Beyond the Small-Angle Approximation: A Computational Study of the Period of a Simple Pendulum

The familiar formula for the period of a simple pendulum, T0 = 2π√︁L/g, is derived by assuming that the angular displacement is small. This study examines how the formula’s accuracy changes as the initial swing angle increases. Starting from the nonlinear equation of motion, the period is expressed using a complete elliptic integral and evaluated numerically. The predictions are compared with both the small-angle formula and a higher-order approximation. For an ideal pendulum of length 1.00 m in a gravitational field of 9.81 m s−2, the small-angle formula underestimates the period by about 0.76% at an initial angle of 20◦, 3.84% at 45◦, and 15.28% at 90◦. Under the model used here, its error first reaches 1% at approximately 22.93◦. A correction containing terms through the fourth power of the angle greatly improves the estimate across the tested range. The results show why the assumptions behind a familiar physics formula matter. All numerical values are theoretical calculations, not measurements from a physical experiment.