Banked curves

In a banked curve (or parabolic curve) the road is inclined towards the inside of the bend. The idea is brilliant in its simplicity: if the road is inclined, the normal reaction $\vec{N}$ — always perpendicular to the road surface — is no longer vertical, and its horizontal component can contribute to the centripetal force, reducing (or eliminating) the need for friction.

Consider the ideal case, without friction. Only the weight $m\vec{g}$ (vertical) and the normal reaction $\vec{N}$ (perpendicular to the surface, inclined at $\alpha$) act on the road. Resolving $\vec{N}$:

$\begin{cases} N\cos\alpha = mg & \text{(vertical: supports the weight)} \\ N\sin\alpha = \dfrac{mv^2}{r} & \text{(horizontal: centripetal force)} \end{cases}$

Dividing the two equations term by term, $N$ cancels out, leaving a clean relation between angle, speed and radius:

Key formula — Ideal banked curve

$\tan\alpha = \frac{v^2}{gr} \quad\Rightarrow\quad v = \sqrt{gr\tan\alpha}$

At this speed — called the design speed — the normal reaction alone is enough to make the car turn: no friction, no lateral stress on the tyres. This is why, on velodrome tracks and motorway bends, the banking is calculated for a specific speed.

In the real world

At speeds different from the design speed, some friction is needed to compensate: too slow and there is a tendency to slide towards the inside, too fast and towards the outside. Banking does not eliminate friction, but shifts the equilibrium point.

The numerical application is worked out in Banked curve without friction and in Banking angle of the curve.

Links

Topics: Dynamics Concepts: Centripetal force · Normal force Skills: Vector resolution Objects: Banked curve

Related exercises: Maximum speed on a flat curve · Banking angle of the curve · Flat curve, dry and wet