Approximate period for large-amplitude oscillations of a simple pendulum based on quintication of the restoring force

A new approximate formula for estimating the period of the simple pendulum has been presented in this paper. The formula was derived based on 'quintication' of the restoring force of the pendulum, which replaces the original pendulum equation with an equivalent cubic–quintic oscillator. The coefficients of the equivalent oscillator are amplitude-dependent and were derived based on quasi-static equilibrium principles. Then, an approximate solution for the equivalent cubic–quintic oscillator was applied to derive the approximate formula used to estimate the pendulum period for the entire range of possible amplitudes of angular displacement, i.e. $0^{\circ }<{\phi }_0<{180}^{\circ }.$ The formula is simple and very accurate with a maximum relative error of 0.421% for amplitudes up to 179.9°. (paper)