Banking of Tracks

Since a leveled road's safe turning speed is capped entirely by the (often unreliable, weather-dependent) coefficient of friction, real roads and railway tracks instead raise the outer edge relative to the inner edge — called banking, with the incline angle $\theta$ called the banking angle.

On a banked road, the normal force $N$ (now tilted at angle $\theta$ from the vertical) has a horizontal component, $N\sin\theta$, pointing toward the centre — this itself can supply some or all of the needed centripetal force, reducing (or, at the design speed, entirely removing) the reliance on friction. Force balance at the ideal design speed (ignoring friction): vertical, $N\cos\theta=mg$; horizontal (centripetal), $N\sin\theta=\dfrac{mv^2}{r}$. Dividing:

$\tan\theta=\frac{v^2}{rg}\quad\Rightarrow\quad v=\sqrt{rg\tan\theta}.$ …