--- title: "Block A of mass \\(m_A\\) rests on a horizontal, frictionless track. A lightweight string is attached to the left side of Block A, runs horizontally to the left, and is wrapped around a solid, disk-shaped pulley. The pulley has rotational inertia \\(I\\) and radius \\(R\\) and is free to rotate on a fixed, frictionless horizontal axle. Block B of mass \\(m_B\\) is stationary on the track a distance \\(D\\) to the right of Block A. A small fan mounted on top of Block A is turned on and exerts a constant rightward force of magnitude \\(F_0\\) on the block. The system is released from rest, and Block A accelerates to the right, causing the string to unwind from the pulley without slipping. Exactly as Block A travels the distance \\(D\\) and reaches Block B, the string fully unwinds and detaches from the pulley. Block A then collides completely inelastically with Block B. The mass of the fan is included in \\(m_A\\)." description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]" url: "https://nerd-notes.com/ubq/109546/" date_modified: "2026-04-01T07:59:37+00:00" --- # Block A of mass \(m_A\) rests on a horizontal, frictionless track. A lightweight string is attached to the left side of Block A, runs horizontally to the left, and is wrapped around a solid, disk-shaped pulley. The pulley has rotational inertia \(I\) and radius \(R\) and is free to rotate on a fixed, frictionless horizontal axle. Block B of mass \(m_B\) is stationary on the track a distance \(D\) to the right of Block A. A small fan mounted on top of Block A is turned on and exerts a constant rightward force of magnitude \(F_0\) on the block. The system is released from rest, and Block A accelerates to the right, causing the string to unwind from the pulley without slipping. Exactly as Block A travels the distance \(D\) and reaches Block B, the string fully unwinds and detaches from the pulley. Block A then collides completely inelastically with Block B. The mass of the fan is included in \(m_A\). Block A of mass \(m_A\) rests on a horizontal, frictionless track. A lightweight string is attached to the left side of Block A, runs horizontally to the left, and is wrapped around a solid, disk-shaped pulley. The pulley has rotational inertia \(I\) and radius \(R\) and is free to rotate on a fixed, frictionless horizontal axle. Block B of mass \(m_B\) is stationary on the track a distance \(D\) to the right of Block A. A small fan mounted on top of Block A is turned on and exerts a constant rightward force of magnitude \(F_0\) on the block. The system is released from rest, and Block A accelerates to the right, causing the string to unwind from the pulley without slipping. Exactly as Block A travels the distance \(D\) and reaches Block B, the string fully unwinds and detaches from the pulley. Block A then collides completely inelastically with Block B. The mass of the fan is included in \(m_A\). ![A horizontal track. On the left end of the track, a circular pulley is mounted on a fixed axis. Above the pulley are labels 'I' and 'R'. A string is wrapped around the pulley and extends horizontally to the right, where it attaches to the left face of a rectangular block labeled 'Block A'. On top of Block A is a small box representing a fan, with a right-pointing arrow originating from it labeled 'F_0'. To the right of Block A is another rectangular block labeled 'Block B'. A dimension line below the blocks indicates the initial distance between the right edge of Block A and the left edge of Block B is 'D'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774412334-yX2sXv.jpg) **Part a)** **Derive** an expression for the magnitude of the rightward acceleration of Block A before the string detaches. Express your answer in terms of \(m_A\), \(I\), \(R\), \(F_0\), and fundamental constants. *(3 points)* **Part b)** **Derive** an expression for the speed of Block A immediately before the string detaches. Express your answer in terms of \(m_A\), \(I\), \(R\), \(F_0\), \(D\), and fundamental constants. *(2 points)* **Part c)** Immediately after the string detaches, Block A collides and sticks to Block B. **Derive** an expression for the speed of the combined blocks immediately after the collision. Express your answer in terms of \(m_A\), \(m_B\), \(I\), \(R\), \(F_0\), \(D\), and fundamental constants. *(2 points)* **Part d)** Suppose the original disk-shaped pulley is replaced with a hoop-shaped pulley of the exact same mass and radius. **Indicate** whether the speed of the combined blocks immediately after the collision will be greater, smaller, or the same compared to the original scenario. - [ ] Greater - [ ] Smaller - [ ] The same **Justify** your answer using physical reasoning or functional dependence from your derived expressions. *(3 points)* **Part e)** **Sketch** a graph of the speed \(v\) of Block A as a function of time \(t\) from the moment it is released from rest until it reaches distance \(D\). Sketch two curves on the same axes: - One representing the system with the original **disk-shaped** pulley (label this curve "Disk"). - One representing the system with the **hoop-shaped** pulley (label this curve "Hoop"). *(3 points)* *The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109546/*