×
Pendulum Physics — Complete Theory
📖 What is a Pendulum?
A pendulum is a weight suspended from a pivot so that it can swing freely. When displaced from equilibrium, gravity provides a restoring force, causing oscillation.
📐 Simple Harmonic Motion (Small Angle Approximation)
For small angles (θ < 15°), a pendulum follows simple harmonic motion:
θ(t) = θ₀ · cos(√(g/L) · t)
⚡ The Period Formula
T = 2π √(L/g)
Where T = period (seconds), L = length (meters), g = gravity (9.81 m/s² on Earth).
Key insight: Period depends only on length and gravity — NOT on mass or amplitude (for small angles).
📈 Frequency & Angular Frequency
f = 1/T = (1/2π) √(g/L)
ω = 2πf = √(g/L)
💡 Energy in a Pendulum
E = mgh = mgL(1 - cosθ)
E = ½ m v² + mgh
Total mechanical energy is conserved in ideal conditions (no damping). Energy transfers between kinetic and potential forms.
📊 Physical Example
Example: A pendulum with L = 1 m on Earth (g = 9.81 m/s²):
T = 2π √(1/9.81) ≈ 2.01 seconds
This means it takes about 2 seconds for one complete back-and-forth swing.
🌍 Real-World Applications
- Pendulum clocks (timekeeping)
- Seismometers (earthquake detection)
- Metronomes (musical tempo)
- Foucault pendulum (demonstrates Earth's rotation)
⚡ Key Takeaways
✔ Longer pendulum = slower swing (larger period)
✔ Stronger gravity = faster swing (smaller period)
✔ Mass DOES NOT affect the period
✔ Damping causes amplitude to decay over time
✔ For angles > 15°, the small-angle approximation becomes less accurate
