---
title: "A uniform solid cylinder of mass \\(M\\) and radius \\(R\\) is launched up an incline of angle \\(\\theta\\) and rolls without slipping, momentarily stops at time \\(t_1\\), and rolls back down to its launch position at time \\(2t_1\\). The positive rotational direction is defined such that the cylinder has a positive angular velocity (\\(\\omega > 0\\)) during its ascent, as shown in the graph of angular velocity \\(\\omega\\) versus time \\(t\\). Which of the following statements correctly relates the features of the graph to the static friction force exerted on the cylinder by the incline?"
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/124287/"
date_modified: "2026-09-28T14:04:39+00:00"
---

# A uniform solid cylinder of mass \(M\) and radius \(R\) is launched up an incline of angle \(\theta\) and rolls without slipping, momentarily stops at time \(t_1\), and rolls back down to its launch position at time \(2t_1\). The positive rotational direction is defined such that the cylinder has a positive angular velocity (\(\omega > 0\)) during its ascent, as shown in the graph of angular velocity \(\omega\) versus time \(t\). Which of the following statements correctly relates the features of the graph to the static friction force exerted on the cylinder by the incline?

A uniform solid cylinder of mass \(M\) and radius \(R\) is launched up an incline of angle \(\theta\) and rolls without slipping, momentarily stops at time \(t_1\), and rolls back down to its launch position at time \(2t_1\). The positive rotational direction is defined such that the cylinder has a positive angular velocity (\(\omega > 0\)) during its ascent, as shown in the graph of angular velocity \(\omega\) versus time \(t\). Which of the following statements correctly relates the features of the graph to the static friction force exerted on the cylinder by the incline?

![A Cartesian coordinate graph with a horizontal time axis labeled t and a vertical angular velocity axis labeled \omega. The horizontal axis has three tick marks with text labels: 0 at the origin, t_1 at the midpoint, and 2t_1 at the right end. The vertical axis has three tick marks with text labels: -\omega_0 below the horizontal axis, 0 at the origin, and \omega_0 above the horizontal axis. A single solid straight line segment of uniform thickness starts at coordinates (0, \omega_0), extends downward with a constant negative slope, intersects the horizontal axis precisely at (t_1, 0), and ends at coordinates (2t_1, -\omega_0). The graph has bare axes with no gridlines. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604279-mcXHTz.jpg)

- **A.** Because the value of \(\omega\) is positive for \(t < t_1\) and negative for \(t > t_1\), the static friction force is directed down the incline during the ascent and up the incline during the descent.
- **B.** Because the graph crosses \(\omega = 0\) at \(t = t_1\), the cylinder is instantaneously at rest and experiences zero static friction force at that instant.
- **C.** Because the slope \(\dfrac{d\omega}{dt}\) is negative for all \(t\), the torque about the center of mass must be negative, which requires the static friction force to be directed down the incline for the entire motion.
- **D.** Because the slope \(\dfrac{d\omega}{dt}\) is constant and negative throughout the interval, the net torque about the center of mass is constant, requiring the static friction force to have a constant magnitude directed up the incline at all times, including at \(t = t_1\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124287/*