Car on a flat bend

A car travelling round a flat bend needs a horizontal centripetal force, directed towards the centre of the bend. On a flat (unbanked) road the only horizontal force available is the friction between tyres and tarmac: it is friction, and nothing else, that “pulls” the car towards the centre and stops it carrying straight on.

The reasoning is a comparison between what is needed and what is available. The centripetal force required to corner at speed $v$ on radius $r$ is

$F_c = m\,\frac{v^2}{r}$

The maximum friction the tarmac can provide is $F_{a,\max} = \mu\,R = \mu\,mg$ (on a flat road the constraint reaction carries the whole weight). The car holds the road as long as $F_c \leq F_{a,\max}$, that is, as long as

$m\,\frac{v^2}{r} \leq \mu\,mg$

The mass cancels, giving the maximum cornering speed:

Key formula — Maximum speed on a flat bend

$\ev{v_{\max} = \sqrt{\mu\,g\,r}}$

Notably, $v_{\max}$ does not depend on the mass: a lorry and a small car run off the road at the same speed (for equal $\mu$ and $r$), because both the required and the available force grow in proportion to the mass. Beyond $v_{\max}$ friction is no longer enough and the car skids outwards. The numerical applications are in Car on a flat bend and Flat bend with strong friction.

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Topics: Dinamica · Attrito Concepts: Forza centripeta · Attrito statico Skills: Applicazione delle leggi di Newton

Related exercises: Why circular motion is accelerated · Maximum speed on a flat bend · Dry and wet flat bend