We present a new learning activity that enables students to apply eigenvalue theory to investigate a practical modeling problem. We demonstrate how to build a spring-coupled pair of pendula and describe how students can measure the movements of these pendula using open-source image processing software. We then illustrate how to analyze this position data and provide supporting theory that enables students to accurately model this system using eigenvalues. This hands-on activity enhances student motivation and prepares students to model other complex systems using linear ordinary differential equations.