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Two blocks are positioned on surfaces, each inclined at the same angle of 43.0 degrees with respect to the horizontal. The blocks are connected by a rope which rests on a frictionless pulley at the top of the inclines as shown, so the blocks can slide together. The mass of the black block is 6.15 kg, and the coefficient of kinetic friction for both blocks and inclines is 0.260. Assume static friction has been overcome and that everything can slide. What is must be the mass of the white block if both blocks are to slide to the LEFT at an acceleration of 1.5 m/s^2?

Two blocks are positioned on surfaces, each inclined at the same angle of 43.0 degrees with respect to the horizontal. The blocks are connected by a rope which rests on a frictionless pulley at the top of the inclines as shown, so the blocks can slide together. The mass of the black block is 6.15 kg, and the coefficient of kinetic friction for both blocks and inclines is 0.260. Assume static friction has been overcome and that everything can slide. What is must be the mass of the white block if both blocks are to slide to the LEFT at an acceleration of 1.5 m/s^2?

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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Two blocks are positioned on surfaces, each inclined at the same angle of 43.0 degrees with respect to the horizontal. The blocks are connected by a rope which rests on a frictionless pulley at the top of the inclines as shown, so the blocks can slide together. The mass of the black block is 6.15 kg, and the coefficient of kinetic friction for both blocks and inclines is 0.260. Assume static friction has been overcome and that everything can slide. What is must be the mass of the white block if both blocks are to slide to the LEFT at an acceleration of 1.5 m/s^2?

### Mechanics of Inclined Planes with Pulleys #### Diagrams and Descriptions The diagram above demonstrates a classic physics problem involving two masses on inclined planes connected by a pulley. This setup is commonly used to explore the concepts of tension, gravitational force, friction, and the laws of motion. **Diagram Details:** 1. **Inclined Planes:** - There are two inclined planes, each forming an angle \( \theta \) with the horizontal ground. - The angles on both sides are equal, symbolized by \( \theta \). 2. **Masses:** - Two blocks of differing masses are positioned on these inclined planes. - One block is on the left incline, and the other is on the right incline. - The block on the right is shown with a textured surface, likely indicating a different mass compared to the left block. 3. **Pulley System:** - A pulley is located at the apex where the two inclined planes meet. - A frictionless pulley connects the two blocks with a string or cord running over it. - The pulley allows the blocks to move in contrasting directions, i.e., if one block moves up, the other moves down. **Concepts to Explore with This Diagram:** 1. **Newton's Second Law of Motion:** - The diagram can be used to set up Newton's second law for each mass. The force equations will involve components of gravitational force (\(Mg \ sin \ \theta \)) along the inclined plane. 2. **Tension in the String:** - The tension in the string is the same on both sides of the pulley. This can be used to solve the equations for acceleration and tension. 3. **Free-body Diagrams:** - Drawing free-body diagrams for each block will help visualize the forces acting along the planes, including gravitational forces, tension, and possibly normal force and friction. 4. **Equations of Motion:** - We can derive the equations of motion by balancing forces along the direction of movement for both masses. - For mass \(m_1\) on the left incline with angle \( \theta \): \(m_1 g \ sin \ \theta - T = m_1 a\) - For mass \(m_2\) on the right incline with angle \( \theta \): \(T - m_2 g \ sin \ theta =
Transcribed Image Text:### Mechanics of Inclined Planes with Pulleys #### Diagrams and Descriptions The diagram above demonstrates a classic physics problem involving two masses on inclined planes connected by a pulley. This setup is commonly used to explore the concepts of tension, gravitational force, friction, and the laws of motion. **Diagram Details:** 1. **Inclined Planes:** - There are two inclined planes, each forming an angle \( \theta \) with the horizontal ground. - The angles on both sides are equal, symbolized by \( \theta \). 2. **Masses:** - Two blocks of differing masses are positioned on these inclined planes. - One block is on the left incline, and the other is on the right incline. - The block on the right is shown with a textured surface, likely indicating a different mass compared to the left block. 3. **Pulley System:** - A pulley is located at the apex where the two inclined planes meet. - A frictionless pulley connects the two blocks with a string or cord running over it. - The pulley allows the blocks to move in contrasting directions, i.e., if one block moves up, the other moves down. **Concepts to Explore with This Diagram:** 1. **Newton's Second Law of Motion:** - The diagram can be used to set up Newton's second law for each mass. The force equations will involve components of gravitational force (\(Mg \ sin \ \theta \)) along the inclined plane. 2. **Tension in the String:** - The tension in the string is the same on both sides of the pulley. This can be used to solve the equations for acceleration and tension. 3. **Free-body Diagrams:** - Drawing free-body diagrams for each block will help visualize the forces acting along the planes, including gravitational forces, tension, and possibly normal force and friction. 4. **Equations of Motion:** - We can derive the equations of motion by balancing forces along the direction of movement for both masses. - For mass \(m_1\) on the left incline with angle \( \theta \): \(m_1 g \ sin \ \theta - T = m_1 a\) - For mass \(m_2\) on the right incline with angle \( \theta \): \(T - m_2 g \ sin \ theta =
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