Two masses are attached to a pulley and released from rest, with M2 on the ground. M1 begins to fall and M2 rises. There is friction in the axle that produces a constant torque of t Nm on the rotating object. Find the speed of M1 just before it hits the ground in terms of the given information (M1, M2, M,, R, T, H, t, and our usual constants) Two solid cylinders are glued together to form a pulley,

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**Title: Mechanics Problem Involving Pulley and Masses**

**Introduction:**

This example explores the dynamics of a system where two masses are connected to a pulley and are released from a state of rest. The system includes friction in the axle, which results in a constant torque being applied to the rotating object. We aim to determine the speed of mass M1 just before it impacts the ground.

**Problem Statement:**

Two masses are attached to a pulley and released from rest, with mass M2 on the ground. As the system evolves, M1 begins to fall while M2 rises. There is friction in the axle that produces a constant torque \( \tau \) Nm on the rotating object. Determine the speed of M1 just before it hits the ground in terms of the given variables (M1, M2, \(M_p\), \(R\), \(T\), \(H\), \( \tau \), and our typical physical constants).

### Components of the System:

1. **Masses and Pulley:**
   - Two masses: M1 and M2
   - A complex pulley composed of two solid cylinders:
     - **Cylinder 1:** Radius \( R \) and mass \( M_p \), thickness \( T \)
     - **Cylinder 2:** Radius \( 2R \) and mass \( 2M_p \), thickness \( 2T \)

2. **Diagram Description:**

   - **Pulley System (Left Side):** Two masses (M1 and M2) are connected to a pulley, with M2 on the ground and M1 at an elevation \( H \).
   - **Pulley Description (Right Side):** The pulley consists of two solid cylinders glued together:
     - Cylinder 1 has a radius \( R \) and a mass \( M_p \).
     - Cylinder 2 has a radius \( 2R \) and a mass \( 2M_p \).
     - Thickness specifications for the cylinders are \( T \) for Cylinder 1 and \( 2T \) for Cylinder 2.

### Diagram Analysis:

- **Left Diagram:**
   - The pulley is shown with both masses M1 (falling) and M2 (rising).
   - Both masses are attached via strings that pass through the pulley's edge.
   - The height \( H \) is indicated as the initial vertical distance between M1 and the ground.

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Transcribed Image Text:**Title: Mechanics Problem Involving Pulley and Masses** **Introduction:** This example explores the dynamics of a system where two masses are connected to a pulley and are released from a state of rest. The system includes friction in the axle, which results in a constant torque being applied to the rotating object. We aim to determine the speed of mass M1 just before it impacts the ground. **Problem Statement:** Two masses are attached to a pulley and released from rest, with mass M2 on the ground. As the system evolves, M1 begins to fall while M2 rises. There is friction in the axle that produces a constant torque \( \tau \) Nm on the rotating object. Determine the speed of M1 just before it hits the ground in terms of the given variables (M1, M2, \(M_p\), \(R\), \(T\), \(H\), \( \tau \), and our typical physical constants). ### Components of the System: 1. **Masses and Pulley:** - Two masses: M1 and M2 - A complex pulley composed of two solid cylinders: - **Cylinder 1:** Radius \( R \) and mass \( M_p \), thickness \( T \) - **Cylinder 2:** Radius \( 2R \) and mass \( 2M_p \), thickness \( 2T \) 2. **Diagram Description:** - **Pulley System (Left Side):** Two masses (M1 and M2) are connected to a pulley, with M2 on the ground and M1 at an elevation \( H \). - **Pulley Description (Right Side):** The pulley consists of two solid cylinders glued together: - Cylinder 1 has a radius \( R \) and a mass \( M_p \). - Cylinder 2 has a radius \( 2R \) and a mass \( 2M_p \). - Thickness specifications for the cylinders are \( T \) for Cylinder 1 and \( 2T \) for Cylinder 2. ### Diagram Analysis: - **Left Diagram:** - The pulley is shown with both masses M1 (falling) and M2 (rising). - Both masses are attached via strings that pass through the pulley's edge. - The height \( H \) is indicated as the initial vertical distance between M1 and the ground. -
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