What is Banking of Roads?
Banking of a road is the engineering practice of raising the outer edge of a curved road slightly above the inner edge. This creates a sloped surface that helps vehicles negotiate turns safely by providing a component of the normal force to act as the centripetal force required for circular motion.
Why do we need it?
When a vehicle rounds a flat curve, it relies solely on the friction between its tires and the road to provide centripetal force. If the road is slippery (rain, ice) or the speed is too high, the vehicle may skid. Banking reduces the dependence on friction, allowing for safer high-speed cornering.
Derivation of Banking Angle
Consider a vehicle of mass $m$ moving at a speed $v$ on a track banked at an angle $\theta$ with a radius of curvature $r$. Two forces act on the vehicle:
- The gravitational force $mg$ acting downwards.
- The normal reaction $N$ acting perpendicular to the road surface.
Resolving the normal reaction $N$ into two components:
- Vertical component: $N \cos \theta$, which balances the weight $mg$, so $N \cos \theta = mg$.
- Horizontal component: $N \sin \theta$, which provides the necessary centripetal force $\frac{mv^2}{r}$.
Therefore, we have:
- $N \cos \theta = mg$
- $N \sin \theta = \frac{mv^2}{r}$
Dividing equation (2) by equation (1): $\frac{N \sin \theta}{N \cos \theta} = \frac{mv^2 / r}{mg}$
$\tan \theta = \frac{v^2}{rg}$
Thus, the banking angle $\theta$ is given by: $\theta = \tan^{-1}\left(\frac{v^2}{rg}\right)$
Key Takeaways
- Banking minimizes wear and tear on tires by reducing reliance on friction.
- The optimal speed $v$ for a banked curve is directly proportional to the square root of the tangent of the banking angle.
- If $v^2 < rg \tan \theta$, the vehicle tends to slide inward; if $v^2 > rg \tan \theta$, it tends to slide outward.