A jumping cylinder in an incline

The problem of a cylinder of mass m and radius r, with its center of mass out of the cylinder axis, rolling in an incline that makes an angle $\alpha $ respect to the horizontal is analyzed. The equation of motion is solved to obtain the site where the cylinder loses contact with the incline (jumps). Several simplifications are made: the analyzed system consists of an homogeneous disc with a one dimensional straight line of mass parallel to the disc axis at a distance d $<$ r of the center of the cylinder. To compare our results with experimental data, we use a Styrofoam cylinder of radius r = 10.0 $\pm $ 0.05 cm, high h = 5.55 $\pm $ 0.05 cm and a mass m$_{1}$ = 24.45 $\pm $ 0.05 g, to which a 9.50 $\pm $ 0.01 mm diameter and 5.10 $\pm $ 0.001 cm long brass road of mass m$_{2}$ = 30.75 $\pm $ 0.05 g was imbibed parallel to the disc axis at a distance of 5.40 $\pm $ 0.05 cm from it. Then the disc rolls on a 3.20 m long wooden ramp inclined at 30\r{ } and 45\r{ } respect to the horizontal. To determine the jumping site, the movements were recorded with a high-speed video camera (Casio EX ZR100) at 400 frames per second. The experimental results agree well with the theoretical predictions.