Question

An arched bridge is to be designed that cars may pass over safely a a speed of 180km/h without jumping off the road.what should the minimum or maximum value of the radius of curvature of the arch be?

Answer 1

255 m.

Explanation:

If we approximate the bridge to a circular path with radius R, and taking the highest point of the bridge's curvature as the point of reference for the v = 180 km/h speed, then the centripetal force (Fc) in this case may be written as:

`F_c = W - N`, where 'W' and 'N' denote the car's weight and normal forces, respectively.

Also, since `F_c = mv^2/R`, with `m` being the car's mass; then we can write that

`W - N = mv^2/R`.

However, in a limit situation where the car would lose contact with the bridge, we would have N = 0, which means that the centripetal force would be equal to the car's weight. In this case, then:

`mg = mv^2/R`.

Therefore:
`R = v^2/g`.

Taking `g = 9,81 m/s^2` and the speed in m/s:

`R = 50^2/9.81`

`R = 255 m`, approximately.