Question

1. For a car travelling with speed v around a curve of radius r.
a) Determine a formula for the angle at which a road should be banked so that no
friction is required.
b) What is this angle for an expressway off-ramp curve of radius 50 m at a design
speed of 50 km/h?

Answer 1

a) Let's consider the case when the car travels on a frictionless banked road with speed $v$ around a curve of radius $r$. There are two forces that act on the car: weight, $mg$, and the normal force $N$. We can resolve the normal force into the horizontal ($Nsin \theta$ ) and vertical ($Ncos \theta$) components. The horizontal component of the normal force provides the necessary centripetal force, so that if the car has a right speed there is no friction required.

Let's apply the Newton's Second Law of Motion in projections on $x$- and $y$-axis:

$$Nsin \theta = \dfrac{mv^2}{r} (1)$$ $$Ncos \theta = mg (2)$$

Dividing equation (1) by equation (2) we determine the formula for the angle at which a road should be banked so that no friction is required:

$$tan \theta = \dfrac{v^2}{rg},$$ $$\theta = tan^{-1}(\dfrac{v^2}{rg}).$$

b) We can find the angle for an expressway off-ramp curve of radius 50 m at a design speed of 50 km/h from the formula obtained in part (a):

$$\theta = tan^{-1}(\dfrac{(50 \dfrac{km}{h} \cdot \dfrac{1000m}{1km} \cdot \dfrac{1h}{3600s})^2}{50m \cdot 9.8 \dfrac{m}{s^2}}) = 21.5^{\circ}.$$

Answer:

$\theta = 21.5^{\circ}.$