---
title: "A stepped pulley consists of two joined solid cylinders: an outer cylinder of radius \\(R\\) and an inner cylinder of radius \\(r\\). The pulley has a total rotational inertia \\(I\\) and mass \\(M_p\\), and it is mounted on a frictionless horizontal axle. A block of mass \\(m\\) is suspended from a lightweight string wrapped around the right side of the outer cylinder. A student holds a second lightweight string wrapped around the left side of the inner cylinder. The student pulls downward on the second string with a constant force \\(F_A\\). Neither string slips on the pulley."
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url: "https://nerd-notes.com/ubq/110125/"
date_modified: "2026-03-27T05:29:22+00:00"
---

# A stepped pulley consists of two joined solid cylinders: an outer cylinder of radius \(R\) and an inner cylinder of radius \(r\). The pulley has a total rotational inertia \(I\) and mass \(M_p\), and it is mounted on a frictionless horizontal axle. A block of mass \(m\) is suspended from a lightweight string wrapped around the right side of the outer cylinder. A student holds a second lightweight string wrapped around the left side of the inner cylinder. The student pulls downward on the second string with a constant force \(F_A\). Neither string slips on the pulley.

A stepped pulley consists of two joined solid cylinders: an outer cylinder of radius \(R\) and an inner cylinder of radius \(r\). The pulley has a total rotational inertia \(I\) and mass \(M_p\), and it is mounted on a frictionless horizontal axle. A block of mass \(m\) is suspended from a lightweight string wrapped around the right side of the outer cylinder. A student holds a second lightweight string wrapped around the left side of the inner cylinder. The student pulls downward on the second string with a constant force \(F_A\). Neither string slips on the pulley.

![A 2D cross-section view of a stepped pulley consisting of two concentric circles. The outer circle has a radius indicated by a solid line labeled 'R'. The inner circle has a radius indicated by a solid line labeled 'r'. A fixed axle is at the center. A vertical string extends downward from the right edge of the outer circle and is attached to a square block labeled 'm'. A second vertical string extends downward from the left edge of the inner circle. A downward-pointing arrow labeled 'F_A' is at the bottom end of this second string.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774589361-JvQKRc.jpg)

**Part a)** The student pulls downward with a specific force \(F_0\) such that the block moves upward with a constant velocity. On the dots below, which represent the pulley and the block, **draw and label** the forces (not components) exerted on each object. Each force must be represented by a distinct arrow starting on, and pointing away from, the appropriate dot. *(3 points)*

**Part b)** **Derive** an expression for the magnitude of the applied force \(F_0\) required to keep the block moving upward at a constant velocity. Express your answer in terms of \(m\), \(R\), \(r\), and fundamental constants. *(2 points)*

**Part c)** The student now pulls downward with a larger constant force \(F_1\) (where \(F_1 > F_0\)), causing the block to accelerate upward. **Derive** an expression for the linear acceleration \(a\) of the block. Express your answer in terms of \(m\), \(M_p\), \(I\), \(R\), \(r\), \(F_1\), and fundamental constants as appropriate. *(4 points)*

**Part d)** The student repeats the experiment, but this time wraps the pull string around the left side of the **outer** cylinder (radius \(R\)) instead of the inner cylinder. The student pulls downward with the exact same constant force \(F_1\) used in part (c). **Indicate** whether the upward acceleration of the block is greater than, less than, or equal to the acceleration derived in part (c). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using physical principles. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110125/*