Motion of a simple pendulum: a digital technology approach

Abstract

In differential equations textbooks, the motion of a simple pendulum for small-amplitude oscillations is analyzed. This is due to the impossibility of expressing, in terms of simple elementary functions, the solutions of the nonlinear differential equation (NLDE) that models the pendulum, which is why the authors usually choose the linearized differential equation arising from the sin θ ≈ θ approximation for small angles θ. In this article, we use the computational power of Mathematica software and take advantage of its capacity to graphically show the solutions of the NLDE. In this manner, we determine approximations of the periods that result from varying the initial amplitude θ 0, without assuming that it takes small values. From a comparison of the solutions of both types of modelling of a simple pendulum, a criterion for deciding how small θ 0 must be for the linear equation to be adequate for modelling is obtained; this comparison includes an analysis of the pendulum′s motion when the cable length and the gravitational acceleration constant are varied.