Depth and slip ratio dependencies of friction for a sphere rolling on a granular slope

TL;DR

This study experimentally explores how the depth of sinking and slip ratio influence the effective friction experienced by a sphere rolling on a granular slope, revealing linear relationships and dependencies on various parameters.

Contribution

It introduces a systematic analysis of the effective friction coefficient's dependence on sinking depth and slip ratio for spheres on granular slopes.

Findings

Effective friction decreases with slope angle and slip ratio.

Normalized sinking depth linearly relates to the effective friction coefficient.

The baseline friction coefficient decreases linearly with slip ratio.

Abstract

We experimentally investigate the dynamics of a sphere rolling down a granular slope by varying the initial velocity, slope angle, and sphere density. The results show that the sphere rolls down with constant deceleration while sinking into the granular bed. $\delta/R$ (the sinking depth $\delta$ normalized to the sphere radius $R$) is scaled by the sphere density normalized by the bulk density of the granular layer. To evaluate the translational energy dissipation, we introduce an effective friction coefficient $\mu_\mathrm{d}$. We demonstrate that $\mu_\mathrm{d}$ decreases with increasing the slope angle and the slip ratio. Furthermore, systematic measurements over a wide range of sphere densities reveal that $\mu_\mathrm{d}$ increases linearly with $\delta/R$ : $\mu_\mathrm{d}=\beta(\delta/R)+\mu_0$. The value of $\mu_0$ is linearly decreasing with slip ratio and its coefficient $\beta(\simeq0.41)$ does not vary significantly. The results suggest that the normalized depth and slip ratio determine the effective friction of a rolling sphere.