Figure 8.

Surface friction. (a) The forces produced by friction effects between an element of the rod and the substrate are naturally decomposed into a lateral component in the direction ${\mathbf{\text{u}}}_{\parallel }^{\mathrm{w}}=\mathbf{\text{t}}\times {\mathbf{\text{u}}}_{\perp }^{\mathrm{w}}$ and a longitudinal one in the direction ${\mathbf{\text{u}}}_{\times }^{\mathrm{w}}={\mathbf{\text{u}}}_{\perp }^{\mathrm{w}}\times {\mathbf{\text{u}}}_{\parallel }^{\mathrm{w}}$. We note that in general u^w_×≠t. The notation x_⊥, x_∥, x_× denotes the projection of the vector x in the directions u^w_⊥, u^w_∥, u^w_×. (b,c) Kinematic and dynamic quantities at play at any cross section in the case of (b) rolling and slipping and (c) pure rolling motion. Red arrows correspond to forces and torques, green arrows correspond to velocities, and black arrows correspond to geometric quantities.