Worked exercise: banked bend without friction

Problem

A bend of radius $r = 100\ \mathrm{m}$ is banked at angle $\alpha = 15^\circ$. At what speed can a car take it without relying on friction?

Banked bend: weight $m\vec g$ and normal $\vec N$; the horizontal component of $\vec N$ provides the centripetal acceleration $\vec a_c$.

Solution

Method. Only the weight $m\vec g$ (vertical) and the normal $\vec N$ (perpendicular to the road surface) act on the car. Circular motion requires a horizontal centripetal force, directed towards the centre of the bend. Resolving $\vec N$: vertical component $N\cos\alpha = mg$, horizontal component $N\sin\alpha = mv^2/r$: $N\cos\alpha = mg \qquad N\sin\alpha = \frac{mv^2}{r}$

Dividing term by term: $\tan\alpha = \frac{v^2}{gr} \;\Longrightarrow\; v = \sqrt{gr\tan\alpha}$

Numerically: $v = \sqrt{9{,}81\cdot 100\cdot 0{,}268} \approx 16{,}2\ \mathrm{m/s} \approx 58\ \mathrm{km/h}$

$\ev{v = \sqrt{gr\tan\alpha} \approx 16{,}2\ \mathrm{m/s} \approx 58\ \mathrm{km/h}}$

At this speed the constraint reaction alone is enough to curve the car: no friction, no strain on the tyres.

Links

Topics: Dinamica Concepts: Forza centripeta · Forza normale · Accelerazione centripeta · Seconda legge di Newton Skills: Scomposizione vettoriale Methods: Metodo del diagramma di corpo libero Objects: Curva sopraelevata