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18-29. The 10-lb sphere starts from rest at 0= 0° and rolls without slipping down the cylindrical surface which has a radius of 10 ft. Determine the speed of the sphere's center of mass at the instant = 45°. 0.5 ft

18-29. The 10-lb sphere starts from rest at 0= 0° and rolls without slipping down the cylindrical surface which has a radius of 10 ft. Determine the speed of the sphere's center of mass at the instant = 45°. 0.5 ft

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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**Title: Motion of Rolling Spheres on Cylindrical Surfaces** **Problem Description:** 18–29. A 10-lb sphere starts from rest at \( \theta = 0^\circ \) and rolls without slipping down the cylindrical surface, which has a radius of 10 ft. Determine the speed of the sphere’s center of mass at the instant \( \theta = 45^\circ \). **Diagram Explanation:** The provided image features a diagram illustrating a sphere rolling down a cylindrical surface. The main components and annotations in the diagram are: - A sphere representing the aforementioned 10-lb sphere, positioned initially at the topmost part of the cylindrical surface (\( \theta = 0^\circ \)). - The cylindrical surface perimeter is indicated with a semi-circular arc. - The sphere is shown in two positions: - At \( \theta = 0^\circ \) (topmost point). - At \( \theta = 45^\circ \), where it has rolled a certain distance down the cylindrical surface. - Radius indicators: - The total radius of the cylindrical surface (10 ft). - The radius marked from the surface to the sphere’s center of mass (\( 0.5 \) ft from the surface). - The angle \( \theta \) is depicted by a radial line originating from the cylinder’s center to the sphere’s center of mass. **Instructions for Students:** - Carefully observe the motion and forces acting on the sphere as it rolls down. - Apply principles of rotational dynamics and energy conservation to derive the velocity of the sphere’s center of mass. - Use mathematical equations relevant to potential and kinetic energy transformations. - Ensure to keep track of radius components, as the sphere rolls without slipping, affecting rotational inertia. **Learning Objectives:** - To understand how spherical bodies roll without slipping and the implications for kinetic friction forces. - To solve problems involving rotational motion on curved surfaces. - To apply the energy conservation principle in physical systems comprehensively. --- This transcription serves as an educational aid, helping students visualize and understand the physics of rolling motion through guided problem-solving and analysis.
Transcribed Image Text:**Title: Motion of Rolling Spheres on Cylindrical Surfaces** **Problem Description:** 18–29. A 10-lb sphere starts from rest at \( \theta = 0^\circ \) and rolls without slipping down the cylindrical surface, which has a radius of 10 ft. Determine the speed of the sphere’s center of mass at the instant \( \theta = 45^\circ \). **Diagram Explanation:** The provided image features a diagram illustrating a sphere rolling down a cylindrical surface. The main components and annotations in the diagram are: - A sphere representing the aforementioned 10-lb sphere, positioned initially at the topmost part of the cylindrical surface (\( \theta = 0^\circ \)). - The cylindrical surface perimeter is indicated with a semi-circular arc. - The sphere is shown in two positions: - At \( \theta = 0^\circ \) (topmost point). - At \( \theta = 45^\circ \), where it has rolled a certain distance down the cylindrical surface. - Radius indicators: - The total radius of the cylindrical surface (10 ft). - The radius marked from the surface to the sphere’s center of mass (\( 0.5 \) ft from the surface). - The angle \( \theta \) is depicted by a radial line originating from the cylinder’s center to the sphere’s center of mass. **Instructions for Students:** - Carefully observe the motion and forces acting on the sphere as it rolls down. - Apply principles of rotational dynamics and energy conservation to derive the velocity of the sphere’s center of mass. - Use mathematical equations relevant to potential and kinetic energy transformations. - Ensure to keep track of radius components, as the sphere rolls without slipping, affecting rotational inertia. **Learning Objectives:** - To understand how spherical bodies roll without slipping and the implications for kinetic friction forces. - To solve problems involving rotational motion on curved surfaces. - To apply the energy conservation principle in physical systems comprehensively. --- This transcription serves as an educational aid, helping students visualize and understand the physics of rolling motion through guided problem-solving and analysis.
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