Physics Tutorial: Pendulum Motion

Introduction

• Simple pendulum: massive object (pendulum bob) hung by a string.
• Equilibrium position: vertical hang where it rests.
• Periodic motion: back and forth vibration when displaced from equilibrium.
• Focus: Forces, position, velocity, kinetic and potential energy changes over time.

Force Analysis of a Pendulum

• Forces on Pendulum Bob:
  • Gravity: acts downward (mass * 9.8 N/kg).
  • Tension: directed towards pivot point, direction varies with bob’s position.
  • Air resistance: ignored for simplicity (weak compared to gravity and tension).
• Resolving Forces:
  • Gravity resolved into perpendicular and tangent components.
  • Perpendicular: opposes tension (no net force when bob is at extremes).
  • Tangent: acts as restoring force, directed towards equilibrium.*

Sinusoidal Nature of Pendulum Motion

• Describes how displacement and velocity change over time.
• Position vs. Time:
  • Right of equilibrium: positive displacement.
  • Left of equilibrium: negative displacement.
• Velocity vs. Time:
  • Changes continuously; max speed at equilibrium, zero speed at extremes.
  • Velocity least when displacement is greatest.

Energy Analysis

• Kinetic Energy (KE):
  • Depends on mass and speed (KE = mv²).
  • Increases towards equilibrium, decreases away from it.
• Potential Energy (PE):
  • Gravitational potential energy (PE = mgh).
  • Zero at equilibrium (lowest point), max at highest displacement.
• Energy Conservation:
  • Total mechanical energy (TME) = KE + PE, remains constant.
  • Transformation between PE and KE as bob moves.

The Period of a Pendulum

• Variables Affecting Period:
  • Mass: no significant effect.
  • Length: direct relation (longer length increases period).
  • Arc angle: minimal effect.
• Equation:
  • Experimentally derived: T = 2.0045L^0.5 + 0.0077
  • Theoretical: T = 2(L/g)^0.5 (with g = 9.8 N/kg).

Investigation and Interactivity

• Interactive widgets suggested for hands-on learning.
• Encourage use of Physics Classroom’s Interactives.

Check Your Understanding

• Exercises and questions to reinforce learning.
• Questions involve forces, energy, period calculation, and frequency comparison.

Additional Resources

• Next section: Motion of a Mass on a Spring
• Interactive tools and problem-solving exercises available on the website.

Note

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