---
title: "A yo-yo of total mass \\(M\\) consists of two uniform solid disks, each of radius \\(R\\), joined by a central cylindrical axle of radius \\(r\\) and negligible mass. The rotational inertia of the yo-yo about its central axis of symmetry is \\(I = \\dfrac{1}{2}MR^2\\). A light, inextensible string is wound around the axle, and its upper end is held stationary. The yo-yo is released from rest and falls vertically as the string unwinds without slipping. Which of the following expressions correctly gives the magnitude of the downward linear acceleration \\(a\\) of the yo-yo and the tension \\(T\\) in the string during its descent?"
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url: "https://nerd-notes.com/ubq/120941/"
date_modified: "2026-08-23T04:44:33+00:00"
---

# A yo-yo of total mass \(M\) consists of two uniform solid disks, each of radius \(R\), joined by a central cylindrical axle of radius \(r\) and negligible mass. The rotational inertia of the yo-yo about its central axis of symmetry is \(I = \dfrac{1}{2}MR^2\). A light, inextensible string is wound around the axle, and its upper end is held stationary. The yo-yo is released from rest and falls vertically as the string unwinds without slipping. Which of the following expressions correctly gives the magnitude of the downward linear acceleration \(a\) of the yo-yo and the tension \(T\) in the string during its descent?

A yo-yo of total mass \(M\) consists of two uniform solid disks, each of radius \(R\), joined by a central cylindrical axle of radius \(r\) and negligible mass. The rotational inertia of the yo-yo about its central axis of symmetry is \(I = \dfrac{1}{2}MR^2\). A light, inextensible string is wound around the axle, and its upper end is held stationary. The yo-yo is released from rest and falls vertically as the string unwinds without slipping. Which of the following expressions correctly gives the magnitude of the downward linear acceleration \(a\) of the yo-yo and the tension \(T\) in the string during its descent?

![A schematic diagram showing a yo-yo suspended vertically from a fixed horizontal ceiling. At the top, a horizontal line with diagonal hatching represents the fixed support. A single vertical line representing the string extends straight downward from the support and wraps tangentially around a small inner circle of radius \(r\) at the center of the yo-yo. The yo-yo is shown in side profile as two concentric circles: a large outer circle of radius \(R\) and the small inner circle of radius \(r\). A horizontal dashed segment extends from the center to the outer perimeter of the small circle, labeled \(r\). Another horizontal dashed segment extends from the center to the perimeter of the large circle, labeled \(R\). A straight downward arrow near the center of mass is labeled \(a\). A straight upward arrow along the vertical string line is labeled \(T\). A straight downward arrow extending from the center of mass is labeled \(Mg\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460273-AsgUot.jpg)

- **A.** \(a = \left(\dfrac{r^2}{R^2 + r^2}\right)g\) \quad and \quad \(T = \left(\dfrac{R^2}{R^2 + r^2}\right)Mg\)
- **B.** \(a = \left(\dfrac{2r^2}{R^2 + 2r^2}\right)g\) \quad and \quad \(T = \left(\dfrac{R^2}{R^2 + 2r^2}\right)Mg\)
- **C.** \(a = \left(\dfrac{R^2}{R^2 + 2r^2}\right)g\) \quad and \quad \(T = \left(\dfrac{2r^2}{R^2 + 2r^2}\right)Mg\)
- **D.** \(a = \left(\dfrac{2r}{R + 2r}\right)g\) \quad and \quad \(T = \left(\dfrac{R}{R + 2r}\right)Mg\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120941/*