= Uniform-pressure friction torque on a circular contact
{title2=$\tau_f=2\mu Na/3$}
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> $dF=\mu p\,2\pi s\,ds$. Its resisting <torque> is $d\tau=s\,dF$. <Integration> from zero to $a$ gives $\tau_f=2\mu Na/3$. A different pressure distribution changes the result; for an axisymmetric distribution use $\tau_f=2\pi\mu\int_0^a p(s)s^2ds$.
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