Numerical Solution of the Simple Pendulum Equation Using the Fourth-Order Runge-Kutta Method: A Comparative Study with Analytical Solutions

Abstract

This study investigates the numerical solution of the non-linear pendulum differential equation using the fourth-order Runge-Kutta method. While the simple pendulum is a classic problem in dynamics, it often lacks an explicit analytical solution for arbitrary amplitudes, necessitating numerical approaches. By employing the fourth-order Runge-Kutta algorithm, this research computes the angular displacement and velocity of a pendulum with a length of 4 meters under specific initial conditions. The study performs a comprehensive comparison between the numerical results and analytical solutions to evaluate the accuracy and stability of the method. The implementation was conducted using MATLAB, where the impact of various parameters, including time range and step size (h), were systematically analyzed to optimize performance and precision. Results demonstrate that smaller step sizes (e.g., h=0.001) significantly enhance numerical accuracy, although they require increased computational time. Furthermore, the study examines energy conservation, illustrating the periodic exchange between kinetic and potential energy in the system. Ultimately, this research confirms that the fourth-order Runge-Kutta method is a highly effective and robust tool for solving second-order ordinary differential equations in physical systems. The findings provide valuable insights for applications in horology, gravimetry, and inertial navigation, where precise pendulum modeling is essential.