The ratio of heat capacities, cp/cv, is important in thermodynamics. It appears in the pressure-volume relation for an adiabatic process, for instance, and in formulas for the efficiencies of cyclic processes like internal combustion engines.
This ingenious experiment makes use of a precision glass tube inserted through the stopper of a large glass bottle. Drop a steel ball into the tube, and if the amount of air escaping around the ball is negligible, it oscillates. (Can you explain why it oscillates instead of falling straight through)? From the period of oscillation and other quantities you measure, calculate the ratio of heat capacities.
Repeat the experiment for other gases such as helium, carbon dioxide, and freon.