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A uniform spherical shell of mass M = 9.70 kg and radius R = 0.530 m can rotate about a vertical axis on frictionless bearings (see the figure). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 0.210 kg·m2 and radius r = 0.140 m, and is attached to a small object of mass m = 4.20 kg. There is no friction on the pulley's axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen a distance 1.05 m after being released from rest? Use energy considerations.

A uniform spherical shell of mass M = 9.70 kg and radius R = 0.530 m can rotate about a vertical axis on frictionless bearings (see the figure). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 0.210 kg·m2 and radius r = 0.140 m, and is attached to a small object of mass m = 4.20 kg. There is no friction on the pulley's axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen a distance 1.05 m after being released from rest? Use energy considerations.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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A uniform spherical shell of mass M = 9.70 kg and radius R = 0.530 m can rotate about a vertical axis on frictionless bearings (see the figure). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 0.210 kg·m2 and radius r = 0.140 m, and is attached to a small object of mass m = 4.20 kg. There is no friction on the pulley's axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen a distance 1.05 m after being released from rest? Use energy considerations.

### Description of the Mechanical System **Components:** 1. **Sphere:** - The large pink sphere is labeled with mass \( M \) and radius \( R \). - It is mounted on a vertical stand above a flat surface. 2. **Pulley System:** - The pulley is depicted with two labeled quantities: moment of inertia \( I \) and radius \( r \). - A string passes over the pulley, connecting the sphere to a block. 3. **Block:** - This smaller block is suspended on the right side of the pulley and is labeled with mass \( m \). **Functionality:** - The string is tied around the sphere and extends horizontally towards the pulley. - As the block descends due to gravity, it causes the string to unravel from the sphere, imparting rotational motion to the sphere. - The system illustrates concepts of rotational dynamics, conservation of energy, and mechanical advantage in physics. **Key Concepts:** - **Rotational Dynamics:** Involves analyzing the motion of objects that rotate about an axis. The moment of inertia \( I \) plays a critical role similar to mass in linear dynamics. - **Conservation of Energy:** The system might be used to demonstrate how potential energy from the block’s height is converted into kinetic energy of the sphere and the block, and rotational kinetic energy of the pulley. - **Mechanical Advantage:** By using the pulley, the force exerted is modified, affecting how the system behaves dynamically. This system is commonly used in physics education to demonstrate the interplay of rotational and linear motion, as well as energy conservation concepts.
Transcribed Image Text:### Description of the Mechanical System **Components:** 1. **Sphere:** - The large pink sphere is labeled with mass \( M \) and radius \( R \). - It is mounted on a vertical stand above a flat surface. 2. **Pulley System:** - The pulley is depicted with two labeled quantities: moment of inertia \( I \) and radius \( r \). - A string passes over the pulley, connecting the sphere to a block. 3. **Block:** - This smaller block is suspended on the right side of the pulley and is labeled with mass \( m \). **Functionality:** - The string is tied around the sphere and extends horizontally towards the pulley. - As the block descends due to gravity, it causes the string to unravel from the sphere, imparting rotational motion to the sphere. - The system illustrates concepts of rotational dynamics, conservation of energy, and mechanical advantage in physics. **Key Concepts:** - **Rotational Dynamics:** Involves analyzing the motion of objects that rotate about an axis. The moment of inertia \( I \) plays a critical role similar to mass in linear dynamics. - **Conservation of Energy:** The system might be used to demonstrate how potential energy from the block’s height is converted into kinetic energy of the sphere and the block, and rotational kinetic energy of the pulley. - **Mechanical Advantage:** By using the pulley, the force exerted is modified, affecting how the system behaves dynamically. This system is commonly used in physics education to demonstrate the interplay of rotational and linear motion, as well as energy conservation concepts.
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