PX155 - E3 - circular motion and rolling

$\vec{r}$ is the vector from a moving the centre of rotation $C$ to a point of contact between the wheel and road
velocity of P? $⟹ {\vec{v}}_{p} = \vec{u} + (\vec{ω} \times \vec{r})$
direction of $\vec{ω} \times \vec{r} : ω r (- \vec{k} \times - \vec{j}) = - ω r \vec{i}$
after one rolling rotation, $C$ has moved a distance of $2 π r$ in time $\frac{2 π}{ω}$
so, $u = \frac{2 π r}{\frac{2 π}{ω}} = r ω$
$v_{p} = r ω \vec{i} + (\vec{ω} \times \vec{r}) = r ω \vec{i} - ω r \vec{i} = 0$
$v_{p} = 0 :$ point of contact instantaneously at rest - static friction is irrelevant
if $ω < \frac{u}{r}$ , then kinetic friction applies
- wheels of a car rotating at $ω < \frac{u}{r}$ , they are under-rotating, which makes the car skid along the surface
- if $ω > \frac{u}{r}$ , wheels over-rotate and skid