Normal mode solutions for the perfectly flexible hanging cord problem have been known for over 200 years. More recently, theoretical results for a hanging cord with a point mass attached were presented. Here the theoretical results are tested experimentally using high-precision techniques which are accessible for use in an introductory laboratory. Also included is a generalization of the theory for the case where the rotational inertia of the tip mass is not negligible. An exact expression is obtained which can be solved for the normal mode frequencies. In all cases, the theoretical results are in good agreement with experimental results. Both the theory and experiment are at a level accessible to upper level undergraduate physics students. The problem serves as an example of other problems involving one-dimensional wave propagation with a spatially dependent wave speed resulting in non-harmonic normal mode frequencies.