THE STUDY OF THE SIMPLE GRAVITATIONAL PENDULUM WITH EXCEL SPREADSHEETS

Abstract

This paper presents a didactic tool designed with the help of Excel spreadsheets for the study of the simple gravitational pendulum in oscillation regime. The period of the pendulum is calculated for a certain value of the initial angular amplitude and it is compared with the zero, one and two-order approximations from the expansion series of the period. Moreover, the velocity and tension force of the thread are calculated according to the angular displacement from the input data. It is shown how the complete elliptic integral of the first kind can be evaluated in the spreadsheet, through the numerical algorithm of the trapezium, from the formula of the oscillation period. The tool charts are placed in the same spreadsheet with the input data and the numeric results to rapidly trace the graphical feedback to data change. The oscillation period was graphically rendered based on the initial angular amplitude and the value of the period for the amplitude fixed in the input data has been highlighted on the graph. Another graph overlaps the period-initial angular amplitude one with the graphs of zero and one-order approximations of the period. The other graph shows, at a set value of the amplitude in the input data, the pendulum velocity and the tension force of the thread according to the initial angular displacement. In the tension force-angle graph, the weight force is highlighted and also the maximum range of values for the tension force is highlighted according to the initial angular amplitude. With the help of this graph, it can be observed that, at small oscillations, the tension in the thread varies slowly with the angular displacement around certain values close to the value of the weight force. This justifies the approximations made when establishing the calculation formula for the period of small oscillations. It is thus demonstrated that the spreadsheet allows to both solve the non-linear equation of the simple pendulum and to graphically analyze some measures characteristic for the oscillatory motion. By using the tool in the classroom, students can simulate the simple pendulum oscillatory motion at any value of the initial angular amplitude and can more easily clarify under which conditions the oscillations of the pendulum can be considered as isocrone.