---
title: "A block of mass \\(M\\) is suspended from a light string that is wound around a pulley of mass \\(M_P\\) and radius \\(R\\). The pulley is free to rotate about a frictionless, horizontal axle through its center. In Scenario 1, the pulley is a uniform solid disk with rotational inertia \\(I_1 = \\dfrac{1}{2}M_P R^2\\). In Scenario 2, the pulley is a thin hoop with rotational inertia \\(I_2 = M_P R^2\\). In both scenarios, the block is released from rest and the string does not slip. Which of the following correctly compares the acceleration \\(a\\) of the block and the tension \\(T\\) in the string in Scenario 2 to their values in Scenario 1?"
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url: "https://nerd-notes.com/ubq/109838/"
date_modified: "2026-03-26T05:53:10+00:00"
---

# A block of mass \(M\) is suspended from a light string that is wound around a pulley of mass \(M_P\) and radius \(R\). The pulley is free to rotate about a frictionless, horizontal axle through its center. In Scenario 1, the pulley is a uniform solid disk with rotational inertia \(I_1 = \dfrac{1}{2}M_P R^2\). In Scenario 2, the pulley is a thin hoop with rotational inertia \(I_2 = M_P R^2\). In both scenarios, the block is released from rest and the string does not slip. Which of the following correctly compares the acceleration \(a\) of the block and the tension \(T\) in the string in Scenario 2 to their values in Scenario 1?

A block of mass \(M\) is suspended from a light string that is wound around a pulley of mass \(M_P\) and radius \(R\). The pulley is free to rotate about a frictionless, horizontal axle through its center. In Scenario 1, the pulley is a uniform solid disk with rotational inertia \(I_1 = \dfrac{1}{2}M_P R^2\). In Scenario 2, the pulley is a thin hoop with rotational inertia \(I_2 = M_P R^2\). In both scenarios, the block is released from rest and the string does not slip. Which of the following correctly compares the acceleration \(a\) of the block and the tension \(T\) in the string in Scenario 2 to their values in Scenario 1?

![Two side-by-side diagrams of a block hanging from a pulley. On the left, labeled Scenario 1, the pulley is a solid gray disk. On the right, labeled Scenario 2, the pulley is a dark thin ring (hoop). Both pulleys have the same radius R and are connected to a block of mass M by a string wound around the pulley rim. Both blocks are at the same vertical height.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774504390-ESaD8C.jpg)

- **A.** \(a_2 < a_1\) and \(T_2 > T_1\) because the hoop has a larger rotational inertia, requiring a larger tension to provide the torque for rotation and resulting in a smaller downward acceleration.
- **B.** \(a_2 < a_1\) and \(T_2 < T_1\) because the hoop resists changes in its rotational motion more than the disk, which reduces both the tension in the string and the acceleration of the block.
- **C.** \(a_2 > a_1\) and \(T_2 > T_1\) because the mass of the hoop is further from the axis of rotation, allowing the tension to exert a larger torque and speed up the system more quickly.
- **D.** \(a_2 = a_1\) and \(T_2 = T_1\) because both pulleys have the same mass and radius, and the net force on the block is determined only by the gravitational force on the block and the total mass of the system.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109838/*