Frictional Coupling between Sliding and Spinning Motion

The tangential motion at the contact of two solid objects is studied. It consists of a sliding and a spinning degree of freedom (no rolling). We show that the friction force and torque are inherently coupled. As a simple test system, a sliding and spinning disk on a horizontal flat surface is considered. We calculate, and also measure, how the disk slows down and find that it always stops its sliding and spinning motion at the same moment. We discuss the impact of this coupling between friction force and torque on the physics of granular materials.

Figure 1

(a) A sliding and spinning disk on a flat horizontal surface. (b)–(d) The relative velocity field on the surface of the disk at $\epsilon =0.2$, $\epsilon =1$, and $\epsilon =5$, respectively ($\epsilon =v/R\omega$).

Figure 2

(a) The dimensionless friction force and torque, $\mathcal{F}$ and $\mathcal{T}$, as functions of the dimensionless velocity parameter $\epsilon$. Squares and triangles with error bars represent experimental data (explained in the text). (b) The friction force and torque are coupled: the curve shows the possible $(\mathcal{F},\mathcal{T})$ pairs.

Figure 3

Function $f(\epsilon )=\epsilon -\mathcal{F}(\epsilon )/2\mathcal{T}(\epsilon )$, the right-hand side of differential Eq. (8). It has zero value at $\epsilon =0$ and ${\epsilon }_0\approx 0.653$ and is positive for $0<\epsilon <{\epsilon }_0$, negative for ${\epsilon }_0<\epsilon$.

Figure 4

(a) The pressure distribution of the normal force acting on a sliding tall cylinder is nonuniform, because it has to counterbalance the torque exerted by the friction force $F$, provided that the cylinder does not topple (schematic side view). (b) As a consequence of the nonuniform normal pressure distribution, if the cylinder is also spinning, the net friction force is not collinear with the velocity; therefore, the trajectory is curved.