Revisiting the physical pendulum with torque

Abstract

We derive the angular frequency of a physical pendulum made from acrylic rods by considering both gravitational torque and the rod's moment of inertia. Unlike the simple pendulum, our model incorporates rod length and diameter, predicting that shorter lengths and wider diameters lead to higher angular frequencies. We tested this by using rods with lengths ranging from 20 to 30 cm and diameters from 2 to 12 mm. Frequencies were extracted from Fast Fourier Transforms (FFTs) of velocity data, with uncertainties estimated via full width at half maximum (FWHM), ranging from 2.64 to 5.28 rad/s. Experimental results generally align with the theory, confirming the model. However, for the 30 cm rod with a 12 mm diameter, measured frequencies were higher than predicted, suggesting potential effects from non-ideal factors such as dynamic stiffening or mass distribution. These results highlight how rod geometry impacts pendulum dynamics beyond idealized models.