This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the problem of the elastic pendulum \[ \begin{aligned} \dfrac{d^{2}z}{dt^2} &= (l_0 + z) \dot{\theta}^2 - \dfrac{kz}{m} + g \sin{\theta} \\ \dfrac{d^{2} \theta}{dt^2} &= -\dfrac{g}{l_0 + z} \cos{\theta} - \dfrac{2\dot{z}\dot{\theta}}{l_0 + z}. \end{aligned} \] Where:
\(z\) is the extension of the pendulum beyond its rest length (in metres).
\(\theta\) is the angle of the pendulum relative to the positive \(x\)-axis.
\(g\) is the acceleration due to gravity (in metres per second squared).
\(l_0\) is the rest length of the pendulum.
The buttons below pertain to the animation immediately below them.