Experimental and analytical study of the motion of a sphere falling along an inclined plane in still water

• Experimental study of a submerged sphere, moving downhill along an inclined plane
• Two regimes of motion are observed: a first accelerated stage, followed by a constant velocity stage (terminal velocity).
• The longitudinal position as function of time, is obtained by direct integration of the dynamical equations of motion.

In this work, the transient regime of motion of a sphere, that falls along an inclined smooth plane, submerged in still water, is studied experimentally and analytically. Several experiments were performed, with spheres of different diameters and densities, and two slope angles. The measurements show that the motion of the sphere presents two well defined stages: a transitional regime, where the sphere rapidly accelerates from rest, and an asymptotic regime, where the sphere moves with constant velocity. Closed expressions, for the position and velocity of the sphere as functions of time, are obtained by direct integration of the motion equations. This theoretical analysis is based on the empirical correlation for the drag coefficient as a function of the Reynolds number of the sphere, that was proposed by Chhabra and Ferreira [2] for the experimental data set of Jan and Chen [1]. The analytical solution shows good agreement with experimental results, for a wide range of experimental conditions.

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