Harmonic motion analysis simplified for introductory physics students

In introductory physics courses, students learn about harmonic oscillation through the classic example of a frictionless mass on a spring moving back and forth on a horizontal surface. While they learn simple harmonic motion, these students rarely explore more realistic systems with dry friction, as it is considered by some teachers to be too conceptually challenging or mathematically onerous.

To help students understand and visualize harmonic motion with dry friction, Roitberg and Drory developed a simple graphical analysis that can be used to solve such systems. The analysis extends simple harmonic motion as a projection of uniform circular motion with a two-dimensional visualization of string being wound around two nails. It mimics how you might coil a rope around your hand and elbow.

“Our analysis treats the motion as a continuous process, which it obviously is, and does so in a simple, very visual and tangible way,” said author Alon Drory.

This analysis can be solved with only simple algebra and trigonometry instead of differential equations as is normally taught. The authors hope this visual approach can make a complex system appear simpler and help students retain the physics longer.

“We hope the paper will inspire other teachers to present harmonic motion with solid-on-solid friction in class, but mostly encourage them to seek analogies and visual simplifications wherever possible in order to reveal the core physical behavior that is buried, too often, under a mathematical tangle,” Drory said.

Source: “Graphical analysis of an oscillator with constant magnitude sliding friction,” by V. Roitberg and Alon Drory, American Journal of Physics (2022). The article can be accessed at https://doi.org/10.1119/5.0073812 .