The simple model to estimate the force exerted on the sphere rolling up a granular slope

Department of Earth and Space Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

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Published online: 1 December 2025

Abstract

Spheres rolling on a granular bed sometimes experience a "stuck" phenomenon: translational motion ceases while rotational motion continues. To explore this phenomenon, we investigate the interaction between the granular slope and the sphere rolling up the slope using data obtained by Fukumoto et al. (PRE 109, 014903, 2024). Their experimental results confirmed the stuck phenomenon. According to these results, both translational and rolling motions exhibited constant deceleration until the translational motion halted. In this study, we propose a simple force model, following the approach proposed by Texier et al. (EPL 123, 54005, 2018). We estimate two forces acting on the sphere at a point slightly shifted from its bottommost point, both before and during the stuck phenomenon: one in the tangential direction and the other in the normal direction at a point on the surface of the sphere defined relative to an angle θ_m from the axis perpendicular to the slope surface. Our analysis indicates that the normalized tangential and normal forces remain nearly constant for the slope angle and are independent of the density of the sphere, both before and during the stuck motion. Finally, by comparing the magnitudes of these forces and θ_m before and during the stuck motion, we find that the point of force application shifts while the forces themselves remain nearly unchanged. This stuck phenomenon occurs when the translational force becomes insufficient owing to a change in θ_m, even though the magnitudes of the forces remain unchanged.