Nonlinearity Affects a Pendulum
2 days ago
- #nonlinear-dynamics
- #approximation
- #pendulum
- The pendulum's equation of motion is nonlinear when using sin θ, which is complex to solve with elementary math.
- For small θ, sin θ ≈ θ approximates the equation to linear form, justified by the small-angle approximation in trigonometry.
- The small-angle threshold lacks a simple answer; commonly cited as less than 10°, but accuracy depends on comparing exact and approximate solutions.
- Nonlinear pendulum solutions have longer periods than linear ones; the period increases with initial displacement θ₀, approximated by a factor involving θ₀².
- An adjusted linear solution can closely match the nonlinear one if the period is stretched appropriately, with differences often minimal in plots.