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A block, initially at rest, has a mass m and sits on a plane inclined at angle theta. It slides a distance d before hitting a spring and compresses the spring by a maximum distance of xf. If the coefficient of kinetic friction between the plane and block is uk, then what is the force constant of the spring?

A block, initially at rest, has a mass m and sits on a plane inclined at angle theta. It slides a distance d before hitting a spring and compresses the spring by a maximum distance of xf. If the coefficient of kinetic friction between the plane and block is uk, then what is the force constant of the spring?

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 block, initially at rest, has a mass m and sits on a plane inclined at angle theta. It slides a distance d before hitting a spring and compresses the spring by a maximum distance of xf. If the coefficient of kinetic friction between the plane and block is uk, then what is the force constant of the spring? 

### Understanding Motion on an Inclined Plane with a Spring #### Diagram Explanation This diagram illustrates the motion of a mass \( m \) on an inclined plane with an initial and final position. The system includes a spring with a spring constant \( k \). - **Initial Position:** - The mass \( m \) is located at distance \( d \) from the spring. - A spring with a constant \( k = ? \), indicating it is unknown, is shown in a compressed state. - Angle of inclination is represented by \( \theta \). - \( x_i \) denotes the initial compression distance of the spring. - **Final Position:** - The mass \( m \) has moved from its initial position and is now compressing the spring further. - The spring remains with the same constant \( k = ? \). - Angle of inclination is represented by \( \theta \). - \( x_f \) denotes the final compression distance of the spring after the mass has moved. Both diagrams show the forces involved when a mass is moved on an inclined plane and the resulting effect on the spring. The distance \( d \) is consistent in both initial and final states, representing the distance the mass travels on the incline. These visual aids are critical for understanding the relationship between the forces due to gravity, the spring compression, and the inclined plane's angle. ### Learning Objectives: - Understand the basic principles of mechanics involving an inclined plane and a spring system. - Learn how the motion of the mass \( m \) impacts the spring's compression. - Analyze the initial and final states of the system to deduce the spring constant \( k \) and other relevant physical quantities. This fundamental physics concept is widely applicable in various engineering and physical sciences disciplines. The diagrams effectively illustrate the interaction of different forces and how they affect the mechanical energy in the system.
Transcribed Image Text:### Understanding Motion on an Inclined Plane with a Spring #### Diagram Explanation This diagram illustrates the motion of a mass \( m \) on an inclined plane with an initial and final position. The system includes a spring with a spring constant \( k \). - **Initial Position:** - The mass \( m \) is located at distance \( d \) from the spring. - A spring with a constant \( k = ? \), indicating it is unknown, is shown in a compressed state. - Angle of inclination is represented by \( \theta \). - \( x_i \) denotes the initial compression distance of the spring. - **Final Position:** - The mass \( m \) has moved from its initial position and is now compressing the spring further. - The spring remains with the same constant \( k = ? \). - Angle of inclination is represented by \( \theta \). - \( x_f \) denotes the final compression distance of the spring after the mass has moved. Both diagrams show the forces involved when a mass is moved on an inclined plane and the resulting effect on the spring. The distance \( d \) is consistent in both initial and final states, representing the distance the mass travels on the incline. These visual aids are critical for understanding the relationship between the forces due to gravity, the spring compression, and the inclined plane's angle. ### Learning Objectives: - Understand the basic principles of mechanics involving an inclined plane and a spring system. - Learn how the motion of the mass \( m \) impacts the spring's compression. - Analyze the initial and final states of the system to deduce the spring constant \( k \) and other relevant physical quantities. This fundamental physics concept is widely applicable in various engineering and physical sciences disciplines. The diagrams effectively illustrate the interaction of different forces and how they affect the mechanical energy in the system.
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