1. Two blocks are positioned on surfaces, each inclined at the same angle of 42.8 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 2.07 kg, and the coefficient of kinetic friction for both blocks and inclines is 0.510. 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?2. Two blocks are positioned on surfaces, each inclined at the same angle of 49.5 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 2.24 kg, and this time there is NO friction. 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? **Inclined Planes with Connected Blocks**### Diagram Description:The diagram illustrates a system involving two blocks connected by a string over a pulley, which is positioned at the apex of two inclined planes. 1. **Inclined Planes**: - The system includes two inclined planes that are mirror images of each other, both inclined at an angle θ relative to the horizontal ground. - There are two angles marked as θ on either side of the inclined planes.2. **Blocks**: - There are two blocks, each placed on a different inclined plane. - The left block appears as an empty square, while the right block is colored or marked darker.3. **Pulley System**: - The blocks are connected by a string which goes over a pulley located at the apex where the two inclined planes meet. - The string is assumed to be perfectly flexible and the pulley frictionless, ensuring smooth motion.### Educational Discussion:This type of setup is commonly used to study various principles of physics, such as equilibrium, friction, tension, and gravitational components of force acting on inclined planes. ### Key Concepts:- **Free Body Diagrams (FBD)**: Each block can be analyzed using an FBD to understand forces acting on them (gravitational force, normal force, frictional force, and tension).- **Equations of Motion**: The forces parallel and perpendicular to the inclined planes contribute to the net acceleration of the blocks. The system can be analyzed to find the relationship between the masses of the blocks and the angle of inclination.- **Tension in the String**: The tension in the string will be equal on both sides assuming the pulley is frictionless. ### Intended Learning Outcomes:- Applying Newton's second law to systems with multiple objects.- Understanding how the components of gravitational force affect motion on inclined planes.- Analyzing and solving problems involving pulleys and inclined planes with different angles.This setup enhances comprehension of mechanics principles through a practical visual example.

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Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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1. Two blocks are positioned on surfaces, each inclined at the same angle of 42.8 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 2.07 kg, and the coefficient of kinetic friction for both blocks and inclines is 0.510. 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?

2. Two blocks are positioned on surfaces, each inclined at the same angle of 49.5 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 2.24 kg, and this time there is NO friction. 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?

**Inclined Planes with Connected Blocks**

### Diagram Description:
The diagram illustrates a system involving two blocks connected by a string over a pulley, which is positioned at the apex of two inclined planes.

1. **Inclined Planes**:
- The system includes two inclined planes that are mirror images of each other, both inclined at an angle θ relative to the horizontal ground.
- There are two angles marked as θ on either side of the inclined planes.

2. **Blocks**:
- There are two blocks, each placed on a different inclined plane.
- The left block appears as an empty square, while the right block is colored or marked darker.

3. **Pulley System**:
- The blocks are connected by a string which goes over a pulley located at the apex where the two inclined planes meet.
- The string is assumed to be perfectly flexible and the pulley frictionless, ensuring smooth motion.

### Educational Discussion:
This type of setup is commonly used to study various principles of physics, such as equilibrium, friction, tension, and gravitational components of force acting on inclined planes.

### Key Concepts:
- **Free Body Diagrams (FBD)**: Each block can be analyzed using an FBD to understand forces acting on them (gravitational force, normal force, frictional force, and tension).
- **Equations of Motion**: The forces parallel and perpendicular to the inclined planes contribute to the net acceleration of the blocks. The system can be analyzed to find the relationship between the masses of the blocks and the angle of inclination.
- **Tension in the String**: The tension in the string will be equal on both sides assuming the pulley is frictionless.

### Intended Learning Outcomes:
- Applying Newton's second law to systems with multiple objects.
- Understanding how the components of gravitational force affect motion on inclined planes.
- Analyzing and solving problems involving pulleys and inclined planes with different angles.

This setup enhances comprehension of mechanics principles through a practical visual example.

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