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

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. $δ/R$ (the sinking depth $δ$ 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 $μ_\mathrm{d}$. We demonstrate that $μ_\mathrm{d}$ decreases with increasing the slope angle and the slip ratio. Furthermore, systematic measurements over a wide range of sphere densities reveal that $μ_\mathrm{d}$ increases linearly with $δ/R$ : $μ_\mathrm{d}=β(δ/R)+μ_0$. The value of $μ_0$ is linearly decreasing with slip ratio and its coefficient $β(\simeq0.41)$ does not vary significantly. The results suggest that the normalized depth and slip ratio determine the effective friction of a rolling sphere.