---
title: "Two solid cylinders, Cylinder A and Cylinder B, both have total mass \\(M\\), radius \\(R\\), and length \\(L\\).  – Cylinder A is made of a uniform material. – Cylinder B is made of a material such that its volume mass density \\(\\rho\\) increases with radial distance \\(r\\) from the central axis according to the function \\(\\rho(r) = \\beta r\\), where \\(\\beta\\) is a positive constant.  Both cylinders are placed at rest at the top of a rough incline of angle \\(\\theta\\) and roll without slipping down the incline."
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url: "https://nerd-notes.com/ubq/117808/"
date_modified: "2026-08-04T07:57:29+00:00"
---

# Two solid cylinders, Cylinder A and Cylinder B, both have total mass \(M\), radius \(R\), and length \(L\).

– Cylinder A is made of a uniform material.
– Cylinder B is made of a material such that its volume mass density \(\rho\) increases with radial distance \(r\) from the central axis according to the function \(\rho(r) = \beta r\), where \(\beta\) is a positive constant.

Both cylinders are placed at rest at the top of a rough incline of angle \(\theta\) and roll without slipping down the incline.

Two solid cylinders, Cylinder A and Cylinder B, both have total mass \(M\), radius \(R\), and length \(L\).

- Cylinder A is made of a uniform material.
- Cylinder B is made of a material such that its volume mass density \(\rho\) increases with radial distance \(r\) from the central axis according to the function \(\rho(r) = \beta r\), where \(\beta\) is a positive constant.

Both cylinders are placed at rest at the top of a rough incline of angle \(\theta\) and roll without slipping down the incline.

![A cross-sectional side view of a flat inclined ramp. The ramp is inclined at an angle \theta to the horizontal. A horizontal dashed line extends from the bottom of the ramp to the left, and an arc with the label \theta marks the angle between the dashed line and the ramp. A solid circle representing a cylinder rests on the ramp near the top. The cylinder has a small dot at its center. No other forces or vectors are shown.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830248-r91dlK.jpg)

**Part a)** Student 1 claims that Cylinder B will reach the bottom of the ramp in less time than Cylinder A. The student argues that because Cylinder B is denser at its outer edge, gravity exerts a greater torque on Cylinder B, causing it to accelerate more rapidly. **Indicate** whether the student's claim about the time to reach the bottom is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your answer qualitatively using physical principles. Address the student's reasoning about torque in your justification. *(3 points)*

**Part b)** The following derivations refer to Cylinder B. *(7 points)*

**Part c)** The graph below shows the velocity \(v\) as a function of time \(t\) for Cylinder B as it rolls down the ramp. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117808/*