---
title: "A solid cylinder of mass \\(M\\) and radius \\(R\\) has a non-uniform volumetric mass density that varies linearly with radial distance \\(r\\) from its central axis according to \\(\\rho(r) = Cr\\), where \\(C\\) is a constant. The cylinder rolls without slipping down a ramp inclined at an angle \\(\\theta\\) to the horizontal. Which of the following expressions represents the linear acceleration of the center of mass of the cylinder as it rolls down the incline?"
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url: "https://nerd-notes.com/ubq/117654/"
date_modified: "2026-08-04T07:49:31+00:00"
---

# A solid cylinder of mass \(M\) and radius \(R\) has a non-uniform volumetric mass density that varies linearly with radial distance \(r\) from its central axis according to \(\rho(r) = Cr\), where \(C\) is a constant. The cylinder rolls without slipping down a ramp inclined at an angle \(\theta\) to the horizontal. Which of the following expressions represents the linear acceleration of the center of mass of the cylinder as it rolls down the incline?

A solid cylinder of mass \(M\) and radius \(R\) has a non-uniform volumetric mass density that varies linearly with radial distance \(r\) from its central axis according to \(\rho(r) = Cr\), where \(C\) is a constant. The cylinder rolls without slipping down a ramp inclined at an angle \(\theta\) to the horizontal. Which of the following expressions represents the linear acceleration of the center of mass of the cylinder as it rolls down the incline?

![A side view showing a flat incline sloping downward to the right at an angle \theta relative to a horizontal dashed ground line. A solid circle representing a cylinder of radius R rests on the inclined surface. An arrow labeled \vec{a} points down along the incline parallel to the surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829771-66cyaB.jpg)

- **A.** \(\dfrac{2}{5} g \sin\theta\)
- **B.** \(\dfrac{1}{2} g \sin\theta\)
- **C.** \(\dfrac{4}{7} g \sin\theta\)
- **D.** \(\dfrac{2}{3} g \sin\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117654/*