Simple Pendulum

🎯 About Simple Pendulum

A simple pendulum is a mass (bob) suspended from a fixed point by a massless, inextensible string. When displaced from its equilibrium position and released, it oscillates under the influence of gravity. This is one of the most fundamental systems in classical mechanics and demonstrates simple harmonic motion for small angles.

💡 Key Physics Concepts

• •
  Restoring Force: F = -mg sin(θ), always directed toward equilibrium
• •
  Simple Harmonic Motion: For small angles (θ < 15°), sin(θ) ≈ θ, giving SHM
• •
  Isochronism: Period is independent of mass and amplitude (for small angles)
• •
  Energy Conservation: KE + PE = constant (without damping)
• •
  Large-Angle Correction: For large angles, T increases as θ₀²/16

📐 Formulas

🔬 Real-World Applications

• →
  Grandfather clocks — Pendulum regulates timekeeping
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  Foucault pendulum — Demonstrates Earth's rotation (1851)
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  Seismographs — Detect and measure earthquakes
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  Metronomes — Keep musical tempo
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  Torsion pendulum — Measures gravitational constant (Cavendish experiment)
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  Pendulum waves — Visual demonstration of wave mechanics

🎮 How to Use the Simulation

• 1.
  Adjust Length — longer pendulum = longer period
• 2.
  Try gravity presets — see how pendulum behaves on Moon vs Jupiter
• 3.
  Mass doesn't affect period (Galileo's discovery!)
• 4.
  Large angles break the small-angle approximation
• 5.
  Enable Force Vectors to see gravity, tension, and net force
• 6.
  Watch the energy bars — KE↔PE exchange in real time
• 7.
  Use Speed slider for slow-motion analysis

🧠 Interesting Facts

• ✦
  Galileo discovered pendulum isochronism by watching a chandelier swing in Pisa Cathedral (1582)
• ✦
  A pendulum 1 meter long has a period of approximately 2 seconds
• ✦
  The Foucault pendulum at the Panthéon in Paris takes ~32 hours to complete a full rotation
• ✦
  On the ISS (microgravity), a pendulum simply won't oscillate — try the ISS preset!