, a force of magnitude \(F_1\) is applied to pull the crate at an angle \(\theta\) above the horizontal. In Scenario 2, a force of magnitude \(F_2\) is applied to push the crate at the same angle \(\theta\) below the horizontal. If both forces represent the maximum magnitudes that can be exerted without causing the crate to slip, and \(\mu_s \tan\theta < 1\), what is the ratio \(\dfrac{F_1}{F_2}\)?

![A two-panel grayscale diagram showing two mechanical scenarios side by side, separated by a thin vertical line. The left panel is labeled Scenario 1 and the right panel is labeled Scenario 2. In each panel, a horizontal floor is represented by a solid line with hatching underneath. A square box of mass \(m\) rests on the floor. In Scenario 1, an arrow labeled \(F_1\) originates from the center of the right face of the box and points upward and to the right at an angle \(\theta\) above the horizontal, with a dashed horizontal reference line and an arc indicating \(\theta\). In Scenario 2, an arrow labeled \(F_2\) originates from the upper-left corner of the box and points downward and to the right into the box at an angle \(\theta\) below the horizontal, with a dashed horizontal reference line and an arc indicating \(\theta\). No other forces, vectors, labels, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602257-3QdDIu.jpg)

- **A.** \(\dfrac{\cos\theta - \mu_s \sin\theta}{\cos\theta + \mu_s \sin\theta}\)
- **B.** \(\dfrac{\cos\theta + \mu_s \sin\theta}{\cos\theta - \mu_s \sin\theta}\)
- **C.** \(\dfrac{\sin\theta - \mu_s \cos\theta}{\sin\theta + \mu_s \cos\theta}\)
- **D.** \(\dfrac{\cos\theta - \mu_s}{\cos\theta + \mu_s}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124177/*