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Pendulum Period Calculations (Numerical Approach)

Keywords: Oscillations & Waves, Pendula, Pendulum Period, Taylor and Power Series, Elliptic Integral, Energy Diagram

Highlights of the activity

This small group activity is designed to help upper-division undergraduate students understand the quantitative relationship between the period and amplitude for a plane pendulum.
Students derive the expression for the period of a plane pendulum from the equations of motion, evaluate the integral expression numerically using series expansion or a computer program (Maple/Mathematica), and plot the experimental and numerical results on the same graph.
The wrap-up discussion focuses on techniques for simplifying the integral expression of the period and on the quantitative agreement of the model and experiment.

Reasons to spend class time on the activity

The potential energy diagram approach has yielded a method of calculating the pendulum period based on the model chosen for the appropriate potential U(theta) ≈ sin(theta). Students now calculate the quantitative prediction of the model. This forms a basis to evaluate the validity of the model based on their experimental data.

The series approximation to the evaluation of the period integral requires many steps, and much intervention and class discussion is required. The calculation represents a new level of sophistication for most students. They practice series expansion skills acquired in the “Idealizations and Symmetries” course. The series expansion method shows that successive approximation really does work.

Maple or similar package can be used to calculate the exact integral. At the time of writing, the evaluation of an elliptic integral in Maple was non-trivial, but not beyond the capabilities of the students. This process models what a professional physicist might do.

(It is daunting for students to be expected to complete both methods, so we stick to just one.)