Inclined Plane and Pulley Systems

Newton's Second Law Applied to a Two-Mass System

In this scenario, two masses (m1 and m2) are connected by a string over a pulley, with m2 resting on an inclined plane at 30° and m1 hanging vertically. The system is subject to kinetic friction on the plane, characterized by the coefficient μk. Newton's Second Law is used to analyze the forces and resulting acceleration.

• Newton's Second Law: States that the net force on an object is equal to its mass times its acceleration: .

• Free-Body Diagram (FBD): Essential for visualizing all forces acting on each mass.

• Forces on m2 (on the incline):

  • Gravity: (acts downward)

  • Normal force: (perpendicular to the plane)

  • Kinetic friction: (opposes motion along the plane)

  • Tension: (along the string, up the incline)

• Forces on m1 (hanging):

  • Gravity: (downward)

  • Tension: (upward)

Component Analysis: For m2, resolve forces parallel and perpendicular to the incline:

• Parallel to incline: (down the plane)

• Friction: (up the plane)

• Tension: (up the plane)

Newton's Second Law (parallel to incline):

Newton's Second Law (for hanging mass):

Note: The tension is the same throughout the string if the pulley is ideal (massless and frictionless).

Example: Multiple-Choice Application

Given the system, students are asked to select the correct application of Newton's Second Law for each mass. Typical answer choices involve correct or incorrect signs and terms for the forces and acceleration.

• Correct equation for m2: (if friction is neglected)

• Correct equation for m1: (if upward is positive and acceleration is downward)

• Sign conventions: The direction of acceleration and force components must be consistent with the chosen coordinate system.

Example Calculation: If kg, kg, , and m/s2, calculate the acceleration.

• First, find the net force on each mass using the equations above.

• Set up simultaneous equations to solve for .

Key Terms and Concepts

• Inclined Plane: A flat surface set at an angle to the horizontal, used to analyze components of forces.

• Pulley: A simple machine that changes the direction of a force applied via a string or rope.

• Kinetic Friction (): The force that opposes the motion of two surfaces sliding past each other.

• Tension: The pulling force transmitted along a string, rope, cable, or similar object.

Summary Table: Forces Acting on Each Mass

Additional info: The original slides focus on conceptual understanding and correct application of Newton's Second Law to multi-body systems, emphasizing the importance of sign conventions, force components, and free-body diagrams. These problems are foundational for later topics such as energy conservation and work-energy principles.