---
title: "A uniform solid cylinder of mass \\(M\\) and radius \\(R\\) is attached at its central axle to an ideal horizontal spring of spring constant \\(k\\). The cylinder rolls without slipping on a horizontal surface as it oscillates in simple harmonic motion. In terms of the given quantities and fundamental constants, what is the period of oscillation of the cylinder?"
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url: "https://nerd-notes.com/ubq/120980/"
date_modified: "2026-08-23T04:44:55+00:00"
---

# A uniform solid cylinder of mass \(M\) and radius \(R\) is attached at its central axle to an ideal horizontal spring of spring constant \(k\). The cylinder rolls without slipping on a horizontal surface as it oscillates in simple harmonic motion. In terms of the given quantities and fundamental constants, what is the period of oscillation of the cylinder?

A uniform solid cylinder of mass \(M\) and radius \(R\) is attached at its central axle to an ideal horizontal spring of spring constant \(k\). The cylinder rolls without slipping on a horizontal surface as it oscillates in simple harmonic motion. In terms of the given quantities and fundamental constants, what is the period of oscillation of the cylinder?

![A schematic diagram showing a horizontal floor and a vertical wall on the left. An ideal horizontal coiled spring with spring constant label \(k\) is fixed to the vertical wall and extends horizontally to the right, attached to the center of a circular wheel representing a uniform solid cylinder. A mass label \(M\) is positioned inside the circle, and a single radius line from the center to the top rim is labeled \(R\). The bottom edge of the circle is in contact with the horizontal line representing the floor. A horizontal double-headed arrow indicates bidirectional rolling motion along the surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460295-plY2k0.jpg)

- **A.** \(2\pi \sqrt{\dfrac{2M}{3k}}\)
- **B.** \(2\pi \sqrt{\dfrac{M}{k}}\)
- **C.** \(2\pi \sqrt{\dfrac{3M}{2k}}\)
- **D.** \(2\pi \sqrt{\dfrac{2M}{k}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120980/*