 Uniform-pressure friction torque on a circular contact

A flat circular contact of radius $a$ carrying total normal force $N$ has uniform pressure $p=N/(\pi a^2)$. In steady relative rotation, a ring of radius $s$ and thickness $ds$ experiences tangential kinetic friction $dF=\mu p2\pi sds$. Its resisting torque is $d\tau =sdF$. Integration from zero to $a$ gives ${\tau }_f=2\mu Na/3$. A different pressure distribution changes the result; for an axisymmetric distribution use ${\tau }_f=2\pi \mu {\int }_0^{a}p(s)s^{2}ds$.