Vehicle on a Banked Road

Relying on friction to provide the centripetal force has two real drawbacks: friction has an upper limit, and on a real road its value is rarely constant, since road surfaces are never perfectly uniform (and can be wet, dusty, or oily in patches). To reduce this dependence on friction, curved roads are deliberately TILTED with the horizontal at some angle $\theta$ -- this is called banking a road.

Considering the vehicle as a point and IGNORING friction for the moment (not eliminating it -- just setting the safest-case baseline), only two forces act: weight mg, vertically downward, and normal reaction N, perpendicular to the (now tilted) road surface. Since the vehicle's actual motion is along a HORIZONTAL circle, the net force must be purely horizontal, which means the VERTICAL component of N must exactly balance mg: $N\cos\theta=mg$. The HORIZONTAL component of N, $N\sin\theta$, is then the entire resultant force, and being directed towards the centre of the track, it is the centripetal force: $N\sin\theta=\frac{mv^2}{r}$. Dividing the second relation by the first eliminates N: $\tan\theta=\frac{v^2}{rg} \qquad \text{--- (1.1)}$

This single relation serves two complementary engineering purposes. (a) For a road of GIVEN radius r and banking angle $\theta$ (both fixed once built), it gives the unique 'most safe speed' $v_s=\sqrt{rg\tan\theta}$ -- not a minimum or maximum, but the one special speed at which the vehicle needs NO help from friction at all. (b) When DESIGNING a road for an intended speed v and radius r, it instead gives the required banking angle, $\theta=\tan^{-1}\left(\frac{v^2}{rg}\right) \qquad \text{--- (1.2)}$

In practice, vehicles never travel at exactly this one safe speed $v_s$, so friction (up to its limit $\mu_sN$) does come into play to extend the usable speed range on either side.

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