A counterweight of mass m = 3.60 kg is attached to a light cord that is wound around a pulley as shown in the figure below. The pulley is a thin hoop of radius R = 7.00 cm and mass M = 1.50 kg. The spokes have negligible mass. M m (a) What is the net torque on the system about the axle of the pulley? magnitude 2.058 x N-m direction to the right along the axis of rotation (b) When the counterweight has a speed v, the pulley has an angular speed = v/R. Determine the magnitude of the total angular momentum of the system about the axle of the pulley. (0.357 kg - m)v (c) Find the magnitude of the acceleration of the counterweight. 5.76 x m/s²

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### Analyzing Torque and Angular Momentum in a Pulley System

A counterweight of mass \( m = 3.60 \, \text{kg} \) is attached to a light cord that is wound around a pulley, depicted in the figure. The pulley is a thin hoop with a radius \( R = 7.00 \, \text{cm} \) and a mass \( M = 1.50 \, \text{kg} \). The spokes have negligible mass.

#### (a) Net Torque on the Pulley

- **Magnitude:** \( 2.058 \, \text{N} \cdot \text{m} \)
- **Direction:** To the right along the axis of rotation.

#### (b) Total Angular Momentum

When the counterweight has a speed \( v \), the pulley's angular speed \( \omega = v/R \). Calculate the total angular momentum of the system about the axle of the pulley:

- **Magnitude:** \( 0.357 \, \text{kg} \cdot \text{m}^2/\text{s} \)

#### (c) Acceleration of the Counterweight

- **Magnitude:** \( 5.76 \, \text{m/s}^2 \)

### Visual Diagram Explanation

The diagram shows:

- A pulley system with a light cord wrapped around it.
- A counterweight \( m \) hanging downwards.
- The pulley is represented as a circle with internal spokes.
- Arrows indicating the direction of rotation and forces acting on the system. 

This setup is used to explore concepts of torque, angular momentum, and acceleration within rotational dynamics.
Transcribed Image Text:### Analyzing Torque and Angular Momentum in a Pulley System A counterweight of mass \( m = 3.60 \, \text{kg} \) is attached to a light cord that is wound around a pulley, depicted in the figure. The pulley is a thin hoop with a radius \( R = 7.00 \, \text{cm} \) and a mass \( M = 1.50 \, \text{kg} \). The spokes have negligible mass. #### (a) Net Torque on the Pulley - **Magnitude:** \( 2.058 \, \text{N} \cdot \text{m} \) - **Direction:** To the right along the axis of rotation. #### (b) Total Angular Momentum When the counterweight has a speed \( v \), the pulley's angular speed \( \omega = v/R \). Calculate the total angular momentum of the system about the axle of the pulley: - **Magnitude:** \( 0.357 \, \text{kg} \cdot \text{m}^2/\text{s} \) #### (c) Acceleration of the Counterweight - **Magnitude:** \( 5.76 \, \text{m/s}^2 \) ### Visual Diagram Explanation The diagram shows: - A pulley system with a light cord wrapped around it. - A counterweight \( m \) hanging downwards. - The pulley is represented as a circle with internal spokes. - Arrows indicating the direction of rotation and forces acting on the system. This setup is used to explore concepts of torque, angular momentum, and acceleration within rotational dynamics.
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