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

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 $\beta (\simeq 0.41)$ does not vary significantly. The results suggest that the normalized depth and slip ratio determine the effective friction of a rolling sphere.