---
title: "A solid cylinder of mass \\(M\\) and radius \\(R\\) is released from rest at the top of an incline of vertical height \\(H\\). The incline makes an angle \\(\\theta\\) with the horizontal. The surface of the incline is rough enough that the cylinder rolls down without slipping. The rotational inertia of a solid cylinder about its center is \\(I = \\dfrac{1}{2}MR^2\\)."
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url: "https://nerd-notes.com/ubq/110267/"
date_modified: "2026-04-01T02:44:19+00:00"
---

# A solid cylinder of mass \(M\) and radius \(R\) is released from rest at the top of an incline of vertical height \(H\). The incline makes an angle \(\theta\) with the horizontal. The surface of the incline is rough enough that the cylinder rolls down without slipping. The rotational inertia of a solid cylinder about its center is \(I = \dfrac{1}{2}MR^2\).

A solid cylinder of mass \(M\) and radius \(R\) is released from rest at the top of an incline of vertical height \(H\). The incline makes an angle \(\theta\) with the horizontal. The surface of the incline is rough enough that the cylinder rolls down without slipping. The rotational inertia of a solid cylinder about its center is \(I = \dfrac{1}{2}MR^2\).

![A right triangular ramp with its horizontal base at the bottom. The angle between the horizontal base and the sloped surface is labeled \(\theta\). A solid circular cylinder is shown resting on the sloped surface near the top. The vertical height of the ramp is labeled \(H\). The cylinder has a curved arrow around its center indicating clockwise rotation, and a straight vector arrow pointing down the incline indicating its direction of translational motion.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775011459-9i9zMq.jpg)

**Part a)** A hollow cylinder of the same mass \(M\) and radius \(R\) (with rotational inertia \(I = MR^2\)) is now released from rest at the top of the same incline and rolls without slipping to the bottom. **Predict** whether the translational speed of the hollow cylinder at the bottom of the incline is greater than, less than, or equal to the translational speed of the solid cylinder at the bottom. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using energy principles. *(2 points)*

**Part b)** To determine the acceleration of the solid cylinder quantitatively, a student must analyze the forces involved. On the dot below, which represents the solid cylinder, **draw** and **label** the forces (not components) that are exerted on the cylinder as it rolls down the incline. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. *(3 points)*

**Part c)** Starting from Newton's second law for both translation and rotation, **derive** an expression for the translational acceleration \(a\) of the center of mass of the solid cylinder. Express your answer in terms of \(\theta\), \(M\), \(R\), and fundamental constants, as appropriate. *(4 points)*

**Part d)** Another student attempts to derive an expression for the acceleration of the hollow cylinder and proposes the equation \(a = g \sin\theta\). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110267/*