Motion of Connected Bodies

Introduction

• Understanding the motion of connected bodies (connected through pulleys)
• Example problem with a speaker and blocks connected via pulley

Key Concept

• Single Rope Concept: If tension (T) is the same in a single rope, then changes in the rope affect tension accordingly.
• Acceleration: Positive in the direction of acceleration, negative in the opposite direction.

Example Problem 1

• System Setup: Masses m1 and m2 connected via a pulley, forces acting downward due to gravity:

  • Mass m1: Force = m1 * g
  • Mass m2: Force = m2 * g

• Steps to Solve:

  1. Identify the heaviest mass; direction of the system's acceleration.
  2. Equations of motion:
     • For mass m1: m1g - T = m1a
     • For mass m2: T - m2g = -m2a
  3. Solving simultaneous equations:
     • Substitute values to find acceleration (a) and tension (T).
     • Example: m1 = 3kg, m2 = 2kg, g = 10m/s². Result: a = 2m/s², T = 24N.

Example Problem 2

• Three Connected Bodies: Masses m1, m2, and m3 connected via pulleys.
  • Forces: m1 * g, m2 * g, m3 * g
• Steps to Solve:
  1. Write equations for each mass.
  2. Solve the equations to find acceleration and tension.
  3. Example results: tension = 24N, acceleration = 2m/s².

Concepts Involving Movable Pulley

• Movable Pulley: Analysis for a system where the pulley itself moves.
  • Key idea: The tension in different segments of the rope is different, but consistent within each segment.
  • Relation of tensions: If the pulley moves, account for the rationed change in segments connected by it.

Advanced Example Problem with Movable Pulley

1. Setup: Complex system with multiple pulleys and masses.
   • Calculate the net force and solve using the same principles as above.
2. Using Acceleration Relations:
   • For a pulley connected with multiple masses, the acceleration of masses is inversely proportional to their segments pulled.

Problem 1: Movable Pulley Example

• Two masses: Mass 1 (2kg), Mass 2 (1kg), move via a pulley system.
  • Write equations of motion for each mass.
  • Calculate tension and acceleration.

Problem 2: Complex Movable Pulley

• Setup where pulleys and connected masses have different accelerations but move coherently as a system.
  • Use equations derived from net forces.
  • Example outcome: Calculate specific values for tension and system's acceleration.

Key Takeaways

• Consistency and Precision: Ensure consistency in sign conventions (positive/negative) based on acceleration direction.
• Step-by-step Approach: Break down complex systems into simpler parts, solve systematically.
• Validation: Verify results by cross-checking all equations.

Homework

• Practice problems with different setups and pulley configurations.
• Comment and validate your answers with peers.

Additional Resources

• Find related notes and further reading material on the provided Telegram channel.