Simulation · Classical Mechanics
Pendulum Simulation
This simulation numerically integrates the full nonlinear pendulum equation of motion, letting you adjust length, gravitational acceleration, initial angle, and damping, and observe the resulting motion, period, and energy in real time — including large-angle effects the small-angle approximation misses.
Interactive simulation
Small-angle period: — s
Current angle: —°
Angular velocity: — rad/s
Kinetic energy: — J
Potential energy: — J
Total energy: — J
The physics behind this simulation
The exact equation of motion for a pendulum, with no small-angle approximation, is:
d²θ/dt² = −(g/L)sinθ − b(dθ/dt)
This simulation numerically integrates this exact nonlinear equation (via a fourth-order Runge-Kutta method) rather than using the small-angle approximation, so it correctly shows how the period lengthens noticeably for large starting angles — a real physical effect the commonly quoted formula below only approximates.
T ≈ 2π√(L/g) (small-angle approximation, valid for θ₀ ≲ 15°)
Common mistakeThe formula T = 2π√(L/g) is only an approximation, valid for small starting angles. Set the initial angle above 30°–40° and watch the actual period (visible in the swing timing) grow noticeably longer than the "Small-angle period" readout predicts — the pendulum genuinely takes longer to complete each swing as amplitude increases, a real nonlinear effect this simulation reproduces but the simple formula does not.
Sources
- Taylor, J.R. (2005). Classical Mechanics. University Science Books, Ch. 5.
- Goldstein, H., Poole, C., Safko, J. (2002). Classical Mechanics, 3rd ed. Addison Wesley, Ch. 6.
Last verified: July 2026 · Source: Taylor (2005) · Confidence: high