---
title: "A uniform solid cylinder of mass \\(M\\) and radius \\(R\\) is held a negligible distance above a rough horizontal surface. The cylinder is given an initial clockwise angular velocity \\(\\omega_0\\) about its central axis and is then dropped so that it lands gently on the surface without bouncing. The coefficient of kinetic friction between the cylinder and the surface is \\(\\mu_k\\). The cylinder initially slips on the surface, but after a time interval \\(\\Delta t\\), it begins to roll without slipping. The rotational inertia of a uniform solid cylinder is \\(I = \\dfrac{1}{2}MR^2\\)."
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url: "https://nerd-notes.com/ubq/110136/"
date_modified: "2026-04-02T07:32:53+00:00"
---

# A uniform solid cylinder of mass \(M\) and radius \(R\) is held a negligible distance above a rough horizontal surface. The cylinder is given an initial clockwise angular velocity \(\omega_0\) about its central axis and is then dropped so that it lands gently on the surface without bouncing. The coefficient of kinetic friction between the cylinder and the surface is \(\mu_k\). The cylinder initially slips on the surface, but after a time interval \(\Delta t\), it begins to roll without slipping. The rotational inertia of a uniform solid cylinder is \(I = \dfrac{1}{2}MR^2\).

A uniform solid cylinder of mass \(M\) and radius \(R\) is held a negligible distance above a rough horizontal surface. The cylinder is given an initial clockwise angular velocity \(\omega_0\) about its central axis and is then dropped so that it lands gently on the surface without bouncing. The coefficient of kinetic friction between the cylinder and the surface is \(\mu_k\). The cylinder initially slips on the surface, but after a time interval \(\Delta t\), it begins to roll without slipping. The rotational inertia of a uniform solid cylinder is \(I = \dfrac{1}{2}MR^2\).

![A side-view diagram showing a solid circle representing the cylinder. The circle is located just barely above a thick solid horizontal line that represents the rough surface. Inside the circle, a curved arrow labeled '\(\omega_0\)' points in a clockwise direction to indicate the initial rotation. Next to the cylinder, text reads 'Mass \(M\), Radius \(R\)'.](https://nerd-notes.com/wp-content/uploads/ubq-diagrams/ubq-frq-generatedstem-fig-1-1775114947-S3LT7r.jpg)

**Part a)** The diagram below represents the cylinder at a time immediately after it contacts the surface and is slipping. On the diagram, **draw** and **label** the forces (not components) that act on the cylinder. Each force must be represented by a distinct arrow starting on, and pointing away from, the point where the force is exerted on the cylinder. *(2 points)*

**Part b)** During the time interval \(\Delta t\) when the cylinder is slipping, it experiences both translational and rotational acceleration. *(3 points)*

**Part c)** **Derive** an expression for the final linear speed \(v_f\) of the center of mass of the cylinder at the exact instant it begins to roll without slipping. Express your answer in terms of \(M\), \(R\), \(\omega_0\), \(\mu_k\), and physical constants, as appropriate. *(3 points)*

**Part d)** During the slipping phase, mechanical energy is dissipated by friction. **Derive** a numerical value for the fraction of the cylinder's initial mechanical energy that is dissipated during this phase. Express your final answer as a simple fraction. *(3 points)*

**Part e)** Suppose the experiment is repeated on a new surface that has a significantly greater coefficient of kinetic friction \(\mu_k\) than the original surface. All other initial conditions are identical. **Predict** whether the final linear speed \(v_f\) of the cylinder at the instant it begins rolling without slipping will be greater than, less than, or equal to the final linear speed derived in part (c). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your reasoning using physical principles. You may reference your derived relationships from the previous parts. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110136/*