---
title: "An irregularly shaped spool of total mass \\(M\\) and outer radius \\(R\\) rolls without slipping down an inclined track from rest. A student releases the spool from several different vertical heights \\(h\\) above a photogate at the base of the track, which measures the translational speed \\(v\\) of the spool. A plot of \\(v^2\\) on the vertical axis versus \\(h\\) on the horizontal axis yields a straight line with slope \\(S\\). In terms of \\(S\\), \\(M\\), \\(R\\), and acceleration due to gravity \\(g\\), which of the following is the rotational inertia \\(I\\) of the spool about its central axis?"
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url: "https://nerd-notes.com/ubq/120910/"
date_modified: "2026-08-23T04:44:14+00:00"
---

# An irregularly shaped spool of total mass \(M\) and outer radius \(R\) rolls without slipping down an inclined track from rest. A student releases the spool from several different vertical heights \(h\) above a photogate at the base of the track, which measures the translational speed \(v\) of the spool. A plot of \(v^2\) on the vertical axis versus \(h\) on the horizontal axis yields a straight line with slope \(S\). In terms of \(S\), \(M\), \(R\), and acceleration due to gravity \(g\), which of the following is the rotational inertia \(I\) of the spool about its central axis?

An irregularly shaped spool of total mass \(M\) and outer radius \(R\) rolls without slipping down an inclined track from rest. A student releases the spool from several different vertical heights \(h\) above a photogate at the base of the track, which measures the translational speed \(v\) of the spool. A plot of \(v^2\) on the vertical axis versus \(h\) on the horizontal axis yields a straight line with slope \(S\). In terms of \(S\), \(M\), \(R\), and acceleration due to gravity \(g\), which of the following is the rotational inertia \(I\) of the spool about its central axis?

![A grayscale line schematic showing a right-triangular incline resting on a flat horizontal surface. A circular spool of outer radius R sits on the incline near the top. A vertical double-headed arrow spanning from the photogate height at the base to the center of the spool at its release point is labeled h. At the base of the incline, a U-shaped photogate sensor straddles the track with a horizontal dashed line indicating the photogate beam. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460253-PswdYl.jpg)

- **A.** \(I = MR^2 \left( \dfrac{g}{S} - 1 \right)\)
- **B.** \(I = MR^2 \left( \dfrac{2g}{S} + 1 \right)\)
- **C.** \(I = MR^2 \left( \dfrac{2g}{S} - 1 \right)\)
- **D.** \(I = MR^2 \left( \dfrac{2g}{S} \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120910/*