Question

009 (part 1 of 2) 10.0 points A civil engineer is asked to design a curved section of roadway that meets the following conditions: With ice on the road, when the coefficient of static friction between the road and rubber is 0.24, a car at rest must not slide into the ditch and a car traveling less than 70 km/h must not skid to the outside of the curve. The acceleration of gravity is 9.81 m/s2. At what angle should the road be banked? Answer in units of. 010 (part 2 of 2) 10.0 points What is the minimum radius of curvature of the curve? Answer in units of m

Answer 1

1) tan(th) = u

=> th = arctan(u)

= arc(0.24)

= 13.5 degrees

2) For the case where the car is at rest, there is no centripetal acceleration, so no radius can be determined. For the case where the ideal speed is 70km/h (19.44m/s), so that the radius of curvature is to be found for v < v(ideal), we have for the net horizontal forces:

ΣF(x) = mv²/r = nsinθ + μncosθ

mv²/r = n(sinθ + μcosθ)----------------->(1)

We have no expression for n (the normal force), but we can write one from the net vertical forces:

ΣF(x) = 0 = ncosθ - mg - μnsinθ

n = mg / (cosθ - μsinθ)------------------------>(2)

Now plug (2) into (1) and solve for r:

mv²/r = [mg /(cosθ - μsinθ)](sinθ + μcosθ)

mv²/r = mg(sinθ + μcosθ) / cosθ - μsinθ)

r = v²(cosθ - μsinθ) / g(sinθ + μcosθ)

= (19.44m/s)²(cos 13.5 ° - 0.24sin13.5 °) / 9.81m/s²(sin 13.5 ° + 0.24cos13.5 °)

= 75.62 m (rounded)