---
title: "A spool of total mass \\(M\\) consists of a central hub of radius \\(r\\) and two outer flanges of radius \\(R\\). The spool has a non-uniform mass distribution resulting in a rotational inertia \\(I\\) about its center of mass. A light string wrapped around the top of the central hub is pulled horizontally to the right with a constant force of magnitude \\(T\\), causing the spool to roll without slipping to the right along a horizontal surface. Which of the following correctly gives the magnitude of the linear acceleration \\(a\\) of the spool’s center of mass, and the direction of the static friction force exerted on the spool by the surface if \\(I > MRr\\)?"
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url: "https://nerd-notes.com/ubq/117667/"
date_modified: "2026-08-04T07:49:42+00:00"
---

# A spool of total mass \(M\) consists of a central hub of radius \(r\) and two outer flanges of radius \(R\). The spool has a non-uniform mass distribution resulting in a rotational inertia \(I\) about its center of mass. A light string wrapped around the top of the central hub is pulled horizontally to the right with a constant force of magnitude \(T\), causing the spool to roll without slipping to the right along a horizontal surface. Which of the following correctly gives the magnitude of the linear acceleration \(a\) of the spool’s center of mass, and the direction of the static friction force exerted on the spool by the surface if \(I > MRr\)?

A spool of total mass \(M\) consists of a central hub of radius \(r\) and two outer flanges of radius \(R\). The spool has a non-uniform mass distribution resulting in a rotational inertia \(I\) about its center of mass. A light string wrapped around the top of the central hub is pulled horizontally to the right with a constant force of magnitude \(T\), causing the spool to roll without slipping to the right along a horizontal surface. Which of the following correctly gives the magnitude of the linear acceleration \(a\) of the spool's center of mass, and the direction of the static friction force exerted on the spool by the surface if \(I > MRr\)?

![A side-view diagram shows a circular spool resting on a thick horizontal line representing the ground. The spool consists of a large outer circle of radius \(R\) and a concentric inner circle of radius \(r\). A small black dot at the common center marks the center of mass. A horizontal dashed line extends from the bottom contact point of the outer circle to the right. A thin string extends horizontally to the right from the topmost point of the inner circle, ending in a rightward arrow labeled \(\vec{T}\). A curved arrow above the spool indicates clockwise rotation \(\alpha\). A rightward arrow near the center of mass indicates linear acceleration \(\vec{a}\). Labels \(R\) and \(r\) are attached to radial line segments from the center to the outer circle edge and inner circle edge, respectively. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-diagram-1-1785829782-2aGJxz.jpg)

- **A.** \(a = \dfrac{TR(R-r)}{I + MR^2}\) ; To the left
- **B.** \(a = \dfrac{TR(R+r)}{I + MR^2}\) ; To the left
- **C.** \(a = \dfrac{TR(R+r)}{I + MR^2}\) ; To the right
- **D.** \(a = \dfrac{TR(R-r)}{I + MR^2}\) ; To the right

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117667/*