Classical Mechanics

Simple Pendulum

WEBGPU ENGINE
RK4 INTEGRATION
WebGPU Context Active

Parameters

Initial Angle (θ₀)45°
String Length (L)5.0 m
Gravity (g)9.81 m/s²
Mass (m)2.0 kg
Damping (b)0.00
Angle (θ)
0.0°
Ang. Vel. (ω)
0.00 rad/s
Kinetic E.
0.0 J
Potential E.
0.0 J
Energy Distribution
Kinetic E. 0%Total E. 0.0JPotential E. 0%
Live Equation
θ¨=−9.815.0sin⁡θ\ddot{\theta} = -\frac{9.81}{5.0}\sin\theta
gL=1.96 s−2\frac{g}{L} = 1.96\,\text{s}^{-2}Tsmall≈4.486 sT_{\text{small}} \approx 4.486\,\text{s}
θ vs t

Governing Dynamics

A simple pendulum consists of a point mass m suspended by a massless string of length L in a uniform gravitational field g.

d2θdt2+gLsin⁡θ=0\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0

For small angles, the approximation sin(θ) ≈ θ simplifies to simple harmonic motion. The period becomes independent of amplitude:

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

This simulation integrates the exact non-linear equation using the Runge-Kutta 4th Order (RK4) method, allowing accurate simulation of large amplitudes and full swings, with realistic air drag.